Proceedings of the Conference Positivity IV - Theory and Applications 9783860055120, 3-86005-512-7

Citation preview

Fakultät Mathematik und Naturwissenschaften

Fachrichtung Mathematik

Institut für Analysis

Proceedings of the Conference

Positivity IV − Theory and Applications

Dresden, July 25 − 29, 2005

Impressum Publisher: Technische Universität Dresden Institut für Analysis Editors: Prof. Dr. Martin R. Weber Prof. Dr. Jürgen Voigt Layout: Ingo Tschichholtz, Helga Mettke Cover: Helga Mettke Printing: Copy Cabana Steffen Kürbis & Paul-Stefan Scholz GbR, Dresden Contact: Prof. Dr. Martin R. Weber Institute of Analysis Department of Mathematics Faculty of Science Technische Universität Dresden 01062 Dresden (Germany Tel.: ++49 (0351) 4 63 35434 / Fax: e-mail: [email protected]

++49 (0351) 4 63 37202

All articles are refereed.

Cover Picture: Helga Mettke

No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission of the copyright owner. ISBN 3-86005-512-7

May 2006

Proceedings of the Conference

Positivity IV − Theory and Applications Dresden (Germany), July 25 - 29, 2005

Organizing Committee:

Martin R. Weber, Jürgen Voigt Anke Kalauch, Ingo Tzschichholtz Helga Mettke

Support from the following institutions is gratefully acknowledged: • Deutsche Forschungsgemeinschaft (DFG) • Sächsisches Staatsministerium für Wissenschaft und Kunst • Gesellschaft von Freunden und Förderern der TU Dresden e.V. • Technische Universität Dresden.

Foto: Dr. M. Hamann

Foreword The International Conference Positivity IV – Theory and Applications was organized at the Department of Mathematics of the Technische Universit¨at Dresden and took place from July 25 until 29, 2005 in Dresden (Germany). It was already the fourth conference devoted to the topic of Positivity after the conferences held in Ankara (Turkey) 1998, Nijmegen (Netherlands) 2001 and Rhodos (Greece) 2003. In many fields of mathematical thinking, but also in the wider field of science, economics and engineering and their interaction the property of positive quantities, functions, solutions and others is well defined and of particular importance. The most abstract formulation of positivity makes use of the positive reals and was systematically developed in the last eighty years, although the idea of positivity is much older. It has turned out that for many purposes vector spaces equipped with a partial order (or equivalently, with a convex cone), ordered normed spaces and the corresponding classes of positive operators form the appropriate setting for research and application. The latter establishes nowadays a considerable independent part of functional analysis very interacting with many mathematical disciplines such as optimization, mathematical economy, semigroup theory, measure and integration theory, stochastics, ordinary and partial differential equations and numerical methods. Altogether 44 mathematicians from Europe, Africa, America and Asia participated in the conference. The complete list is included as one of the next pages. In 11 plenary lectures and 28 short communications a large spectrum of the ideas of positivity as a main stream in functional analysis was covered. The Proceedings of the conference present an overview of the topics discussed and contains papers submitted by the participants. The editors of the Proceedings thank all persons who were engaged to host the conference and to organize the social events, and also thank all colleagues who were involved in the refereeing procedure, in particular, Ingo Tzschichholtz, Anke Kalauch and Helga Mettke. Many of the participants visited the city of Dresden for the first time. So a guided evening walk through the center of Dresden was of great interest the more that the rebuilding of the famous Frauenkirche Church was nearly finished and so the historical center of Dresden could be admired in its full beauty. In the middle of the conference week the participants and accompanying persons went by train to Rathen, a small village in the Saxon Switzerland, about 40 km south of Dresden. From there a small steep straining path lead through the typical pillar-like mountains up to the Bastei Plateau, where all of us were compensated for our exertions by the panorama of the so-called Elbe-sandstone mountains.

IV

Foreword

The conference was supported by Deutsche Forschungsgemeinschaft (DFG), the regional Saxon State Ministry for Science and Art, the Department of Mathematics and the Society of Friends and Supporters of the Technische Universit¨at Dresden. We appreciated continuing the tradition of conferences on positivity and hope to meet next time in Belfast.

Dresden, May 2006

Martin R. Weber and J¨urgen Voigt

Contents III

Foreword

1

Opening speech of the Vice-Dean for Mathematics, Prof. Dr. Volker Nollau

Contributions 5

ALEKHNO, E. A. Spectral Properties of Band Irreducible Operators

15

ALPAY, Şafak and MISIRLIOĞLU, Tunç Invariant Subspace Theorems for Families on a Banach Lattice

21

AMOR, Fethi Ben The Riesz-Kantorovich formula for the modulus of a complex order bounded disjointness preserving operator

29

BOULABIAR, Karim and BUSKES, Gerard A note on bijective disjointness preserving operators

35

BUSKES, Gerard and REDFIELD, R. H. Band preserving operators on lattice-ordered groups

45

CHEN, Z. L. On the order continuity of the regular norm

53

EMEL’YANOV, E. Yu. Positive Asymptotically Regular Operators in L1-spaces and KB-spaces

63

GROBLER, J. J. Bivariate and marginal function spaces

73

KITOVER, A. K. and WICKSTEAD, A. W. Positive operators without invariant sublattices

79

KÖNIG, Heinz Stochastic Processes on the Basis of New Measure Theory

93

KUSRAEV, A. G. and KUTATELADZE, S. S. Boolean Valued Analysis and Positivity

107

METAFUNE, Giorgio and RHANDI, Abdelaziz The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions

115

dePAGTER, B. and RICKER, W. J. R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry

131

STEIN, Martin and VOIGT, Jürgen On modulus semigroups and their generators

135

THIEME, H. R. and VOIGT, J. Stochastic semigroups: their construction by perturbation and approximation

147

VÄTH, Martin Generalized Ideal Spaces and Applications to the Superposition Operator

155

WEBER, Martin R. Finite Elements in Vector Lattices

173

WNUK, Witold and WIATROWSKI, Błazej When are ultrapowers of normed lattices discrete or continuous?

Attachment 183

List of Participants

Opening speech of the Vice-Dean for Mathematics, Prof. Dr. Volker Nollau Ladies and Gentlemen, Dear Colleagues and Guests On behalf of the Technical University Dresden, the Faculty of Mathematics and Natural Sciences and the Department of Mathematics I welcome you to the International Conference Positivity IV. We are very pleased that the Department of Mathematics of the Technical University Dresden was selected to host this conference. Let me shortly outline the history and the development of our university and its department of mathematics. In 2003 our university celebrated its 175th anniversary. Our university was founded in 1828 as a Higher Technical Educational Institute. Its development was and is closely related to the growth of the regional Saxon and German industry and economy - in the early years especially to the formation of the Saxon mechanical engineering industry. In 1851 the Institute was upgraded to the Royal Saxon Technical School. Later the name was changed to Technische Hochschule Dresden and in 1961 reached its final designation as the Technische Universit¨at Dresden. By 1993 the Technical University had incorporated various other institutions of teaching and research, especially the Academy of Forestry, the College of Traffic Science and the Medical Academy as well as the Teacher’s College. At the present time the Technical University Dresden also offers programs in humanities and social sciences as well as in economics, linguistics and other subjects. The total of currently about 33 000 students is roughly evenly split between these new subjects and the traditional disciplines such as engineering, science, forestry and computer science. Since the middle of the nineteenth century Saxony was leading in integrating the teaching of science into the curriculum of the secondary and higher educational system. Also there has always been a close interaction between mathematical research and its engineering applications. In other words, mathematicians working at the university were continually involved in engineering development and progress. This engagement also shaped their teaching activities.

2

Opening speech of the Vice-Dean for Mathematics, Prof. Dr. Volker Nollau

Consequently, the curriculum of Mathematics at the Technische Universit¨at Dresden has always required a minor in an engineering or science subject. Since more than ten years the Department offers the following study tracks Mathematics, Technomathematics, Economathematics and Teaching of Mathematics. Moreover, the Department of Mathematics is responsible for the mathematical training in all other faculties of the university. Our graduates have never encountered difficulties in finding suitable employment throughout Germany and Europe. It is hoped that this pool of highly qualified professionals will also support the development of the regional industrial base. Nevertheless the Department of Mathematics also covers some more theoretical branches of Mathematics. From the mathematicians who worked at our Department and are well known in Analysis, Topology and Functional Analysis let me mention only ¨ (1823 -1901): among other things known for a general remainder in • O. S CHL OMILCH Taylor’s fomula • A. H ARNACK (1851-1888): Inequalities for potential functions relevant in partial differential equations • H. S EIFERT (1907-1996) and W. T HRELFALL (1888-1949): well known for their book ,,Lehrbuch der Topologie”, which was the first monograph devoted to combinatoric topology • F. R ELLICH (1906-1955): Spectral theory. Up to now several mathematicians of the Department in various Institutes have strong relations to functional analysis. Nowadays the Department is subdivided into the Institutes of Algebra, Analysis, Geometry, Mathematical Stochastics, Numerical Analysis, Scientific Computation and the Chair of Didactics, each with their own special research topics. We have an extended cooperation with other universities throughout Europe and all over the world. This is also reflected by a remarkably high percentage of partial studies of our Diploma students abroad. Presently, many Diploma- and PhD-students are involved in the research work of our institutes. To this meeting the Technische Universit¨at Dresden and the Department of Mathematics welcome participants from Great Britain, France, the Czech Republic, Poland, the Russian Federation, Greece, the Netherlands, Slovenia, Turkey, the United States, Canada, South Africa, the People’s Republic of China, Japan, Morocco, Tunisia and Germany. The aims of the Conference on Positivity, which I AM PLEASED TO OPEN have been already spelled out in the announcements. Therefore, let me mention only that it is very important

Opening speech of the Vice-Dean for Mathematics, Prof. Dr. Volker Nollau

3

and that we are happy that a lot of young mathematicians can take part in the conference. This was possible due to financial support we are given by the sponsors of the conference: Deutsche Forschungsgemeinschaft (DFG), Saxon State Ministry for Science and Arts, Society of Friends and Supporters of the Technische Universit¨at Dresden and the Department of Mathematics. Today in the evening a sightseeing walk through the center of Dresden is organized for you, where you will see the most famous buildings and architectural sites of the city. One of the most remarkable events this year will be the inauguration of the reconstructed Frauenkirche Church in October 2005. After its destruction in the last months of the second world war it was rebuilt on funds collected in Germany, Europe and other parts of the world. I am confident that you will have an instructive and enjoyable meeting and still find some time to enjoy our beautiful city of Dresden and its environment.

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 5-14 (2006)

Spectral Properties of Band Irreducible Operators E. A. Alekhno (Minsk, Belarus) Abstract. Number of spectral properties of a band irreducible operator T on a Banach lattice E will be discussed. If T is σ-order continuous, r(T ) is a pole of the resolvent R(., T ), and the band Ec∼ of all σ-order continuous functionals on E is nonzero, then we prove among others that r(T ) > 0, that T has an eigenvector which is a weak unit, and that the adjoint T ∗ of T has a positive order continuous eigenvector. Furthermore, we provide some criteria of primitivity for band irreducible operators in terms of limits of real sequences. Finally, we discuss the question whether the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T ), where T is a band irreducible operator on E. Key words: Banach lattice, Banach function space, positive operator, integral operator, spectrum, primitivity, Hermitian operator MSC 2000: Primary 47B65, 47G10, 47A10, 45C05; Secondary 47A55

Spectral theory on positive operators occupies a major place in the general theory of operators on Banach lattices. Over the last decades, special attention has been paid to spectral properties of ideal and band irreducible operators as a natural extension of the well-known concept of irreducible matrices by Perron-Frobenius (see [1], Chapter 8 or [15], Chapter 1). This study has been really launched thanks to Andˆo and Krieger ([1], p. 431), whom investigated the spectral radius of some integral operators on Dedekind complete Banach lattices. Since then, Schaefer, Sawashima, Niiro, Stecenko, and de Pagter (see [15, 1, 11] for precise reference) contributed in the building of the theory of ideal irreducible operators and answered most questions and problems in this direction. Later, Grobler, Schaefer, Caselles, Abramovich, Aliprantis, and Burkinshaw (see [1, 11] for precise reference) studied the band irreducible operators on Banach lattices and showed that this theory also deserves a special consideration. In this prospect, the main purpose has been to study the extent to which the classical Andˆo-Krieger theorem can be generalized. However, along these lines several other aspects of band irreducible operators have receive almost no attentions. This note tries to make some contribution in this direction. A short synopsis of the content of this note seems to be in order. In the first section, there are conditions of existence of eigenvector and eigenfunctional under assumptions that are much more general than compactness. In the second section, we provide criteria of primitivity and imprimitivity for band irreducible operators, which are analogous to famous criteria from the theory of irreducible matrices. In the final section we focus on the question whether the operator inequalities 0 ≤ S < T imply the spectral radius inequality r(S) < r(T ), where T is a band irreducible operator on a Banach lattice E. For terminology, notions, and properties on the theory of Banach lattices not explained or proved in this note, we refer to [1]; see also [8, 15].

6

1

E. A. Alekhno

Theorem about existence of eigenvector and eigenfunctional

Recall ([1], p. 349) that a positive operator T on a Banach lattice E is ideal irreducible if T has no invariant non-trivial closed ideals and band irreducible if T has no invariant non-trivial bands. In the following theorem main spectral properties of band irreducible operators are adduced. Theorem 1. Let E be a Banach lattice, the dimension of E is at least two and the σ-order continuous dual Ec∼ 6= {0}. Suppose that T is a band irreducible σ-order continuous operator on E such that the point λ0 = r(T ) is a pole of the resolvent R(., T ). Then: (a) the spectral radius r(T ) > 0; (b) the eigenvector x0 corresponding to the eigenvalue r(T ) of the operator T is a weak unit and dim N (r(T )I − T ) = 1; (c) the eigenfunctional x∗0 corresponding to the eigenvalue r(T ) of the operator T ∗ is a strictly positive order continuous functional and dim N (r(T )I − T ∗ ) = 1; (d) λ0 = r(T ) is a simply pole of R(., T ); (e) the residuum of R(., T ) at λ0 = r(T ) is a rank-one operator. Proof. Observe first that the condition Ec∼ 6= {0} and the existence on E of a σ-order continuous band irreducible operator imply [16] that the band Ec∼ separates the points of E. Thus a Riesz dual system hE, Ec∼ i is defined and moreover there exists a strictly positive functional on E. Therefore E has the countable sup property, hence the band En∼ of order continuous functionals is equal to the band Ec∼ σ-order continuous functionals. The eigenvector x0 > 0 corresponding to the eigenvalue r(T ) of the operator T exists ([15], p. 352): T x0 = r(T )x0 . The band irreducibility of the operator T implies r(T ) > 0 hence the assertion (a) holds and x0 is a weak unit. It is no loss of generality to assume that r(T ) = 1. The band En∼ is invariant under T ∗ , let T 0 be a restriction of T ∗ on En∼ . The operator T 0 is band irreducible. Indeed, assume by the way of a contradiction that there exists a non-trivial band B invariant under T 0 , {0} ⊂ B ⊂ En∼ . The band B is σ(En∼ , E)-closed. It is by the equality [10] B ◦◦ ∩ En∼ = B, where the polar is taken in the dual system hE, E ∗ i. The polar B ◦ of the band B with respect to the dual system hE, En∼ i is an ideal invariant under the operator T . If x ∈ E and there exists a net of elements xα from B ◦ such that 0 ≤ xα ↑ |x|, then x∗ |x| = lim x∗ xα = 0 for every functional x∗ ∈ B, α hence x ∈ B ◦ , therefore B ◦ is a band. By bipolar theorem ([8], p. 140) the band B ◦ can’t be trivial and we get a contradiction with the band irreducibility of T . For an arbitrary functional x∗ ∈ En∼ we get x∗ x0 = x∗ (T x0 ) = (T 0 x∗ )x0 ; it follows that ((I − T 0 )x∗ )x0 = 0 and therefore λ0 = 1 is a point of the spectrum of the operator T 0 . Since the inequality r(T 0 ) ≤ 1 holds, we get r(T 0 ) = 1. The inclusion ([1], p. 256) ρ∞ (T ∗ ) ⊆ ρ(T 0 ), where ρ∞ (T ∗ ) is the unbounded connected component of the resolvent set of the operator T ∗ , implies that the number λ0 = 1 is an isolated point of the spectrum σ(T 0 ) of the operator T 0 . For all λ sufficiently close to point λ0 = 1, the inequality |λ − 1|k+1 kR(λ, T 0 )k ≤ |λ − 1|k+1 kR(λ, T ∗ )k holds, where k is the order of the pole λ0 = 1 of the resolvent of the operator T . It follows that lim (λ − 1)k+1 R(λ, T 0 ) = 0.

λ→1

Spectral Properties of Band Irreducible Operators

7

Hence the point λ0 = 1 is a pole of the resolvent of the operator T 0 . This shows that there exists a strictly positive functional x∗0 ∈ En∼ such that T 0 x∗0 = x∗0 . Now assume that there exists an element x ∈ E, which is an eigenvector of T corresponding to the eigenvalue λ0 = 1: T x = x. Then T |x| = |x|. Adding this equality with the equality T x = x we get T x+ = x+ and T x− = x− . The band irreducibility of T implies either x+ = 0 or x− = 0. In other words, every element from the eigenspace of the operator T corresponding to λ0 = 1 belongs either E + or −E + . This shows ([15], p. 66) dim N (I − T ) = 1 and the assertion (b) is proved. If k is the order of a pole of R(., T ) around λ0 = 1, then the coefficient of the Laurent series expansion of R(., T ) around λ0 = 1 by (λ − λ0 )−k presents a positive operator which satisfies the identity ([1], p. 265) T−k = (T − I)k−1 T−1 , where T−1 is a residuum of the function R(., T ) at the point λ0 = 1. Then by k > 1 the equalities x∗0 (T−k x) = x∗0 ((T − I)k−1 T−1 x) = 0 hold. Hence T−k x = 0 for all x ≥ 0 as x∗0 is strictly positive. Whence T−k = 0, which is a contradiction. Thus k = 1 and the assertion (e) follows. Next, the equalities ([1], p. 74, 268) dim(N (I − T )) = dim(N (I − T ∗ )) and R(T−1 ) = N (I − T ) give assertions (c) and (d). For the case of ideal irreducible operators, the result analogous to theorem 1 was established by Sawashima [14]. Note that the existence on a given Banach lattice E of an ideal irreducible operator is the stronger assumption than the assumption about the existence of a band irreducible operator. For example, every Banach function space X admits the band irreducible integral operator, while an ideal irreducible operator on X may not exist; see [6, 7] for details. The existence of an eigenvector at a compact σ-order continuous band irreducible operator (by condition Ec∼ 6= {0}) was first noted in [16]. For the case of a band irreducible integral operator on a Banach function space the analog of theorem 1 was obtained in [7].

2

Primitivity and imprimitivity

As for nonnegative matrices, a band irreducible operator T with r(T ) > 0 is called imprimitive, if the peripheral spectrum of T consists of more than one point, and primitive, if it consists of only one point r(T ). Let T be a band irreducible integral operator on a Banach function space X on a set Ω with a measure µ, R T x(t) = k(t, s)x(s) dµ(s). Ω

Recall that by theorems of Lozanovsky and Andˆo-Krieger ([1], p. 199, 369) r(T ) > 0. Next result [7] is an analog of the classical Frobenius theorem about a general form of imprimitive matrices ([15], p. 22-23). Theorem 2. Let T be a band irreducible integral operator with kernel k(t, s) on a Banach function space X such that the point λ0 = r(T ) is a pole of resolvent R(., T ). Then the peripheral spectrum H of the operator T has the form r(T )Hm , where Hm is the group of all m-th roots of unity. In particular the spectrum σ(T ) of the operator T is invariant under the

8

E. A. Alekhno

rotation on angle 2π and dim N (λI − T ) = dim N (λI − T ∗ ) = 1 for all λ ∈ r(T )Hm . In m case m > 1 there exists a partition of the set Ω on m disjoint sets Ωj with positive measure, m S Ω= Ωj , such that k(t, s) = 0 for t ∈ Ωj , s 6∈ Ωj+1 (in case j = m by definition j + 1 := 1). j=1

On the other hand, if for some m > 1 a partition of Ω mentioned above exists, then Hm ⊆ H. Proof. We can suppose r(T ) = 1. From theorem 1 and Lotz-Schaefer theorem ([15], p. 331) we infer that H consists entirely of finite number of poles. By x0 and x∗0 we denote a positive function and a strictly positive functional (that exist again by theorem 1) such that T x0 = x0 , T ∗ x∗0 = x∗0 . Let us verify that H is the group of all m-th roots of unity by some m. We consider the ideal Ix0 generated by the element x0 . Then T (Ix0 ) ⊆ Ix0 . The ideal Ix0 , equipped with the norm kxkx0 = inf {c : |x| ≤ cx0 }, is a M -space with the unit x0 . By Kakutani-BohnenblustKrein theorem ([8], p. 194-195 or [1], p. 95) the space Ix0 is lattice isometric to the space of continuous functions C(K) on a compact Hausdorff space K, Ix0 3 x ↔ x b ∈ C(K). b c Moreover, we can assume that x b0 = 1. The operator T x b := T x on C(K) is continuous. b Clearly, r(T ) = 1. Fix λ ∈ H. Number λ is an eigenvalue of Tb. Then ([12], lemma 5.1(I)) there exists an operator S on C(K) such that Tb = λ−1 S −1 TbS. Now if Tbx1 = λ1 x1 , λ1 ∈ H, then λ1 x1 = λ−1 S −1 TbSx1 , hence λλ1 Sx1 = TbSx1 , therefore λλ1 is an eigenvalue of the operator Tb and it follows that λλ1 ∈ H. Thus λH ⊆ H for all λ ∈ H. It implies that H is the group of m-th roots of unity, that is H = Hm . Let x1 and x2 be eigenvectors of the operator T corresponding to some λ ∈ H, that is T xi = λxi , and moreover kxi k = 1, i = 1, 2. Then xbi ∈ C(K). Since λ1 ∈ H, there exists an operator S such that Tb = λS −1 TbS. Equalities Sb xi = TbSb xi , i = 1, 2, hold, hence Sb x1 = Sb x2 or ∗ Sb x1 = −Sb x2 , therefore x b1 = x b2 or x b1 = −b x2 . Thus dim N (λI − T ) = dim N (λI − T ) = 1. Let the operator T be imprimitive. Again using the restriction of T on Ix0 and lemma 5.1(II) from [12] we conclude that there exist functions yi ∈ X (i = 1, . . . , m) such that m P y i = x0 . yi ∧ yj = 0, i 6= j, T yi+1 = yi , i=1

Let Ωi = Supp yi . Then Ωi ∩ Ωj = ∅, i 6= j, Ω = T yj+1 =

R Ω

k(t, s)yj+1 (s) dµ(s) =

R

m S

Ωi and

i=1

k(t, s)yj+1 (s) dµ(s) = yj = χΩj yj ,

Ωj+1

that is k(t, s) = 0, if t 6∈ Ωj , s ∈ Ωj+1 for every j = 1, . . . , m or k(t, s) = 0, if t ∈ Ωj , s 6∈ Ωj+1 for j = 1, . . . , m. To proof the converse, suppose that there exists the partition, which is mentioned in the condition, of the set Ω on m disjoint sets. Let us show that Hm belongs to the peripheral spectrum. Let ξk = ε−k T k PΩ1 x0 , where ε is an arbitrary m-th root of unity, PA defined by m−1 P PA x = χA x for an arbitrary measurable set A and x ∈ X. Then function xε = ξk is µk=0

almost not vanishing and T xε = εxε . Since ε is an arbitrary m-th root of unity, we get that Hm belongs to the peripheral spectrum H of the operator T . The proof will be finished if we show that all spectrum of the operator T is invariant under m P 2πj . Determine an operator D on X by an equality D = e m i PΩj . Then the rotation on angle 2π m j=1

Spectral Properties of Band Irreducible Operators

9 2π

D is invertible and it is easy to see the justice of the equality T = e m i DT D−1 . Therefore we 2π 2π get σ(T ) = σ(e m i DT D−1 ) = e m i σ(T ) and the proof is finished. For the case of an compact operator T analog of theorem 2 was mentioned in [17], p. 304. In case of an arbitrary Banach lattice E the next result common to theorem 2 holds. It’s proof is analogous and will be omitted. 0 Theorem 2 . Under assumptions of theorem 1 the peripheral spectrum H of the operator T has the form r(T )Hm . In particular dim N (λI − T ) = dim N (λI − T ∗ ) = 1 for all λ ∈ r(T )Hm and there exist elements yi , i = 1, . . . , m such that yi ∧ yj = 0, i 6= j, T yi+1 = yi m P and yi = x0 . Moreover, if all yi are projection elements, then the spectrum σ(T ) is invariant i=1

under the rotation on angle 2π . On the other hand, if for some m > 1 the collection of elements m yi mentioned above exists, then Hm ⊆ H. By the virtue of theorem 2 a nonnegative kernel k(t, s) such that k(t, s) defines the integral operator on some Banach function space X, is called primitive, if the partition of the set Ω mentioned in theorem 2 doesn’t exist for k(t, s). A question arises naturally, namely, can we assert that some iterated kernel of a primitive kernel is positive (µ × µ-almost)? Otherwise is it true for a primitive kernel an analog of the result about the positivity of some power of a primitive matrix? Let us show that it is not. Example 3. Consider the infinite matrix K = (kij )∞ 1 such that elements kij are nonnegative, ∞ ∞ PP kij2 < ∞ and kij = 0 iff j > 1, j 6= i + 1, this matrix has the form i=1 j=1

   K=  

k11 k12 0 0 k21 0 k23 0 k31 0 0 k34 k41 0 0 0 ... ... ... ...

... ... ... ... ...

   .  

This matrix defines the ideal irreducible compact integral operator T on `2 . The kernel of this operator, that is the matrix K, is primitive. Nevertheless, every power of the operator T is not operator with positive kernel. Common examples exist in every infinite dimensional Banach function space. Nevertheless, the next result holds. Theorem 4. Let X be a Banach function space, T on X be a band irreducible integral operator with kernel k(t, s). Then kernel k(t, s) is primitive iff the equality lim (π × π)(Supp k (n) (t, s)) = 1

n→∞

n holds, where k (n) (t, s) are iterated kernels of operators R T and a probability measure π presents a norming of measure µ, i.e. the relation π(A) = η(s) dµ(s) holds, the function η(s) is µA R almost positive and η(s) dµ(s) = 1. Ω

Proof. The sufficiency is obvious, let us show the necessity. First of all, we can assume that µ is a probability measure. Indeed, consider an integral operator Tπ with the primitive kernel kη (t, s) = k(t, s)η(s) on a Banach function space X with the measure π. Then the equality

10

E. A. Alekhno

(n)

(n)

kη (t, s) = k (n) (t, s)η(s) implies that Supp kη (t, s) = Supp k (n) (t, s) on Ω × Ω. Thus we assume that µ is a probability measure and moreover π = µ. We shall verify first the assertion of the theorem in case when X presents the space L∞ (µ) of all bounded measurable functions, and the kernel k(t, s) of the operator T is bounded. It is no loss of generality to assume that r(T ) = 1. The operator T is weak compact, hence, in virtue of Dunford-Pettis theorem ([8], p. 337), we conclude that the operator T 2 is compact. Therefore λ0 = 1 is a pole of R(., T ). Then theorem 1 and 2 imply that the operator T is primitive and the point λ0 = 1 presents a simply pole of R(., T ). Let an operator P be the residuum at this point. Then P is a positive projection onto the one-dimensional space N (I − T ) of fixed points of the operator T and the equalities T P = P T = P hold. The projection P is presented in the integral form R P x(t) = x0 (t) x00 (s)x(s) dµ(s), Ω

where µ-almost positive functions x0 ∈ L∞ (µ) and x00 ∈ L1 (µ) satisfy the equalities T x0 = x0 and T ∗ x00 = x00 . The operator T is presented in the form T = P + T Q, where Q is defined by the equality Q = I − P . Since ([1], p. 266) P is a spectral projection associated with the point λ0 = 1 we infer that r(T Q|R(Q) ) < 1. Then the inequality k(T Q)n k ≤ k(T Q|R(Q) )n kkQk implies r(T Q) < 1. For powers of the operator T the equality T n = P + T n Q holds. Since r(T Q) < 1, then Gelfand formula ([1], p. 243) implies the convergence kT n Qk → 0 by n → ∞. Operators T n Q are regular integral operators. Denote by d(n) (t, s) kernels of T n Q. Relations R R (n) |d (t, s)| dµ(s)dµ(t) ≤ kT n Qk → 0 Ω Ω

imply that the sequence of functions d(n) (t, s) converges in measure µ × µ to zero. By virtue of the equality k (n) (t, s) = x0 (t)x00 (s) + d(n) (t, s) we obtain the convergence of functions’ sequence k (n) (t, s) in measure µ × µ to the µ × µ-almost positive on Ω × Ω function x0 (t)x00 (s). The desired assertion for operators on L∞ (µ) with bounded kernels is proved. Now consider the Rgeneral case of a Banach function space X and an integral operator T . An operator Rx(t) = x(s) dµ(s) on L∞ (µ) is integral with kernel r(t, s) = 1. Consider an Ω

operator B defined by a kernel b(t, s) = min{k(t, s), r(t, s)}. This operator acts on L∞ (µ) and, since Supp b(t, s) = Supp k(t, s), it is primitive. Then, as showed above, for kernels b(n) (t, s) of operators B n the equality lim (µ × µ)(Supp b(n) (t, s)) = 1 holds. Finally, notice n→∞

that b(n) (t, s) ≤ k (n) (t, s) and the proof is finished. For the case of a separable measure µ theorem 4 was obtained in [4]. Note that the sequence (π × π)(Ωn ) from theorem 4 doesn’t convergence “strictly monotonically” to zero even in a finite dimensional space, that is when the operator T defined by a matrix (see [5] for details). In case of an arbitrary Banach lattice E the next theorem holds. 0 Theorem 4 . Under assumptions of theorem 1 next assertions are equivalent: (a) the operator T is primitive; (b) for every non-zero positive functional x∗ ∈ E ∗ and element x > 0 lim x∗ (T n x) > 0; n→∞

(c) for every non-zero positive functional x∗ ∈ Ec∼ and element x > 0 lim inf x∗ (T n x) > 0. n→∞

Spectral Properties of Band Irreducible Operators

3

11

When does 0 ≤ S < T imply r(S) < r(T )?

It is well known that if for an irreducible matrix T the inequalities 0 ≤ S < T hold, then r(S) < r(T ). A question arises naturally, namely, when does the analogous result hold in an arbitrary Banach lattice E? As it will be shown below, the assumption that r(T ) is a pole R(., T ) is sufficient and, even in some sense, necessary. So, the next theorem holds. Theorem 5. Under assumptions of theorem 1 the inequalities 0 ≤ S < T imply the inequality r(S) < r(T ). Proof. Assume by the way of a contradiction that r(S) = r(T ) > 0. Then [9] the point λ0 = r(S) is a pole of R(., S). Therefore r(S)x = Sx ≤ T x

(?)

for some x > 0. Let x∗0 be a strictly positive functional such that T ∗ x∗0 = r(T )x∗0 . From relation (?) we infer r(T )x∗0 x = r(S)x∗0 x ≤ x∗0 (T x) = r(T )x∗0 x, hence T x = r(S)x = r(T )x, therefore x is a weak unit. The equality Sx = T x implies S = T , and we get a contradiction. The assumption that r(T ) is a pole, is essential. Indeed, an ideal irreducible order continuous operator Q on the space L1 exists with r(Q) = 0 ([15], p. 353). Then I ≤ I + Q and r(I + Q) = 1. The assumption about order continuity of the operator T is also essential. It is sufficient to take in the previous example the operator Q such that Q is a band irreducible compact operator rank-one with r(Q) = 0 (see [2]). Now let us pay attention to the case of integral operators. Next example shows that, even with the assumption of the integrality of the operator T , theorem 5 doesn’t hold without the assumption concerning a pole at the point λ0 = r(T ). Recall that a bounded operator Q on a Hilbert space H is Hermitian, if Q∗ = Q. For a Hermitian operator Q the equality ([1], p. 465) r(Q) = kQk holds. Example 6. Consider the space `2 . Let S be an operator on `2 defined by the diagonal matrix with elements sii = 1 − 21i on the diagonal. Clearly, r(S) = 1. If we show that there exists a sequence of real numbers an > 0 such that the operator T , defined by the infinity matrix  1  a1 a2 a3 . . . 2  a1 3 0 0 ...  4   7  a2 0 0 ...  8  ,  a3 0 0 15 . . .  16 ... ... ... ... ... acts on `2 and r(T ) = 1, then, as a result, 0 ≤ S < T , the operator T is ideal irreducible integral and r(S) = r(T ) = 1. By induction we find a decreasing sequence of numbers an > 0 such that for all n spectral radii of matrices   1 a . . . a 1 n 2   a1 3 . . . 0 4     ... ... ... ... 1 an 0 . . . 1 − 2n+1

12

E. A. Alekhno

are strictly less of unit. Then

∞ P i=1

a2i ≤ 34 . Show that by such choice of an r(T ) = 1. Let Tn be

operators defined by matrices  1

a1 . . . an 0 0  a1 3 . . . 0 0 0 4   ... ... ... . . . . . . . ..  1  an 0 . . . 1 − n+1 0 0 2  1  0 0 ... 0 1 − 2n+2 0 ... ... ... ... ... ... 2

... ... ... ... ... ...

    .   

Then r(Tn ) = 1. In fact, the inequality r(Tn ) ≥ 1 is obvious. For the converse kTn xk2 = kTn Pn+1 xk2 + kTn (I − Pn+1 )xk2 ≤ kPn+1 xk2 + k(I − Pn+1 )xk2 = kxk2 , where x = (x1 , x2 , . . .) ≥ 0, Pn are projections on first n coordinates, hence it follows relations r(Tn ) = kTn k ≤ 1. Thus r(Tn ) = 1. Further, the sequence Tn converges to the operator T in L(`2 , `2 ). Actually ¶ µ ∞ ∞ P P 2 2 2 2 ai kxk2 , k(T − Tn )xk = k( ai xi+1 , 0, . . . , 0, an+1 xn+2 , . . .)k ≤ an+1 + {z } | i=n+1 i=n+1 n

hence kT − Tn k2 ≤ a2n+1 +

∞ P i=n+1

a2i → 0 by n → ∞.

Thus r(T ) = kT k = lim kTn k = lim r(Tn ) = 1. n→∞

n→∞

We can choose the sequence an in example 6 like that: for an ideal irreducible integral operator T with r(T ) = 1 there exists an element x > 0 such that T x < x. Note that in example 6 an element x > 0 such that T x > x doesn’t exist. Next theorem suggested the idea of the construction of example 6. Theorem 7. Let T and K be Hermitian operators on L2 such that 0 ≤ K < T , the operator K is compact and the point λ0 = r(T ) doesn’t belong to the point spectrum σp (T ) of the operator T . Then r(T − K) = r(T ). Proof. From the equality σr (T ) = σp (T ), where σr (T ) is the residual spectrum of the operator T , it follows that the point λ0 = r(T ) belongs to the continuous spectrum of the operator T . By Weyl theorem ([3], p. 320) we have r(T ) ∈ σ(T −K), hence r(T ) ≤ r(T −K), therefore r(T ) = r(T − K). Thus for Hermitian operators T ≥ 0 on L2 the condition that r(T ) is a pole of R(., T ), namely r(T ) ∈ σp (T ), is necessary in order that the inequalities 0 ≤ S < T imply the inequality r(S) < r(T ). Note that, as a matter of fact, we can formulate theorem 7 for a wide class of operators on some Banach lattice E, for which Weyl theorem about a perturbation of the spectrum is truth. Note also that conditions, when for an irreducible Hermitian operator T and a diagonal operator D on `2 either r(T + D) = r(T ) or r(T + D) > r(T ) hold, are studied in [13]. Same examples as example 6, exist and in more classical Banach lattices, namely in `1 and `∞ . To the proof this assertion, we need a next lemma, which generalizes lemma V.6.4 from [15]. The proof of it is analogous and will be omitted.

Spectral Properties of Band Irreducible Operators

13

Lemma 8. Let E be a Banach lattice, S and T be operators on E such that 0 ≤ S ≤ T and r(S) = r(T ) hold. Then, there exists an element z > 0 such that the ideal Iz generated by z is invariant under S and T and r(Sz ) = r(Tz ) = r(T ), where Sz and Tz are restrictions of operators S and T on M -space Iz , respectively. Example 9. Let S and T be operators from example 6. Use lemma 8 to find the element z > 0 such that the ideal Iz is invariant under S and T and moreover r(Sz ) = r(Tz ) = 1. Define an operator T∞ on `∞ by the rule T∞ x = z1 T (xz). For an arbitrary element x ∈ `∞ (T∞ x)i =

1 (T (xz))i zi

=

1 zi

∞ P

aij xj zj =

j=1

∞ P j=1

zj a x zi ij j z

holds, where K = {(aij )} is the kernel of the operator T . Therefore {( zji aij )} is the kernel of the operator T∞ and thus, the operator T∞ is band irreducible integral. From equalities n e = z1 T n z and k xz k`∞ = kxkz , where e = (1, 1, . . .), that are holding for all natural n and all T∞ elements x ∈ Iz , we obtain 1

1

1

1

n n kT∞ k = k z1 T n zk`n∞ = kT n zkzn = kTzn k n → 1,

whence r(T∞ ) = 1 (as a matter of fact, we can show that σ(T∞ ) = σ(Tz )). Analogously we establish that r(S∞ ) = 1, where S∞ is an operator on `∞ defined by rule S∞ x = z1 Sz (xz). Thus on the space `∞ there exist a band irreducible integral operator T and a positive operator S such that S < T and r(S) = r(T ). Considering the restrictions of operators S ∗ and T ∗ on `1 , we obtain the analogous example in space `1 . Note that by a more “precise” choice of an element z, we can get even the ideal irreducibility of an operator T . The author is grateful to the referee for many suggestions for the improvement of an earlier version of this paper. The work is supported by the Belarussian Foundation of Fundamental Research (Grant No F05-120).

References [1] Abramovich Y.A., Aliprantis C.D.: An invitation to the operator theory. Graduate studies in Mathematics. — Vol. 50. — 2002. [2] Abramovich Y.A., Aliprantis C.D., Burkinshaw O.: On the spectral radius of positive operators. Math. Z. — 1992. — No 211. — P. 593-607. Corrigendum: Math. Z. — 1994. – No 215. – P. 167-168. [3] Ahiezer N.I., Glazman I.M.: Theory of linear operators in Hilbert spaces. (Russian) Izdat. “Nauka”, Moscow, 1966. [4] Alekhno E.A.: On iterations of kernels of primitive non-negative linear integral operators. (Russian) Doklady of the National Academy of Sciences of Belarus. — 2002. — Vol. 46, No 5. — P. 12-15.

14

E. A. Alekhno

[5] Alekhno E.A.: On some properties of irreducible positive operators in Banach lattices. (Russian) Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series.— 2004. – No 1. – P. 17-20. [6] Alekhno E.A., Zabreiko P.P.: Quasi-positive elements and non-decomposable operators in ideal spaces. I. (Russian) Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series. — 2002. — No 4. — P. 5-9. [7] Alekhno E.A., Zabreiko P.P.: Quasi-positive elements and non-decomposable operators in ideal spaces. II. (Russian) Proceedings of the National Academy of Sciences of Belarus. Physics and Mathematics series. — 2003. — No 1. — P. 5-10. [8] Aliprantis C.D., Burkinshaw O.: Positive operators. Academic Press, 1985. [9] Caselles V.: On the peripheral spectrum of positive operators. Isr. J. Math. — 1987. — Vol. 58, No 2. — P. 144-160. [10] Luxemburg W.A.J., Zaanen A.C.: Notes on Banach function spaces, VIII. Indag. Math. — 1964. — Vol. 26. — P. 104-119. [11] Meyer-Nieberg P.: Banach lattices. Springer-Verlag, 1991. [12] Niiro F., Sawashima I.: On the spectral properties of positive irreducible operators in an arbitrary Banach lattices and problems of H. H. Schaefer. Sci. Papers College Arts Sci. Univ. Tokyo. — 1966. — No 16. — P. 145-183. [13] Savchenko S.V.: Stability of the upper bound of the spectrum under perturbations by diagonal matrices for a class of selfadjoint operators. (Russian) Uspekhi Mat. Nauk. — 1998. — Vol. 53, No 2. — P. 163-164. Translation in Russian: Math. Surveys. — 1998. — Vol. 53, No 2. — P. 406-407. [14] Sawashima I.: On spectral properties of some positive operators. Nat. Sci. Rep. Ochanomizu Univ. — 1964. — No 15. — P. 53-64. [15] Schaefer H.H.: Banach lattices and positive operators. Springer-Verlag, 1974. [16] Schaefer H.H.: On theorems of de Pagter and Andˆ o-Krieger. Math. Z. — 1986. — No 192. — P. 155-157. [17] Zaanen A.C.: Introduction to operator theory in Riesz spaces. Springer-Verlag, 1997.

Egor A. Alekhno Belarussian State University Faculty of Mechanics and Mathematics Minsk, Belarus E-mail: [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 15-20 (2006)

Invariant Subspace Theorems for Families on a Banach Lattice S¸afak Alpay and Tunc¸ Mısırlıo˘glu Abstract. We give an invariant subspace theorem for a quasinilpotent compact-friendly operator on a Banach lattice whose ideal center contains sufficiently many operators and extend this result to a family of positive operators. We then give an invariant subspace theorem for a semigroup of positive operators defined on a topological vector space with a basis. Key words: Invariant subspace, Markushevich basis, separating orthomorphism. MSC 2000: 46B42, 47B60, 47B65

1 Preliminaries Let E be a Riesz space. Orth (E) will denote the orthomorphisms of E and Z(E) will denote the ideal center of E. We follow the notation and terminology of [1]. We refer the reader to the same source for all undefined terminology in this note. Definition: A Riesz space E is said to have separating orthomorphisms if the following holds: if x ∧ y = 0, then there exists π ∈ Orth(E) such that π(x) = x and π(y) = 0. Or equivalently, if for all x ∈ E there exists π ∈ Orth(E) such that π(x+ ) = x+ and π(x− ) = 0. We note that if x ∧ y = 0 and π ∈ Orth(E) satisfies π(x) = x and π(y) = 0, then the orthomorphism π1 = |π| ∧ I satisfies π1 (x) = x and π1 (y) = 0. Hence, we may assume that 0 ≤ π ≤ I in the definition. An order ideal I in E has the extension property if every π0 in Z(I) has an extension π ∈ Z(E). The following Theorem is due to de Pagter [4]. Theorem: Consider the following three conditions in an Archimedean Riesz space E. a) If 0 ≤ u ≤ v in E, then u = π(v) for some 0 ≤ π in Orth (E). b) Every principal order ideal in E has the extension property. c) E has separating orthomorphisms. Then a ⇒ b ⇒ c. Moreover, if E is, in addition, uniformly complete, then a, b, and c are equivalent. Every σ-Dedekind complete Riesz space has both the extension property and separating orthomorphisms. However, every order ideal in a uniformly complete Riesz space E has the extension property if and only if E is Dedekind complete. This is due to Wickstead [5]. A sequence {xn } of a Banach space X is called a Schauder basis if for each x ∈ X there ∞ X exists a unique sequence of scalars {αn } such that x = αn xn , where the series is convergent n=1

in norm.

16

S. Alpay, T. Misirlioglu

Let X be a Hausdorff topological vector space with dual X ′ . A sequence (xn , fn ) in X × X ′ is called a Markushevich basis if the span of (xn ) is dense in X, fn (xn ) = 1 and fn (xm ) = 0 for n 6= m, and {fn } separates the points of X. Clearly, a Schauder basis in a Banach space is also a Markushevich basis but the converse is not true in general. A Hausdorff topological vector space with a Markushevich basis can be partially ordered by the relation x ≤ y if and only if fn (x) ≤ fn (y) for all n. Recall that an operator T : X → X is called quasinilpotent at x0 if lim k T n x0 k1/n = 0. n→∞ According to [3] an operator T on a Hausdorff topological vector space X is weakly quasinilpo1 tent at x0 if |f (T n x0 )| n → 0 for each f ∈ X ′ , where X ′ is the topological dual of X. For a Banach space X, a family C of linear operators on X is said to be quasinilpotent at a point x0 ∈ X if lim k C n x0 k1/n = 0, and finitely quasinilpotent at a point x0 ∈ X if every n→∞ finite subcollection of C is quasinilpotent at x0 . The notion of weak quasinilpotency of an operator at a vector x0 can be generalized to families of operators. This was done in [2] to prove an invariant subspace theorem for a family of continuous operators on a discrete Archimedean locally convex solid Riesz space X with dim(X) > 1. We borrow this definition from [2] and define a non-empty set C of continuous linear operators on a topological vector space X to be weakly quasinilpotent at x0 ∈ X if 1 |f (C n (x0 ))| n → 0 for each f ∈ X ′ , where X ′ is the topological dual and |f (C n (x0 ))| = sup{|f (T1 · · · Tn (x0 ))| : Ti ∈ C, i = 1, · · · , n} The commutant of a family C will be denoted by C ′ . For a positive operator T : E → E, we denote by RC (T ) the collection of all positive operators S : E → E such that ST ≥ T S. In accordance with this notation we also let RC (C) = {S : S ≥ 0 and

ST ≥ T S

for each

T ∈ C}.

For a collection C of positive operators on E, the smallest semigroup that contains C will be denoted by SC . Given a family C, the collection DC of positive operators is defined as follows: DC = {D : D ≥ 0, ∃{T1 , · · · , Tk } ⊆ RC (C), {S1 , · · · , Sk } ⊆ SC s.t. D ≤

k X

Ti Si }.

i=1

We refer the reader to [1] for properties of SC and DC .

2 Results The following Theorem was given in [1] as Theorem 10.57. Theorem: If a non-zero compact-friendly operator B : E → E on a Dedekind complete Banach lattice is quasinilpotent at some x0 > 0, then there exists a non-trivial closed ideal that is invariant under RC(B). The next result is a generalization of the preceding Theorem. We show that Dedekind completeness is not needed and that E having separating orthomorphisms is sufficient. The proof is a modification of the proof of Theorem 10.57 in [1]. Proposition 1: Let E be a Banach lattice with separating orthomorphisms. If a non-zero compact-friendly operator B : E → E is quasinilpotent at some x0 > 0, then there exists a non-trivial closed ideal that is invariant under RC (B).

Invariant Subspace Theorems for Families on a Banach Lattice

17

Proof. For each 0 < x, we denote by Jx the ideal generated by the orbit RC (B)x, that is, Jx = {y ∈ E :

|y| ≤ Ax

for some A ∈ RC (B) }.

As x ∈ Jx , Jx is a non-zero ideal and Jx is RC (B)-invariant. Therefore, if for some vector x > 0 the ideal Jx is not norm dense in E, then J x is a non-trivial closed RC (B)-invariant ideal. Consequently, we assume from now on that J x = E for each x > 0. B is compact-friendlly, hence there exist non-zero operators R, K, C : E → E with R positive, K positive compact, and satisfying RB = BR, |Cx| ≤ R|x|, and |Cx| ≤ K|x| for each x ∈ E. Since C 6= 0, there exists some x1 > 0 such that one of the vectors (Cx1 )+ or (Cx1 )− is non-zero. Suppose that (Cx1 )+ 6= 0. Then there exists an orthomorphism M, 0 ≤ M ≤ I, of E with MCx1 > 0. Let x2 = MCx1 and π1 = MC. We note that the operator π1 is dominated by the compact operator K and the positive operator R. Since Jx2 is dense in E and C is non-zero, there exists y ∈ Jx2 and A1 ∈ RC (B) such that 0 < y < A1 x2 and Cy 6= 0. E has separating orthomorphisms, thus there exists an operator M1 , 0 ≤ M1 ≤ I, in Z(E), with y = M1 A1 x2 . Suppose that (CM1 A1 x2 )+ 6= 0. We choose M2 in Z(E) with 0 ≤ M2 ≤ I, such that M2 CM1 A1 x2 = (CM1 A1 x2 )+ > 0. Letting x3 = M2 CM1 A1 x2 and π2 = M2 CM1 A1 , we note that the operator π2 is dominated by the compact operator KA1 and the positive operator RA1 . We now repeat the preceding arguments with the vector x2 replaced by x3 . There exists some z in Jx3 and an operator A2 in RC (B) such that 0 < z ≤ A2 x3 and Cz 6= 0. We choose M3 in Z(E), 0 ≤ M3 ≤ I, with z = M3 A2 x3 . Suppose (CM3 A2 x3 )+ is not zero. We choose M4 in Z(E), 0 ≤ M4 ≤ I with M4 CM3 A2 x3 = (CM3 A2 x3 )+ > 0. Now let π3 = M4 CM3 A2 . Consider the operator π3 π2 π1 . It is non-zero as π3 π2 π1 x1 = π3 x3 6= 0. Each operator πi is dominated by a compact positive operator and therefore π3 π2 π1 is compact by Theorem 2.34 in [1]. Also for each x ∈ E, we have |π3 π2 π1 x| ≤ RA2 RA1 R|x|. If S = RA2 RA1 R, then S ∈ RC(B) since RC (B) is a semigroup. Consider C = {B}. The family DB is finitely quasinilpotent at x0 by Lemma 10.43 in [1]. RC (B) is contained in the family DB . Therefore RC (B) is finitely quasinilpotent at x0 and contains the operator S that dominates a non-zero compact operator π3 π2 π1 . That RC (B) has a non-trivial closed invariant ideal follows from Theorem 10.44 in [1]. A Theorem due to Drnovsek [1, Theorem 10.50] says that if C is a family of positive operators on a Banach lattice which is finitely quasinilpotent at a non-zero positive vector and its commutant C ′ contains an operator that dominates a non-zero compact operator, then the families C and RC (C) have a common non-trivial closed invariant ideal. The next result generalizes the preceding and shows DC and RC (DC ) have a common nontrivial closed ideal. Proposition 2: Let C be a non-zero collection of positive operators on a Banach lattice E with separating orthomorphisms. Suppose C is finitely quasinilpotent at some x0 > 0 and the commutant C ′ of C contains a positive operator T0 which dominates an operator L which is dominated by a compact positive operator K (i.e., there exist L and K (positive, compact) with |Lx| ≤ T0 |x| and |Lx| ≤ K|x| for all x ∈ E). Then C and RC (C) have a common non-trivial closed invariant ideal.

18

S. Alpay, T. Misirlioglu

Proof: We know that the family D C is finitely quasinilpotent at x0 by Lemma 10.43 in [1]. For x > 0, we denote by [DC x] the ideal generated by the orbit DC x of the family DC , i.e., [DC x] = {y ∈ E : |y| ≤ Dx for some D ∈ DC }. Since I ∈ DC , x ∈ [DC x] and [DC x] 6= {0} if x > 0. It is easy to show that [DC x] is invariant under DC . Therefore, we may assume that [DC x] = E for x > 0. As L 6= 0, there exists x1 with Lx1 6= 0. Thus, either (Lx1 )+ or (Lx1 )− is non-zero. Suppose (Lx1 )+ 6= 0. Since E has separating orthomorphisms, there exists an orthomorphism V , 0 ≤ V ≤ I, such that V Lx1 = (Lx1 )+ . Let x2 = V Lx1 and M1 = V L. Observe that the operator M1 is dominated by the compact operator K and the operator T0 . Recall that [DC x2 ] = E. Hence, there exists y with 0 < y ≤ D1 x2 for some D1 ∈ DC such that Ly 6= 0. Since E has separating orthomorphisms, there exists U1 , 0 ≤ U1 ≤ I, with y = U1 D1 x2 . Ly 6= 0 therefore there exists an orthomorphism V1 , 0 ≤ V1 ≤ I, with V1 Ly > 0. That is, V1 LU1 D1 x2 > 0. Let x3 = V1 LU1 D1 x2 and M2 = V1 LU1 D1 . As |M2 z| = |V1 LU1 D1 z| ≤ |LU1 D1 z| ≤ K|U1 D1 z| ≤ KD1 |z| for each z ∈ E, we see that the operator M2 is dominated by the compact operator KD1 . M2 is also dominated by the positive operator T0 D1 . As [DC x3 ] = E, we must have Lz 6= 0 for some z with 0 < z < D2 x3 , where D2 ∈ DC . Then z = U2 D2 x3 for some 0 ≤ U2 ≤ I and, as Lz 6= 0, we have V2 LU2 D2 x3 > 0 for some V2 with 0 ≤ V2 ≤ I. Let M3 = V2 LU2 D2 . Then M3 is dominated by the compact operator KD2 and also by the positive operator T0 D2 . We have T0 D2 T0 D1 T0 ∈ DC since DC is a multiplicative semigroup and T0 ∈ C ′ . On the other hand, |M3 M2 M1 x| ≤ T0 D2 T0 D1 T0 |x| for each x ∈ E. Thus M3 M2 M1 is compact by Theorem 2.34 in [1]. Hence DC contains an operator T0 D2 T0 D1 T0 which dominates the compact operator M3 M2 M1 6= 0. Hence DC has a non-trivial closed invariant ideal by Theorem 10.44 in [1]. Since C and RC (C) are contained in DC , it follows that C and RC (C) have a common non-trivial closed invariant ideal. Next, we give an invariant subspace result for a semigroup of operators on a Banach space with a Schauder basis. The proof goes along the same lines as the proof of a similar result proved for a single operator by Abramovich et al. cf. Theorem 10.66 in [1] and [3]. Proposition 3: Let J be a multiplicative semigroup of positive continuous operators defined on a Banach space X. If {xn } is a Schauder basis for X and J is finitely quasinilpotent at some x0 > 0, then J has a non-trivial closed invariant subspace. Proof: Let {fn } be the sequence \of coefficient functionals associated with the basis {xn }. If T x0 = 0 for each T ∈ J , then T −1 (0) is a non-trivial closed subspace that is invariant T ∈J

under J . Thus, we can assume that T x0 6= 0 for some T ∈ J . Hence T xk 6= 0 for some k. Without loss of generality, we can assume 0 ≤ xk ≤ x0 . Let P : X → X be the continuous projection onto the subspace spanned by xk defined by P x = fk (x)xk . Then 0 ≤ P x ≤ x for each 0 ≤ x ∈ X. We claim that P ST xk = 0 for each S ∈ J . To prove this, let P ST xk = αxk for some non-negative scalar α ≥ 0. Since all the operators involved are positive, we have 0 ≤ αn xk ≤ (P ST )nxk ≤ (ST )n xk ≤ (ST )n x0 .

Invariant Subspace Theorems for Families on a Banach Lattice

19

Using the positivity of the functional fk , we have 0 ≤ αn = fk (αn xk ) ≤ fk ((ST )n x0 ). Consequently, 0 ≤ αn ≤ ||fk || ||(ST )n x0 ||. Since ST ∈ J and J is finitely quasinilpotent at x0 , from 1

1

0 ≤ α ≤ ||fk || n ||(ST )nx0 || n 1

and lim ||(ST )n x0 || n = 0, we have α = 0. n→∞

Finally, we consider the subspace Y generated by {ST xk : S ∈ J }. We have Y 6= {0} since T xk 6= 0 . Since J is a multiplicative semigroup, Y is invariant under J . Thus, it remains to show Y 6= X. For each y ∈ Y , we have fk (y) = fk (P y) = 0, and consequently, fk (y) = 0 for all y ∈ Y . As fk (xk ) 6= 0, Y is a non-trivial closed J -invariant subspace of X. The following is a generalization of the preceding result to topological vector spaces with Markushevich basis. Proposition 4: Let J be a multiplicative semigroup of positive continuous operators on a Hausdorff topological vector space X with a Markushevich basis (xn , fn ). If J is weakly quasinilpotent at some x0 > 0, then J has a non-trivial closed invariant subspace. \ −1 Proof: If T x0 = 0 for each T ∈ J , then T (0) is a non-trivial closed subspace that is T ∈J

invariant under J . Thus we can assume that T x0 6= 0 for some T ∈ J . Since span{xk } = X, it follows that T xk 6= 0 for some k. Without loss of generality, we can assume that 0 ≤ xk ≤ x0 . Consider the projection operator P on X defined by P x = fk (x)xk . Then 0 ≤ P ≤ I. We claim that P ST xk = 0 for each S ∈ J . To prove this, let P ST xk = αxk for some non-negative scalar α. Since all operators are positive, we have 0 ≤ αn xk ≤ (P ST )nxk ≤ (ST )n xk ≤ (ST )n x0 . Using the positivity of the functional fk , we have 0 ≤ αn ≤ fk (αn xk ) ≤ fk ((ST )n x0 ) and consequently, we get 1

0 ≤ α ≤ fk ((ST )n x0 ) n → 0. Hence, α = fk (P ST xk ) = 0. Let Y = span {ST xk : S ∈ J }. As T xk 6= 0, we have Y 6= {0}. Since J is a multiplicative semigroup, Y is invariant under J . Thus, it remains to show Y 6= X. For each y ∈ Y , we have fk (y) = fk (P y) = 0 and consequently fk (y) = 0 for all y ∈ Y . As fk (xk ) 6= 0, xk 6∈ Y and Y is a non-trivial closed invariant subspace under the semigroup J . The authors would like to thank the referee for many valuable comments.

20

S. Alpay, T. Misirlioglu

References [1] Abramovich, Y.A. and Aliprantis, C.D., An invitation to operator theory, Graduate Studies in Math., Vol. 50, Amer. Math. Soc., (2002). [2] C ¸ aglar, M., Invariant subspaces of positive operators on Riesz spaces and observation on CD0 (K)-spaces, Ph.D. Thesis, Middle East Technical University, (2005). ¨ [3] Ercan, Z. and Onal, S., Invariant subspaces for positive operators acting on a Banach space with Markushevich basis, Positivity 8, (2004), 123-126. [4] de Pagter, B., f -algebras and orthomorphisms, Ph.D. Thesis, University of Leiden (1981). [5] Wickstead, A.W., Extensions of orthomorphisms. J. Austral. Math. Soc. 29, (1980), 87-98.

S¸afak Alpay Department of Mathematics Middle East Technical University 06531 Ankara, Republic of Turkey [email protected] R. Tunc¸ Mısırlıo˘glu Department of Mathematics Istanbul Technical University 34469 Maslak-Istanbul, Republic of Turkey [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 21-28 (2006)

The Riesz-Kantorovich formula for the modulus of a complex order bounded disjointness preserving operator Fethi Ben Amor (La Marsa) Abstract. In this note we give an alternative proof of the Riesz-Kantorovich formula for the modulus of an order bounded disjointness preserving operator on complex vector lattices. Our approach is intrinsic and constructive. Also, we don’t assume the vector lattices under consideration to be uniformly complete. Key words: complex vector lattice, disjointness preserving operator, modulus, Riesz-Kantorovich formula. MSC 2000: 06F20,07B65

1 Introduction Let L and M be Archimedean vector lattices such that the vector space complexifications LC = L + iL and MC = M + iM are complex vector lattices. The modulus |T | of an order bounded operator T from LC into MC is defined by |T | = sup {(cos θ) U + (sin θ) V : 0 ∈ [0, 2π]} whenever {(cos θ) U + (sin θ) V : 0 ∈ [0, 2π]} has a supremum in the ordered vector space Lb (L, M) of all order bounded operators from L into M. Moreover, if the set {|T (z)| : z ∈ LC , |z| ≤ x} has a supremum in M for every x ∈ L+ , then the modulus |T | exists and satisfies the RieszKantorovich formula |T | (x) = sup {|T z| : z ∈ LC , |z| ≤ x}

for all x ∈ L+

Meyer proved in [11] that if both L and M additionally are uniformly complete and T preserves disjointness then the set {|T (z)| : z ∈ LC , |z| ≤ x} has a supremum in M for every x ∈ L+ . In this situation, the modulus of T exists in Lb (L, M) and satisfies the above Riesz-Kantorovich formula. Moreover, |T | (x) = |T (x)|

for all x ∈ L+

Later, Arendt in [3] obtained the Meyer’s result but only for complex Banach lattices. The proofs of Arendt and Meyer are based on the Axiom of Choice (i.e. Zorn’s Lemma). Quite

22

F. B. Amor

recently, Grobler and Huijsmans in [6] explicitly and successfully avoided representation theorems which employ the Axiom of Choice. Indeed, using representation results based on functional calculus on uniformly complete Archimedean vector lattices, which is derived without the Axiom of Choice, Grobler and Huijsmans established the above Meyer’s result. Very recently the author and Boulabiar proved in [4] the following result. Theorem 1. Let L and M be Archimedean vector lattices such that LC and MC are complex vector lattices, and let T = U + iV ∈ Lb (LC , MC ) be a disjointness preserving operator. Then the modulus |T | = sup {(cos θ) U + (sin θ) V : θ ∈ [0, 2π]} of T exists in Lb (L, M) and satisfies |T | (x) = |T x| ,

for all x ∈ L+

Their approach is straightforward (i.e., free of the Riesz-Kantorovich formula), constructive (i.e., free of the Axiom of Choice), and intrinsic (i.e., free of any representation). They also drop the hypotheses of uniform completeness on L and M. In this note, taking into account this result and using the same approach, we establish the Riesz-Kantorovich formula for an order bounded disjointness preserving operator T from LC into MC . For terminology, notations and properties not explained or proved in this paper, the reader can consult the classical monographs [2],[9],[12] and [15].

2 Disjointness preserving subsets To avoid unnecessary repetition we will assume throughout this note that all ordered vector spaces and vector lattices considered are Archimedean. Let L an M be vector lattices, and let Lb (L, M) be the ordered vector space of all order bounded operators from L into M. An operator T ∈ Lb (L, M) is said to be disjointness preserving if |T (x)| ∧ |T (y)| = 0 in M whenever |x| ∧ |y| = 0 in L. Meyer proved in [10] that if T ∈ Lb (L, M) is disjointness preserving then the negative part T − , the positive part T + and the absolute value |T | of T in Lb (L, M) exist and satisfy T + x = (T x)+ ,

T − x = (T x)− ,

|T | x = |T x| ,

for all x ∈ L+ .

An elementary proof of this fundamental result due to Bernau can be found in [2] or [12]. Also, de Pagter furnished another elegant proof of Meyer’s result in [13]. For more background on disjointness preserving operators on vector lattices, we refer to [1], [7] and [12]. In the following definition, we introduce a class of subsets of Lb (L, M). Definition 1. Let D be a nonempty subset of Lb (L, M). We call D a disjointness preserving subset of Lb (L, M) if |x| ∧ |y| = 0 in L implies |Sx| ∧ |T y| = 0 in M for all S, T ∈ D. It is easily checked that D is a disjointness preserving subset of Lb (L, M) if and only if the pair {S, T } is a disjointness preserving subset of Lb (L, M) for all S, T ∈ D. Also, observe that operators in a disjointness preserving subset of Lb (L, M) are automatically disjointness preserving. We proceed with the following proposition.

The Riesz-Kantorovich formula for the modulus of a complex order . . .

23

Proposition 2. Let L and M be Archimedean vector lattices, D be a disjointness preserving subset of Lb (L, M), and S, T ∈ D. Then the supremum S ∨ T and the infimum S ∧ T of S and T in Lb (L, M) exist and satisfy (S ∨ T ) x = Sx ∨ T x and

(S ∧ T ) x = Sx ∧ T x,

for all x ∈ L+ .

Proof. We only prove the ‘supremum’ equality. Let x, y ∈ L such that |x| ∧ |y| = 0. Since S and T are members of a disjointness preserving subset, we get |Sx| ∧ |Sy| = |Sx| ∧ |T y| = |T x| ∧ |Sy| = |T x| ∧ |T y| = 0. This yields 0 ≤ |Sx − T x| ∧ |Sy − T y| ≤ (|Sx| + |T x|) ∧ (|Sy| + |T y|) = 0, so that, S − T is an (order bounded) disjointness preserving. Hence, the negative part (S − T )− of S − T in Lb (L, M) exists and we have (S − T )− x = (Sx − T x)− ,

for all x ∈ L+ .

Therefore, the supremum S ∨ T of S and T in Lb (L, M) exists and is given by S ∨ T = (S − T )− + S. Hence, if x ∈ L+ then (S ∨ T ) x = (S − T )− x + Sx = (Sx − T x)− + Sx = Sx ∨ T x. This completes the proof of the proposition. The next proposition characterizes disjointness of two operators contained in a disjointness preserving subset. Proposition 3. Let L, M be Archimedean vector lattices, D be a dijointness preserving subset of Lb (L, M), and S, T ∈ D. The following are equivalent. (i) S ⊥ T . (ii) Sf ⊥ T f for all f ∈ L. (iii) Sf ⊥ T g for all f, g ∈ L. Proof. Observe first that {|S| , |T |} is a disjointness preserving subset of Lb (L, M) . (i) ⇒ (ii) Let f ∈ L and observe that |Sf | ∧ |T f | = |S| |f | ∧ |T | |f | = (|S| ∧ |T |) (|f |) = 0 so Sf ⊥ T f . (ii) ⇒ (iii) Let f, g ∈ L. Then 0 ≤ |Sf | ∧ |T g| = |S| |f | ∧ |T | |g| ≤ |S| (|f | + |g|) ∧ |T | (|f | + |g|) = 0. Therefore |Sf | ∧ |T g| = 0. (iii) ⇒ (i) If |Sf | ∧ |T g| = 0 for all f, g ∈ L then (|S| ∧ |T |) f = |S| f ∧ |T | f = |T f | ∧ |Sf | = 0 for all f ∈ L+ . Hence |T | ∧ |S| = 0.

24

F. B. Amor

3 Disjointness preserving operators on complex vector lattices We start this section by some notions on vector space complexifications of ordered vector spaces. Our references on this topic are [14] and [15]. Let E be an ordered vector space, and EC = E + iE be the classical vector space complexification of E. The modulus |z| of z = x + iy ∈ EC is defined by |z| = sup {(cos θ) x + (sin θ) y : θ ∈ [0, 2π]} whenever this supremum exists in E. If the modulus |z| of z exists in E for all z ∈ EC then EC is referred to as a complex vector lattice. Observe that a necessary condition for EC in order to be a complex vector lattice is that E is a vector lattice. This necessary condition however is not sufficient, unless the vector lattice E is, for example, uniformly complete as it is shown by Luxemburg and Zaanen in [8] (see also [5] by Beukers, Huijsmans and de Pagter). This seems to be rather amazing since complex vector lattices need not be the vector space complexifications of uniformly complete Archimedean vector lattices. An example in this direction is provided next. Example 1. Let E be the vector space of all real-valued step functions on the real interval [0, 1]. 1/2 Under the pointwise ordering, E is an Archimedean vector lattice. Observe that (x2 + y 2) ∈ E for all z = x + iy ∈ EC . It follows that the modulus
1/2 |z| = sup {(cos θ) x + (sin θ) y : θ ∈ [0, 2π]} = x2 + y 2 exists in E for all z = x + iy ∈ EC . Hence, EC is a complex vector lattice, though E is not uniformly complete.

In this note, we do not assume the uniform completeness on (real) vector lattices under consideration. From now on, L and M stand for vector lattices such that LC and MC are complex vector lattices. The definition of disjointness preserving operators on vector lattices extends in an obvious way to the complex vector lattices case. Indeed, an operator T ∈ Lb (LC , MC ) is said to be disjointness preserving if |T (w)|∧|T (z)| = 0 in M for all w, z ∈ LC such that |w|∧|z| = 0 in L. The first result in this section gives a characterization of complex disjointness preserving operators in terms of disjointness preserving subsets of Lb (L, M). Proposition 4. Let L and M be Archimedean vector lattices such that LC and MC are complex vector lattices, and let U, V ∈ Lb (L, M). Then T = U + iV is disjointness preserving if and only if {U, V } is a disjointness preserving subset of Lb (L, M). Proof. Assume first that T = U + iV is disjointness preserving and let x, y ∈ L such that |x| ∧ |y| = 0. It follows that 0 ≤ |U (x)| ∧ |U (y)| ≤ |U (x) + iV (x)| ∧ |U (y) + iV (y)| = |T (x)| ∧ |T (y)| = 0, and then U preserves disjointness. In a same way we prove that V is also a disjointness preserving operator and that |U (x)| ∧ |V (y)| = 0, for all x, y ∈ L such that |x| ∧ |y| = 0. This proves that {U, V } is a disjointness preserving subset of Lb (L, M).

The Riesz-Kantorovich formula for the modulus of a complex order . . .

25

Conversely, suppose that {U, V } is a disjointness preserving subset of Lb (L, M). Let w = u + iv, z = x + iy ∈ LC such that |w| ∧ |z| = 0 in L. Since 0 ≤ |u| ∧ |x| ≤ |z| ∧ |w| = 0, we get |u| ∧ |x| = 0. In the same way, |u| ∧ |y| = |v| ∧ |x| = |v| ∧ |y| = 0. Since {U, V } is a disjointness preserving subset of Lb (L, M), we derive that the sets {U (u) , V (u) , U (v) , V (v)} and {U (x) , V (x) , U (y) , V (y)} are disjoint in M. On the other hand, |T (w)| ≤ |U (u)|+|V (u)|+|U (v)|+|V (v)| and |T (z)| ≤ |U (x)|+|V (x)|+|U (y)|+|V (y)| . Consequently, |T (z)| ∧ |T (w)| = 0, so that, T is disjointness preserving and we are done.

4 The main result As previously pointed out, L and M in this paper are Archimedean vector lattices such that
δ LC and MC are complex vector lattices. Since Lb L, M is Dedekind complete, the following complex version of the Freudenthal’s spectral theorem is available (here M δ denotes the Dedekind completion of M). See Theorem 36.1 in [16].
Lemma 5. Let T = U + iV ∈ Lb (LC , MC ) and put W = |U| + |V | ∈ Lb L, M δ . Then for any ε > 0, there exist disjoint components W1 , ...Wn of W and complex number γ 1 ...γ n such that |γ k | ≤ 1 for all k = 1, ...n and n n P P T − γ W ≤ εW as well as 0 ≤ |T | − |γ k | Wk ≤ εW k k k=1

k=1

The next lemma is a particular case of the main result.

Lemma 6. Let T ∈ Lb (L, M) be a disjointness preserving operator. Then |T z| ≤ |T | |z|

for all z ∈ LC .

Proof. Let z = x + iy ∈ LC .

n h π io |T z| = |T x + iT y| = sup cos θ (|T x|) + sin θ (|T y|) : θ ∈ 0, 2 n h π io ≤ sup cos θ (|T | |x|) + sin θ (|T | |y|) : θ ∈ 0, 2 n h π io ≤ sup |T | (cos θ (|x|) + sin θ (|y|)) : θ ∈ 0, 2 ≤ |T | |z|

Theorem 7. Let T = U + iV ∈ Lb (LC , MC ) be a disjointness preserving operator. Then |T z| ≤ |T | |z|

for all z ∈ LC .


Proof. Observe first that W = |U| + |V | ∈ Lb L, M δ is a lattice homomorphism and assume that W > 0. By lemma5, for any ε > 0, there exist disjoint components W1 , ...Wn of W and complex number γ 1 ...γ n such that |γ k | ≤ 1 for all k = 1, ...n and n n P P T − ≤ εW and 0 ≤ |T | − γ W |γ k | Wk ≤ εW k k k=1

k=1

26

F. B. Amor

If we write γ k = αk + iβ k , with αk , β k ∈ R for all k = 1, ..., n, n n P P U − ≤ T − ≤ εW α W γ W k k k k k=1

and

k=1

n n P P V − ≤ T − ≤ εW β W γ W k k k k k=1

k=1

Since 0 ≤ Wk ≤ W for all k = 1, ..., n, and W is a lattice homomorphism, {W1 , ...Wn } is a disjointness preserving subset of mutually disjoint operators. We deduce that Wk x ⊥ Wl y

for all k 6= l in {1, ..., n} and all x, y ∈ L.

It follows that Wk z ⊥ Wl z

for all k 6= l in {1, ..., n} and all z ∈ LC .

For all z ∈ LC :   n n n P P P = γ (W z) = |γ k | |Wk z| γ W z k k k k k=1 k=1 k=1  n  n P P ≤ |γ k | Wk |z| = |γ k | Wk |z| k=1

then

k=1

    n   n n P P P |γ k | Wk |z| γ k Wk z − γ k Wk z + |T z| − |T | |z| = |T z| − k=1 k=1 k=1  n  P |γ k | Wk |z| − |T | |z| + k=1  n    n P P |γ k | Wk |z| − |T | |z| γ k Wk z + ≤ |T z| − k=1 k=1    n  n P P |γ k | Wk |z| − |T | |z| γ k Wk z + ≤ T z − k=1

and

(|T z| − |T | |z|)

+

k=1

 n   n  P P ≤ T z − γ k Wk z + |γ k | Wk |z| − |T | |z|   k=1  n k=1 n P P β k Wk z ≤ Uz − αk Wk z + V z − k=1 k=1   n P + |γ k | Wk − |T | |z| k=1  n   n  P P ≤ U − αk Wk |z| + V − β k Wk |z| k=1  k=1  n P + |γ k | Wk − |T | |z| k=1

≤ 3εW |z|

since F is Archimedean, (|T z| − |T | |z|)+ ≤ 0 and then |T z| ≤ |T | |z| .

The Riesz-Kantorovich formula for the modulus of a complex order . . .

27

We end this note with the following. Corollary 8. Let T ∈ Lb (LC , MC ) be a disjointness preserving operator. Then |T | x = sup {|T z| : |z| ≤ x}

for all x ∈ L+

Proof. Let x ∈ L+ . By the precedent result, |T z| ≤ |T | |z| ≤ |T | x

for all z ∈ LC such that |z| ≤ x.

and by theorem.1 |T | x = |T x| . We conclude that {|T z| : |z| ≤ x} has a supremum in M and satisfy |T x| = sup {|T z| : |z| ≤ x}. It follows that |T | x = sup {|T z| : |z| ≤ x} and we are done.

References [1] Y. A. Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Memoirs Amer. Math. Soc. 143, no 679 (2000). [2] C. D. Aliprantis and O. Burkinshaw, Positive Operators, Academic Press Orlando 1985. [3] W. Arendt, Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32, 199-215 (1983). [4] F. Ben amor-K. Boulabiar, On the modulus of disjointness preserving operators on complex vector lattices, Algebra univ., A paraitre [5] F. Beukers, C.B. Huijsmans and B. de Pagter, Unital embedding and complexification of f -algebras, Math. Z. 183, 131-144 (1983). [6] J. J. Grobler and C. B. Huijsmans, Disjointness preserving operators on complex Riesz spaces, Positivity 1, 155-164 (1997). [7] M. Henriksen and F. A. Smith, A look at biseparating maps from an algebraic point of view, Real Algebraic Geometry and Ordered Structures, 125-144, Contemp. Math., 253, Amer. Math. Soc., 2000. [8] W. A. J. Luxembourg and A. C. Zaanen, The linear modulus of an order bounded linear transformation, Indag. Math. 33, 422-434 (1971). [9] W. A. J. Luxembourg and A. C. Zaanen, Riesz Spaces I, North-Holland Amsterdam 1971. [10] M. Meyer, Le stabilateur d’un espace vectoriel r´eticul´e, C. R. Acad. Sci. Paris Serie I 283, 249-250 (1976). [11] M. Meyer, Les homomorphismes d’espaces vectoriels r´eticul´es complexes, C. R. Acad. Sci. Paris Serie I 292, 793-796 (1981).

28

F. B. Amor

[12] P. Meyer-Nieberg, Banach Lattices, Springer Verlag. Berlin-Heidelberg-New York 1991. [13] B. de Pagter, A note on disjointness preserving operators, Proc. Amer. Math. Soc. 90, 543-549 (1984). [14] H. H. Schaefer, Banach Lattice and Positive Operators, Springer Verlag BerlinHeidelberg-New York 1974. [15] A. C. Zaanen, Riesz Spaces II, North-Holland Amsterdam 1983. [16] A. C. Zaanen, Introduction to operator theory in Riesz spaces, Springer, BerlinHeidelberg-New York, 1997.

Fethi Ben Amor IPEST, Universit´e de Carthage BP 51, 2070-La Marsa, Tunisia

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 29-33 (2006)

A note on bijective disjointness preserving operators Karim Boulabiar (Carthage) and Gerard Buskes (Mississippi) Abstract. We use the theory of vector lattices to obtain new proofs for results by Araujo-BeckensteinNarici and Jeang-Wong for bijective disjointness preserving linear operators on certain algebras of continuous functions. Key words: Continous functions; Disjointness preserving; Orthomorphisms MSC 2000: 06F25, 13J25, 47B65

In [3], Araujo, Beckenstein, and Narici proved that if T is a bijective disjointness preserving linear operator from C (X) onto C (Y ), where X and Y are completely regular, and if the inverse linear operator T −1 of T preserves disjointness as well, then there exist w ∈ C (Y ) and a homeomorphism τ from the realcompactification υY of Y onto υX such that (T f ) (y) = w (y) f (τ (y))

(f ∈ C (X) , y ∈ Y ) .

There seems to be no uniformity as to the the nomenclature in the current literature. Indeed, a bijective linear map T from C (X) onto C (Y ) such that both T and T −1 preserve disjointness is called a biseparating linear map in [3], a d-isomorphism in [1], and a bidisjointness preserving operator in [6]. We further mention that representations of the type above seem to have their earliest appearance in the paper [2]. Jeang and Wong established in [9] that if T is a bijective disjointness preserving linear operator from C0 (X) into C0 (Y ) ( where X and Y are locally compact Hausdorff) then there exists a bounded function w ∈ C (Y ) and a homeomorphism τ from Y onto X such that (T f ) (y) = w (y) f (τ (y))

(f ∈ C (X) , y ∈ Y ) .

Notice that in the latter result Jeang and Wong do not assume T −1 to preserve disjointness. The compact version of the theorem by Jeang and Wong was obtained earlier by Jarosz [8]. The proofs of the representation theorems in the literature cited above use Banach algebra and topology techniques, and are at times rather involved. The purpose of this note is to show how these results may receive a unified treatment by placing the theory of vector lattices at the root of their proofs. In doing so, we apply the following theorem on disjointness preserving operators due to Hart (see Theorem 2.1 in [6]), which predates the results above by a decade. Recall that Orth (L) denotes the unital f -algebra of all orthomorphisms on an Archimedean Riesz space L (see[11]). Theorem 1 (Hart). Let L and M be Archimedean Riesz spaces and let T be an order bounded bijective disjointness preserving linear operator from L onto M. Then there exists a unique

30

K. Boulabiar, G. Buskes

algebra and lattice isomorphism Te from Orth (L) onto Orth (M) such that   Teπ (T f ) = T πf (π ∈ Orth (L) , f ∈ L) .

The version for f -algebras of Theorem 1 is discussed next. Let A be an Archimedean semiprime f -algebra [11]. For each f ∈ A, let π f denote the orthomorphism on A defined by π f g = f g for all g ∈ A. The map π from A into Orth (A) defined by πf = π f

(f ∈ A)

is an injective algebra and lattice homomorphism. Hence A and the range πA of A are isomorphic as f -algebras, and A can be considered as an f -subalgebra of Orth (A). In particular, if f ∈ A and π ∈ Orth (A) then πf can be viewed as the product in Orth (A) of f and π. Keeping in mind the latter identifications, the next result follows quickly from Theorem 1. Corollary 2. Let A and B be Archimedean semiprime f -algebras and let T be a bijective order bounded disjointness preserving linear operator from A onto B. Then there exists a unique algebra and lattice isomorphism Te from Orth (A) onto Orth (B) such that   T (f g) = (T f ) Teg (f, g ∈ A) .

The following notational conventions are used throughout the sequel. For a topological space X, we denote by C (X) the Archimedean unital f -algebra of all continuous real-valued functions on X. The subalgebra of C (X) of all bounded functions is denoted by Cb (X). If in addition X is locally compact then C0 (X) denotes the algebra of all continuous functions that vanish at infinity (in [5], the notations C ∗ (X) and C∞ (X) are used instead of Cb (X) and C0 (X) , respectively). Furthermore, υX denotes the realcompactification of X when X is completely regular [5]. Finally, the unit element of C (X) is denoted by 1X , i.e. 1X (x) = 1 for all x ∈ X. We now use Corollary 2 to obtain the aforementioned theorem of Araujo, Beckenstein, and Narici. Theorem 3 (Araujo, Beckenstein and Narici). Let X and Y be completely regular topological spaces and let T be a bijective disjointness preserving linear operator from C (X) onto C (Y ) such that T −1 also preserves disjointness. Then there exist w ∈ C (Y ) and a homeomorphism τ from υY onto υX such that (T f ) (y) = w (y) f (τ (y))

(f ∈ C (X) , y ∈ Y ) .

Proof. It is shown in [4] that every universally σ-complete projection band in C (X) is essentially one-dimensional (see [1] for the definition). Since C (Y ) is uniformly complete, it follows from Corollary 15.3 in [1] that T is order bounded. Moreover, C (X) coincides with Orth (C (X)) as C (X) is an f -algebra with unit [12]. By Theorem 2 there exists a unique algebra and lattice isomorphism Te from C (X) onto C (Y ) such that   T (f g) = (T f ) Teg (f, g ∈ C (X)) . Furthermore, it is well known that there exists a homeomorphism τ from the realcompactification υX of X onto the realcompactification υY of Y such that Tef = f ◦ τ for all f ∈ C (X) [5, Section 10]. Putting w = T 1X , we get the desired result.

A note on bijective disjointness preserving operators

31

We need the next lemma to obtain the result by Jeang and Wong mentioned in the introduction. The lemma in question likely can be found in the literature, but for completeness sake we provide a proof. Lemma 4. Let X be a locally compact Hausdorff space. Then Orth(C0 (X)) = Cb (X). Proof. It is known that Orth(CK (X)) is algebra and lattice isomorphic to C(X), where CK (X) is the space of continuous functions on X with compact support [12]. The natural (restriction) map from Orth(C0 (X)) to Orth(CK (X)) whose value on T ∈ Orth(C0 (X)) is T˜ = T |CK (X) ˜ ) for all f ∈ CK (X). Let g be is injective. Indeed, let T, S ∈ Orth(C0 (X)) and let T˜(f ) = S(f positive in C0 (X). From the order continuity of T and S and the fact that  + !  + ! 1 1 T˜ g − 1X = S˜ g − 1X (n ∈ N) , n n it follows that T = S. Thus Orth(C0 (X)) is algebra and lattice isomorphic to the f -algebra of those functions p in C(X) for which pf ∈ C0 (X) for all f ∈ C0 (X). By an easy argument using the norm completeness of C0 (X) this completes the proof. ˇ Let βX denote the Stone-Cech compactification of a completely regular topological space X. Recall that every fonction f in Cb (X) extends uniquely to a function f β in C (βX) so that the map β defined from Cb (X) into C (βX) by β (f ) = f β for all f ∈ Cb (X) is an algebra and lattice isomorphism [5]. The result by Jeang and Wong now follows. Theorem 5 (Jeang and Wong). Let X and Y be locally compact Hausdorff topological spaces and let T be a bijective disjointness preserving linear operator from C0 (X) onto C0 (Y ). Then there exist w ∈ Cb (Y ) and a homeomorphism τ from Y into X such that (T f ) (y) = w (y) f (τ (y))

(f ∈ C0 (X) , y ∈ Y ) .

Proof. From Lemma 4 above, Orth (C0 (X)) equals Cb (X). Since both C0 (X) and C0 (Y ) (endowed with their uniform norms) are Banach lattices, Theorem 2.3 in [7] yields that T is order bounded. Hence by Theorem 2 there exists a unique algebra and lattice isomorphism Te from Cb (X) onto Cb (Y ) such that   T (f g) = (T f ) Teg (f ∈ C0 (X) and g ∈ Cb (X)) . Then there exists a homeomorphism τ from βY onto βX such that Teg = g β ◦ τ

(g ∈ Cb (X)) .

Therefore

T (f g) = (T f ) g β ◦ τ




32

K. Boulabiar, G. Buskes

for all f ∈ C0 (X) and all g ∈ Cb (X). ¿From the
latter it easily follows that for each positive β f ∈ C0 (X) and each y ∈ Y we have  that f ◦ τ (y) 6= 0 if and only if T (f )(y) 6= 0. Indeed, p
f β ◦ τ (y) = 0 and if f β ◦ τ (y) = 0 then p  p T (f ) (y) = T ( f )(y) f β ◦ τ (y) = 0.


Vice versa, suppose f β ◦ τ (y) 6= 0 and choose (from the fact that T is bijective) a positive function g ∈ C0 (X) such that T (g)(y) 6= 0; then g β ◦ τ (y) 6= 0 by what we just derived,

T (f g)(y) = T (f )(y) g β ◦ τ (y) = T (g)(y) f β ◦ τ (y), and hence T (f )(y) 6= 0. For y ∈ Y define
w (y) = T f (y)/ f β ◦ τ (y)
for any
f such that f β ◦ τ
(y) 6= 0. Then w is well-defined, for let f, g ∈ C0 (X) for which f β ◦ τ (y) 6= 0 and g β ◦ τ (y) 6= 0. Then

(T f ) g β ◦ τ = T (f g) = T (gf ) = (T g) f β ◦ τ , so



T f (y)/ f β ◦ τ (y) = T g(y)/ g β ◦ τ (y).

Clearly w ∈ C (Y ) and w(y) 6= 0 for all y ∈ Y . Also,
(T f ) (y) = w (y) f β ◦ τ (y) (y ∈ Y )

whenever f β ◦ τ (y) 6= 0. If f is positive in C0 (X) and f β ◦ τ (y) = 0 for some y ∈ Y  p f β ◦ τ (y) = 0 and hence T (f )(y) = 0. Thus the equality then also
(T f ) (y) = w (y) f β ◦ τ (y)

(y ∈ Y )

holds for each positive function f in C0 (X) and hence for all f . Now in the end we can see that τ actually maps Y to X and (T f ) (y) = w (y) (f ◦ τ ) (y)

(y ∈ Y ) .

The proof is complete. We end this note with the compact version of Corollary 5. Corollary 6 (Jarosz). Let X and Y be compact Hausdorff topological spaces and let T be a bijective disjointness preserving operator from C (X) onto C (Y ). Then there exist w ∈ C (Y ) and a homeomorphism τ from Y into X such that (T f ) (y) = w (y) f (τ (y))

(f ∈ C (X) , y ∈ Y ) .

Acknowledgments. Both authors gratefully acknowledge support from the NSF-Grant INT0423522 and the second author gratefully acknowledges support from the NWO bezoekersbeurs B 61-548 while visiting the University of Delft in the winter of 2004.

A note on bijective disjointness preserving operators

33

References [1] Y. A . Abramovich and A. K. Kitover, Inverses of disjointness preserving operators, Memoirs Amer. Math. Soc., 143 (2000), no 679. [2] Y. A. Abramovich, A. I. Veksler, and A. V. Koldunov, Operators preserving disjointness, Dokl. Acad. Nauk USSR, 248 (1979), 1033-1036. [3] J Araujo, E. Beckenstein, and L Narici, Biseparating maps and homeomorphic realcompactifications, J. Math. Anal. Appl., 12 (1995), 258-265. [4] K. Boulabiar, G. Buskes, and M. Henriksen, A generalization of a theorem on biseparating maps, J. Math. Anal. Appl., 280 (2003), 334-349. [5] L. Gillman and M Jerison, Rings of Continuous Functions, Springer-Verlag, Berlin, 1976. [6] D. R. Hart, Some properties of disjointness preserving operators, Indag. Math., 88 (1985), 183-197. [7] C. B. Huijsmans and B. de Pagter, Invertible disjointness preserving operators, Proc. Edinburgh Math. Soc., 37 (1993), 125-132. [8] K. Jarosz, Automatic continuity of separating linear isomorphisms, Can. Math. Bull., 33 (1990), 139-144. [9] J. S. Jeang and N. C. Wong, Weighted composition of C0 (X)’s, J. Math. Anal. Appl., 201 (1996), 981-993. [10] P. Meyer-Nieberg, Banach Lattices, Springer-Verlag, Berlin, 1997. [11] B. de Pagter, f -Algebras and Orthomorphisms, Thesis, Leiden, 1981. [12] A. C. Zaanen, Examples of orthomorphisms, J. Approx. Theory, 13 (1975), 192-204.

Karim Boulabiar IPEST, University of Carthage BP51, 2078-La Marsa, Tunisia Gerard Buskes Department of Mathematics University of Mississippi MS-38677, USA

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 35-44 (2006)

Band preserving operators on lattice-ordered groups G. Buskes (Mississippi) and R. H. Redfield (New York) Abstract. We study conditions that force the inverse of band preserving one-to-one group homomorphisms on lattice-ordered groups to have band preserving inverses. Key words: Disjointness preserving; Band preserving; Lattice-ordered Group MSC 2000: 06F15, 47B65

1 Introduction Let G and H be vector lattices. A linear map f : G → H is called disjointness preserving if it satisfies the condition: (1.1)

∀x, y ∈ G, |x|∧|y| = 0 =⇒ |f (x)|∧|f (y)| = 0;

f is said to be band preserving if it satisfies (1.2)

∀x, y ∈ G, |x|∧|y| = 0 =⇒ |f (x)|∧|y| = 0.

There is an extensive literature on these types of operators on Archimedean vector lattices. Two problems have been leading the developments in the theory of these operators, Problem A and Problem B in the memoir of Abramovich and Kitover [3]. Problem A asks whether the inverse of a bijective disjointness preserving operator is also disjointness. Problem B asks whether, if a bijective disjointness preserving has a disjointness preserving inverse, its domain and range are order isomorphic. Every band preserving operator is disjointness preserving and if a bijective band preserving operator has a disjointness preserving inverse then that inverse is band preserving. Recently in [4], Abramovich and Kitover have solved Problem A for band preserving operators, even when f is only injective. For Problem B, while they proved many positive results, they gave a negative answer in general by describing a method for constructing examples of vector lattices that are not order isomorphic, in spite of the existence of a bijective disjointness preserving map with disjointness preserving inverse between them. In contrast with the rich literature on disjointness preserving and band preserving operators on Archimedean vector lattices, the literature on similar maps for lattice-ordered groups (ℓgroups), aside from exploratory investigations for abelian lattice-ordered groups by ConradDiem [10] and Keimel-Bigard [9], is not well developed. Even more so, a study of what we will define as disjointness preserving group homomorphisms appears to be fully absent. In this note, we consider the situation for lattice-ordered groups. In particular, we find several characterizations of one-to-one band preserving operators on lattice-ordered groups that have band preserving inverses, and then we use these characterizations to show, e.g., that if

36

G. Buskes, R. H. Redfield

the group is projectable, then all one-to-one band preserving operators have band preserving inverses. These results are in the style of similar characterizations in [5]. We will use the following terminology and notation. A function f : A → B from one partially ordered set to another is said to be order preserving if ∀x, y ∈ A, x ≥ y =⇒ f (x) ≥ f (y) [11, p. 391]. If A and B are lattice-ordered groups and f is a group homomorphism, then [12, p. 20] f is order preserving if and only if ∀x ∈ A, x ≥ 0 =⇒ f (x) ≥ 0. In this situation, one can define various notions of disjointness preserving. For instance, one can require that ∀x, y ∈ A, x ∧ y = 0 ⇒ f (x) ∧ f (y) = 0; it is not difficult to show that this condition is equivalent to requiring that f be a lattice-homomorphism. A weaker condition is (1.1) for ℓgroups: ∀x, y ∈ A, |x|∧|y| = 0 ⇒ |f (x)|∧|f (y)| = 0; such a function is said to be disjointness preserving. There are other conditions that have the effect of restricting lateral movement as well as preserving disjointness. For instance, for a lattice-ordered group G, a group homomorphism f : G → G is said to be polar-preserving [11, p. 391] if it satisfies the condition: ∀x, y ∈ G, x ∧ y = 0 ⇒ f (x) ∧ y = 0. (Note that a second application of the condition shows that such functions are always lattice-homomorphisms.) We want to consider a similar but less restrictive condition, viz. the analogue of (1.2) for ℓ-groups: ∀x, y ∈ G, |x|∧|y| = 0 ⇒ |f (x)|∧|y| = 0. In the theory of vector lattices, convex subspaces that are closed with respect to arbitrary suprema are called bands, and as noted above an operator satisfying (1.2) is called band preserving. For this reason, we adopt the same terminology for lattice-ordered groups. Note that f is polarpreserving if and only if it is band preserving and order preserving. If a band preserving operator f is one-to-one, it has an inverse function f −1 : f (G) → G; we say that f −1 is band preserving if ∀x, y ∈ G, |f (x)| ∧ |f (y)| = 0 ⇒ |x| ∧ |f (y)| = 0. Finally recall that if G is an ℓ-group and S ⊆ G, then S ⊥ = {g ∈ G |g| ∧ |s| = 0 for all s ∈ S} is a convex ℓ-subgroup of G [11, Theorem 13.1]. To place the results of our paper in a context, we briefly sketch the development that lead up to the Abramovich-Kitover Theorem for inverses of one-to-one band preserving operators. Huijsmans and de Pagter first proved (unpublished) that a bijective, order bounded, and band preserving operator has a band preserving inverse. That result had been preceded by a result of Arendt [7], who showed that a bijective, order bounded, disjointness preserving map between Banach lattices has a disjointness preserving inverse. In the meantime, Abramovich [1] had shown that band preserving operators on Banach lattices are automatically order bounded. Huijsmans and Wickstead, in [14], showed that inverses of order bounded disjointness preserving operators between Archimedean vector lattices are disjointness preserving and that inverses of band preserving operators on a vector lattice are band preserving if the vector lattice has the principal projection property or is relatively uniformly complete. Abramovich and Kitover provided an example in [2] of a bijective disjointness preserving operator on a normed vector lattice that does not have a disjointness preserving inverse. Somewhat later they provided an example of a Dedekind complete vector lattice and a bijective disjointness preserving operator on it without a disjointness preserving inverse. Much of the above culminated in the very beautiful memoir [3], to which we refer for an extensive bibliography and more detailed account of the history of the problem. For the more rigid band preserving operators on Archimedean vector lattices, Abramovich and Kitover showed that the inverse of a one-to-one band preserving op-

Band preserving operators on lattice-ordered groups

37

erator on an Archimedean vector lattice also is band preserving. Their proof relied heavily on topological methods applied on the Stone space of the Boolean algebra of bands. Here we offer a theory that generalizes the results proved by Abramovich and Kitover in [5], while maintaining the flavour of that paper. At the same time, we emphasize that not only do we prove our results for lattice-ordered groups rather than vector lattices but also we do not assume our groups to be Archimedean. The latter has considerable negative consequences for applicability to our setting of the techniques used in the results obtained by the authors above. For instance, the simple and often used fact that an order bounded disjointness preserving operator f on Archimedean vector lattice, |x| ≤ |y| ⇒ |f (x)| ≤ |f (y)| is not necessarily valid in the non-Archimedean setting (see Example 5.1 in this paper). On the other hand, there are positive consequences (in this paper), since we provide a positive answer to Problem A for band preserving group homomorphisms on a variety of non-Archimedean ℓgroups. We also provide what seems to be a novel characterization of one-to-one band preserving group homomorphisms with band preserving inverses in terms of minimal prime subgroups.

2 Band Preserving Functions with Band Preserving Inverses The object of this section is to characterize those one-to-one band preserving group homomorphims that have band preserving inverses (Theorem 1). We begin by making some general observations about ℓ-groups. Proposition 1 Let a and b be elements of an ℓ-group G. If |a| ∧ |b| = 0, then |a + b| = |a − b|. Proof Note that our hypothesis ensures that x ∧ y = 0 for any distinct x, y in {a+ , a− , b+ , b− }. Then a + b = (a+ + b+ ) − (a− + b− ) = (a+ ∨ b+ ) − (a− ∨ b− ) and (a+ ∨ b+ ) ∧ (a− ∨ b− ) = 0. So by [12, p. 75], (a+b)+ = a+ ∨b+ and (a+b)− = a− ∨b− and hence |a+b| = a+ ∨b+ ∨a− ∨b− . As well, a − b = (a+ + b− ) − (a− + b+ ) = (a+ ∨ b− ) − (a− ∨ b+ ) and (a+ ∨ b− ) ∧ (a− ∨ b+ ) = 0. So |a − b| = a+ ∨ b− ∨ a− ∨ b+ , and hence |a + b| = |a − b|. Proposition 2 Let x and y be elements of an ℓ-group G. If y ⊥ ) x⊥ , then y ⊥ ∩ x⊥⊥ 6= {0}. Proof Since y ⊥ ) x⊥ , there exists 0 < z ∈ y ⊥ such that z 6∈ x⊥ . Then 0 < z ∧ |x| ∈ y ⊥ , and since 0 < z ∧ |x| ≤ |x|, z ∧ |x| ∈ x⊥⊥ . We next list several conditions that characterize band preserving group homomorphisms and note a property of such functions that will be extremely useful in the sequel. Proposition 3 Suppose that G is an ℓ-group and that f : G → G is a group homomorphism. Then the following statements are equivalent. 1. f is band preserving; 2. for all x, y ∈ G, if y ∈ x⊥ , then f (y) ∈ x⊥ ;

38

G. Buskes, R. H. Redfield

3. for all x ∈ G, f (x⊥ ) ⊆ x⊥ ; 4. for all w, x ∈ G, if w ⊥ ⊆ x⊥ , then w ⊥ ⊆ f (x)⊥ . 5. for all x ∈ G, f (x)⊥ ⊇ x⊥ . Proof It is easy to see that (1), (2), and (3) are all equivalent and that (4) implies (5). (1) =⇒ (4): Suppose that w ⊥ ⊆ x⊥ and let y ∈ w ⊥ . Then y ∈ x⊥ and thus |y| ∧ |x| = 0. So by (1), |y| ∧ |f (x)| = 0, and hence y ∈ f (x)⊥ . (5) =⇒ (1): If |x| ∧ |y| = 0, then y ∈ x⊥ and hence by (5), y ∈ f (x)⊥ . So |f (x)| ∧ |y| = 0, and thus f is band preserving. Proposition 4 Let G be an ℓ-group and suppose that f : G → G be a band preserving group homomorphism. For all y ∈ G, f (y)⊥ = f (|y|)⊥. Proof Since y + ∧ y − = 0 and f is band preserving, |f (y + )| ∧ |f (y − )| = 0. So by Proposition 1, |f (|y|)| = |f (y + ) + f (y − )| = |f (y + ) − f (y −)| = |f (y)|, and hence f (y)⊥ = |f (y)|⊥ = |f (|y|)|⊥ = f (|y|)⊥. We can now derive a host of conditions characterizing one-to-one band preserving group homomorphisms having band preserving inverses. Theorem 1 Suppose that G is an ℓ-group and that f : G → G is a one-to-one band preserving group homomorphism. Then the following statements are equivalent: 1. f −1 is band preserving; 2. for all x ∈ G, f (x)⊥ = x⊥ ; 3. for all 0 < x ∈ G, f (x)⊥ = x⊥ ; 4. for all x ∈ f (G), f −1 (x)⊥ = x⊥ ; 5. for all w, x ∈ G, 0 < w < x =⇒ f (w)⊥ ⊇ f (x)⊥ ; 6. for all w, x ∈ G, 0 < w < x =⇒ |f (w| ∧ |f (x)| > 0; 7. for all x, y ∈ G, x⊥ ⊆ y ⊥ =⇒ f (x)⊥ ⊆ f (y)⊥ ; 8. for all 0 < x, y ∈ G, x⊥ ⊆ y ⊥ =⇒ f (x)⊥ ⊆ f (y)⊥ . Proof (1) =⇒ (2): If f (x)⊥ 6= x⊥ , then Proposition 3 implies that f (x)⊥ ) x⊥ . So by Proposition 2, there exists 0 < w ∈ f (x)⊥ ∩ x⊥⊥ . Then |f (x)| ∧ |w| = 0 and thus since f is band preserving, |f (x)| ∧ |f (w)| = 0. So by (1), we have |x| ∧ |f (w)| = 0, and since w ∈ x⊥⊥ , w ⊥ ⊇ x⊥ . Then f (w) ∈ w ⊥ and thus |w|∧|f (w)| = 0. So by (1), |f (w)| = |f (w)|∧|f (w)| = 0 and hence f (w) = 0. Then since f is one-to-one, w = 0, a contradiction. So f (x)⊥ = x⊥ . (2) =⇒ (1): If |f (x)| ∧ |f (y)| = 0, then by (2), f (y) ∈ f (x)⊥ = x⊥ and hence |x| ∧ |f (y)| = 0. So (1) holds. (2) ⇐⇒ (3): That (2) =⇒ (3) is obvious. Conversely, if (3) holds, then certainly f (0)⊥ = 0⊥ . Furthermore, for any x 6= 0, |x| > 0 and thus by (2) and Proposition 4, f (x)⊥ = f (|x|)⊥ = |x|⊥ = x⊥ .

Band preserving operators on lattice-ordered groups

39

(2) ⇐⇒ (4): Suppose that (2) holds and assume that y ∈ f (G), i.e., that y = f (x) for some x ∈ G. By (2), we have f −1 (y)⊥ = x⊥ = f (x)⊥ = y ⊥ , and hence (4) holds. Conversely if (4) holds and x ∈ G, then x⊥ = f −1 (f (x))⊥ = f (x)⊥ , and thus (2) holds. (2) =⇒ (5): If 0 < w < x, then by (2), f (w)⊥ = w ⊥ ⊇ x⊥ = f (x)⊥ . (5) =⇒ (6): If 0 < w < x, then by (5), |f (w)|⊥ = f (w)⊥ ⊇ f (x)⊥ = |f (x)|⊥ . But then if |f (w)| ∧ |f (x)| = 0, f (w) ∈ f (x)⊥ ⊆ f (w)⊥ and hence f (w) = 0, a contradiction. So |f (w)| ∧ |f (x)| > 0. (6) =⇒ (2): Suppose that 0 < x ∈ G but that f (x)⊥ 6= x⊥ . Since f (x)⊥ ⊇ x⊥ by Proposition 3, we may apply Proposition 2 and conclude that there exists 0 < w ∈ f (x)⊥ ∩x⊥⊥ . Clearly, replacing w by w ∧ x if necessary, we may assume that 0 < w < x. So (6) implies that |f (w)| ∧ |f (x)| > 0. As well, since w ∈ f (x)⊥ , w ⊥ ⊇ f (x)⊥⊥ and hence since f is band preserving, Proposition 3 implies that |f (w)|⊥ = f (w)⊥ ⊇ f (x)⊥⊥ = |f (x)|⊥⊥ . So |f (x)| ∈ |f (w)|⊥ and thus |f (x)| ∧ |f (w)| = 0. But this is a contradiction; so we conclude that f (x)⊥ = x⊥ . Furthermore, certainly 0⊥ = f (0)⊥ . And if 0 6= x ∈ G, then 0 < |x| ∈ G and Proposition 4, in conjunction with the above argument, implies that f (x)⊥ = f (|x|)⊥ = |x|⊥ = x⊥ . So (2) holds. (2) =⇒ (7): If x⊥ ⊆ y ⊥ , then by (2), f (x)⊥ = x⊥ ⊆ y ⊥ = f (y)⊥ . (7) =⇒ (2): If f (x)⊥ 6= x⊥ , then since f (x)⊥ ⊇ x⊥ by Proposition 3, we may apply Proposition 2 and conclude that there exists 0 < w ∈ f (x)⊥ ∩ x⊥⊥ . Then x⊥ = x⊥⊥⊥ ⊆ w ⊥ , and hence by (7), f (x)⊥ ⊆ f (w)⊥ . So w ∈ f (x)⊥ ⊆ f (w)⊥ and thus w ∧ |f (w)| = 0. Then since f is band preserving, |f (w)| = |f (w)| ∧ |f (w)| = 0, thus f (w) = 0, and hence, since f is one-to-one, w = 0, a contradiction. So f (x)⊥ = x⊥ and this (2) holds. (7) ⇐⇒ (8): Only (8) =⇒ (7) requires explanation. So suppose that (8) holds and that ⊥ x ⊆ y ⊥ . If y = 0, then f (x)⊥ ⊇ G = f (y)⊥ . If x = 0, then, since G = x⊥ ⊆ y ⊥ , y ⊥ = G and hence y = 0; so again f (x)⊥ ⊇ G = f (y)⊥ . Otherwise, 0 < |x|, |y| ∈ G and (7), together with Proposition 4, implies that f (x)⊥ = f (|x|)⊥ ⊆ f (|y|)⊥ = f (y)⊥ .

3 Local Components Our object is to determine some interesting classes of ℓ-groups for which inverses of one-to-one band preserving operators are also band preserving. For the first case, we say that an ℓ-group G is has local components provided that for all x, u ∈ G, if x 6∈ u⊥⊥ , then there exists 0 6= x1 ∈ u⊥ such that x ∈ x1 ⊥ + x1 ⊥⊥ . (Note that vector lattices with this property are said to have “sufficiently many components” [5]). For these ℓ-groups, we have the following. Theorem 2 Suppose that G is an ℓ-group with local components and that f : G → G is a one-to-one group homomorphism. If f is band preserving, then so is f −1 . Proof We will apply condition (3) of Theorem 1. To this end, suppose that 0 < x ∈ G and that f (x)⊥ 6= x⊥ . By Proposition 3, f (x)⊥ ) x⊥ , and hence by Proposition 2, there exists 0 < w ∈ f (x)⊥ ∩ x⊥⊥ . Then w ∧ |f (x)| = 0. As well, since x 6∈ w ⊥⊥ and G has local components, x = a + b for 0 ≤ a ∈ w ⊥⊥ and 0 ≤ b ∈ w ⊥ . Note that if a = 0, then x ∧ w = b ∧ w = 0, thus w ∈ x⊥ ∩ x⊥⊥ and thus w = 0, a contradiction; so a > 0. Since

40

G. Buskes, R. H. Redfield

f preserves addition, f (x) = f (a) + f (b), and since f is band preserving and a ∧ b = 0, |f (a)| ∧ |f (b)| = 0. Then by the proof of Proposition 1, f (x)⊥ = |f (x)|⊥ = |f (a) + f (b)|⊥ = (f (a)+ ∨ f (a)− ∨ f (b)+ ∨ f (b)− )⊥ ⊆ (f (a)+ ∨ f (a)− )⊥ = |f (a)|⊥ = f (a)⊥ . Now suppose that 0 < z ∈ w ⊥ . Then w ∧ z = 0, and since 0 < a ∈ w ⊥⊥ , a ∧ z = 0. So since f is band preserving, |f (a)| ∧ z = 0, and it follows that f (a) ∈ w ⊥⊥ . But since w ∧ |f (x)| = 0, f (x) ∈ w ⊥ so that |f (a)| ∧ |f (x)| = 0. But then f (a) ∈ f (x)⊥ ⊆ f (a)⊥ and hence f (a) = 0. So since f is one-to-one, a = 0, a contradiction of our observation that we must have a > 0. So f (x)⊥ = x⊥ and hence by Theorem 1, f −1 is band preserving. We next list some cases in which Theorem 2 applies. Recall [11, p. 93] that an ℓ-group G is strongly projectable if for all X ⊆ G, G = X ⊥ + X ⊥⊥ and is projectable if for all x ∈ G, G = x⊥ + x⊥⊥ . (Note [17, p. 136] that a strongly projectable vector lattice is said to “have the projection property” and a projectable vector lattice is said to “have the principle projection property.”) Borrowing terminology from the theory of vector lattices, we say that G has sufficiently many projections if for all subsets X of G such that X ⊥ 6= {0}, there exists 0 6= x ∈ X ⊥ such that G = x⊥ + x⊥⊥ . Corollary 1 Suppose that G is an ℓ-group and that f : G → G is a one-to-one band preserving group homomorphism. 1. If G is strongly projectable, then f −1 is band preserving. 2. If G is projectable, then f −1 is band preserving. 3. If G has sufficiently many projections, then f −1 is band preserving. Proof If G is strongly projectable, then if it projectable and if it is projectable, then it has sufficiently many projections. So suppose that G has sufficiently many projections and that x, u ∈ G are such that x 6∈ u⊥⊥ . Since u⊥⊥ 6= G, u⊥ 6= {0} and hence since G has sufficiently many projections, there exists 0 6= x1 ∈ u⊥ such that G = x1 ⊥ + x1 ⊥⊥ . Then x ∈ x1 ⊥ + x1 ⊥⊥ and thus G has local components. So by Theorem 2, f −1 is band preserving.

4 Minimal Prime Subgroups In this section, we characterize band preserving, one-to-one operators with band preserving inverses in terms of the ℓ-group’s minimal prime subgroups. Recall first that a prime subgroup of G is a subgroup that is maximal with respect to not containing some element g ∈ G, and by [11, Theorem 14.9], the following statements are equivalent: (i) P is a minimal prime subgroup of G; S (ii) P = {w ⊥ w ∈ / P };

(iii) for all y ∈ P , y ⊥ 6⊆ P .

Band preserving operators on lattice-ordered groups

41

Note that if P is a normal prime subgroup of G, then with respect to the usual quotient order [11, p. 44], G/P is a totally ordered group [11, Theorem 8.4]. With this definition in mind, we have the following results. Proposition 5 Suppose that G is an ℓ-group and that f : G → G is a band preserving homomorphism. If P is a minimal prime subgroup of G, then f (P ) ⊆ P . Proof Suppose that P is a minimal prime and that x ∈ P . If f (x) 6∈ P , then f (x)⊥ ⊆ P by (ii) above. But by Proposition 3 (4), x⊥ ⊆ f (x)⊥ , and thus we have x ∈ P and x⊥ ⊆ P . Then by (iii) above, P cannot be a minimal prime. This a contradiction and hence f (x) ∈ P , i.e., f (P ) ⊆ P . Note that, in the case of Archimedean vector lattices, the condition f (P ) ⊆ P actually forces an order bounded operator to be band preserving (see Theorem 4.2 in [15]). Proposition 6 Suppose that G is an ℓ-group and that f : G → G is a one-to-one band preserving homomorphism. If f −1 is band preserving, then f −1 (P ) ⊆ P for all minimal primes P. Proof Suppose that f −1 is band preserving, that P is a minimal prime, and that x ∈ f −1 (P ). Then f (x) ∈ P and hence by (ii) above, f (x)⊥ 6⊆ P . But by Theorem 1 (3), since f −1 is band preserving, x⊥ = f (x)⊥ and thus x⊥ 6⊆ P . So, by (ii) above, x ∈ P . Proposition 7 Suppose that G is an ℓ-group and that x ∈ G. Then \ x⊥ = {P | P is a minimal prime and x 6∈ P }.

T Proof Let W = {P | P is a minimal prime and x 6∈ P } and note that by (ii) above, x⊥ ⊆ W . Now suppose that y 6∈ x⊥ . Then |y| ∧ |x| > 0 and there exists a minimal prime subgroup P such that |y| ∧ |x| 6∈ P . Then x 6∈ P and hence W ⊆ P . But as well, y 6∈ P and thus y 6∈ W . So x⊥ ⊇ W . We now arrive at the main result of this section. Proposition 8 Suppose that G is an ℓ-group and that f : G → G is a one-to-one band preserving homomorphism. Then f −1 is band preserving if and only if f −1 (P ) ⊆ P for all minimal primes P . Proof By Proposition 6, it suffices to suppose that for all minimal primes P , f −1 (P ) ⊆ P and to show that f −1 is band preserving. But with this assumption, for all minimal primes P , x 6∈ P if and only if f (x) 6∈ P . So Proposition 7 implies that \ x⊥ = {P | P is a minimal prime and x 6∈ P } \ = {P | P is a minimal prime and f (x) 6∈ P } = f (x)⊥ ,

and hence by Theorem 1, f −1 is band preserving.

42

G. Buskes, R. H. Redfield

5 Examples We devote this section to examples that illustrate our previous work. These examples show that without some sort of restrictions, band preserving group homomorphisms may behave pathologically. In particular, we construct (1) a bijective band preserving operator on a non-Archimedean vector lattice that preserves disjointness and is order bounded but does not preserve magnitude (we alluded to this example at the end of the introduction), (2) a one-to-one band preserving operator on a non-Archimedean vector lattice whose inverse is not band preserving, and (3) a one-to-one band preserving operator whose domain and range are not isomorphic. To construct these examples we will make use of the lexicographic order defined as follows. ← P − If T1 , T2 , . . . are totally ordered groups (vector spaces over R), then Ti denotes the lexico← P − Q graphic sum of the Ti . The elements of Ti are the elements of the product Ti with finite support, addition (and scalar multiplication) are defined coordinatewise, and the order is defined ← P − as follows. For 0 6= a ∈ Ti , let µ[a] denote the largest index i such that ai 6= 0 and define the ← P − order by letting: a > 0 ⇐⇒ a 6= 0 and aµ[a] > 0. It is easy to check that with this order, Ti is a totally ordered group (real vector space). For a finite set of structures, T1 , . . . , Tn , we use ← − ← − the notation: T1 × · · · × Tn . ← − ← − Example 1 Let V be the vector lattice V = R × R × R, and define f : V → V by letting f (a, b, c) = (b, a, c). Certainly f is an automorphism of the underlying vector space, and since V is totally ordered, f must be disjointness preserving. As well, it is clear that f ([(a, b, c), (k, m, n)]) ⊆ [(0, 0, min{c, 0} − 1), (0, 0, max{n, 0} + 1] and hence that f is order bounded. However, |(2, 0, 0)| < |(0, 1, 0)| but |(f (2, 0, 0)| = |(0, 2, 0)| > |(1, 0, 0)| = |f (0, 1, 0)|. ← P − Example 2 Let T be the vector lattice T = ∞ 1 and let W be the vector lattice W = T × T ← − with the coordinatewise order, and let V = W × T . Define f : V → V by letting f (α, β, γ) = (f1 (α, β, γ), f2(α, β, γ), f3(α, β, γ)), where f1 (α, β, γ) = α, f3 (α, β, γ)n = γn+1, and  γ1 if n = 1 f2 (α, β, γ)n = βn − 1 if n > 1. It is easy to see that f is  a bijective band preserving vector lattice homomorphism. But consider 1 if n = 1 τ ∈ T defined by τn = and P ⊆ V defined by P = {(0, β, 0) | β ∈ T }. Then it 0 if n > 1 is easy to see that P is a minimal prime of V such that (0, 0, τ ) ∈ f −1 (P ) but (0, 0, τ ) 6∈ P . So by Proposition 8, f −1 is not band preserving. ← P − Example 3 If {Ai }i∈I is a collection of totally ordered groups, then i∈I Ai is the set of all elements in the product which have finite support with lexicographic total order: v > 0 if and only if vm > 0, where m is the largest index such that vm 6= 0. It is easy to see [6, p. 6] ← P − ← − that i∈I Ai is a totally ordered group. If I = {1, 2}, we use the notation A1 × A2 . Now let ← P − ← − ← − T1 = Z × R; for i ≥ 2, let Ti = R × Z; and let T = ∞ i=1 Ti . Note that if v ∈ T , then for each

Band preserving operators on lattice-ordered groups

43

index i, v has coordinate vi = (vi,1 , vi,2 ). Then S = {v ∈ T | v1 = (0, 0)} is an o-subgroup of T . As well, for v ∈ T , we may define v ∈ T by letting  if i = 1  (0, 0) (v1 , 2, v1 , 1) if i = 2 vi =  (vi − 1, 1, vi − 1, 2) if i ≥ 2

and then define f : T → T by letting f (v) group homomorphism, and since its domain  T  (−1, 1) if i = 1 If ui = , then ui = (0, 0) if i 6= 1

= v. It is easy to see that f is a one-to-one is totally ordered, f must be band preserving. (1, −1) if i = 2 , and hence while u > 0, (0, 0) if i 6= 2  (1, 0) if i = 1 f (u) < 0; so f does not preserve order. Furthermore, if wi = , then (0, 0) if i 6= 1 0 < w ∈ T and the closed interval [0, w] consists of just the two elements 0 and w, while for any 0 < s ∈ S, the closed interval [0, s] has infinitely many elements. It follows that T and S cannot be order-isomorphic and hence, since f (T ) = S, that T and f (T ) cannot be lattice-isomorphic.

Note that if in Example 1 we considered the ℓ-group G = T ×R (with coordinatewise order) in place of T , then G would not be totally ordered and the function L : G → G, defined by letting L(v, r) = (v, r), would be a one-to-one band preserving group homomorphism such that G is not isomorphic to L(G). Acknowledgment. The first author gratefully acknowledges support from the NWO bezoekersbeurs B 61-548 while visiting the University of Delft in the winter of 2004.

References [1] Abramovich, Y.A., Multiplicative representation of disjointness preserving operators, Nederl. Akad. Wetensch. Indag. Math. 45 (1983), no. 3, 265–279. (N.S.) 3 (1992), no. 2, 179– 183. [2] Abramovich, Y. A.; Kitover, A. K., A solution to a problem on invertible disjointness preserving operators, Proc. Amer. Math. Soc. 126 (1998), no. 5, 1501–1505. [3] Abramovich, Y. A.; Kitover, A. K., Inverses of disjointness preserving operators, Mem. Amer. Math. Soc. 143,(2000), no. 679. [4] Y. A. Abramovich & A. K. Kitover, Inverses and regularities of band preserving operators, Indag. Math. (N. S.) 13 (2002), 143-167. [5] Y. A. Abramovich & A. K. Kitover, A characterization of operators preserving disjointness in terms of their inverses, Positivity and its Applications (Ankara, 1998), Positivity 4 (2000), 205-212. [6] Marlow Anderson & Todd Feil, lattice-ordered Groups: An Introduction, Reidel, Dordrecht, 1988.

44

G. Buskes, R. H. Redfield

[7] Arendt, W., Spectral properties of Lamperti operators, Indiana Univ. Math. J. 32 (1983), no. 2, 199–215. [8] Alain Bigard, Klaus Keimel & Samuel Wolfenstein, Groupes et Anneaux R´eticul´es, Lecture Notes in Mathematics 608, Springer-Verlag, Berlin, 1977. [9] Bigard, Alain; Keimel, Klaus, Sur les endomorphismes conservant les polaires d’un groupe r´eticul´e archim´edien, Bull. Soc. Math. France 97,1969, 381–398. [10] Conrad, P. F.; Diem, J. E., The ring of polar preserving endomorphisms of an abelian lattice-ordered group, Illinois J. Math. 15 1971,222–240. [11] Michael R. Darnel, Theory of lattice-ordered Groups, Marcel Dekker, Inc., New York, 1995. [12] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, 1963. [13] A. M. W. Glass, Partially Ordered Groups, World Scientific, Singapore, 1999. [14] Huijsmans, C. B.; Wickstead, A. W., The inverse of band preserving and disjointness preserving operators, Indag. Math. (N.S.) 3 (1992), no. 2, 179–183. [15] Luxemburg, W.A..J. , Some aspects of the theory of Riesz spaces, The University of Arkansas Lecture Notes, Volume 4, 1979. [16] Luxemburg, W. A. J.; de Pagter, B. Maharam extensions of positive operators and fmodules, Positivity 6 (2002), no. 2, 147–190. [17] W.A.J. Luxemburg & A.C. Zaanen, Riesz Spaces I, North-Holland, Amsterdam-London, 1971.

G. Buskes Department of Mathematics University of Mississippi MS-38677, USA R. H. Redfield Department of Mathematics 198 College Hill Road, Clinton NY 13323, USA

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 45-51 (2006)

On the order continuity of the regular norm Z. L. Chen (Chengdu Sichuan) Abstract. We present here some sufficient conditions for the regular norm on Lr (E, F ) being order continuous. Particularly we deduce a characterization of the order continuity of the regular norm using L- and M-weak compactness of regular operators. Some related results are also obtained. Key words: Banach lattice, regular norm, order continuous norm, regular operator MSC 2000: 46B42, 47B65

1

Introduction

For Banach lattices E and F , we use L(E, F ) to denote the space of all continuous linear operators from E into F , and Lr (E, F ) denotes the space of all regular operators from E into F , which is the linear span of the set L+ (E, F ) of all positive operators from E into F . With respect to the operator norm k · k the space Lr (E, F ) is not complete (see for example [1]), but there exists a natural norm on Lr (E, F ), the regular norm k · kr , which turns Lr (E, F ) into a Banach space (see [6] for details). Namely, kT kr = inf { kSk : S ∈ L+ (E, F ), ±T ≤ S }, and, in particular, kT k ≤ kT kr . If Lr (E, F ) is a vector lattice then (Lr (E, F ), k·kr ) is a Banach lattice and kT kr = k|T |k for all T ∈ Lr (E, F ). For instance if F is Dedekind complete, then Lr (E, F ) is a Dedekind complete Banach lattice under the regular norm. The natural and important questions are: If Lr (E, F ) is a vector lattice (i.e. a Banach lattice), when is the regular norm k · kr on Lr (E, F ) order continuous? The purpose of this work is to present some results involving the order continuity of the regular norm on Lr (E, F ). Some related results are included as well. Recall that an operator T : E → F is called L-weakly compact if T ball(E) is an L-weakly compact set in F , i.e. kyn k → 0 for each disjoint sequence (yn )∞ 1 contained in the solid hull of T ball(E). Also T is called M-weakly compact if kT xn k → 0 for each disjoint sequence (xn )∞ 1 ⊂ ball(E), where ball(E) denotes the unit ball of E. See for example [6]. We prefer to [2] and [6] for any unexplained terms from the theory of Banach lattices and operators.

2

Some general results

We start with a necessary condition for the order continuity of the regular norm on spaces of regular operators.

46

Z. L. Chen

Proposition 2.1. Let E and F be Banach lattices. If the regular norm k · kr on Lr (E, F ) is order continuous, then the norms both on E 0 and F are order continuous norm. Proof. If the regular norm k · kr on Lr (E, F ) is order continuous, for each increasing sequence 0 0 0 (yn )∞ 1 ⊂ [0, y] ⊂ F , taking x ∈ E+ with kx k = 1 and define S, Sn : E → F by Sn x = x0 (x)yn ,

Sx = x0 (x)y

for

x∈E

then Sn , S ∈ L+ (E, F ) and 0 ≤ Sn ↑ ≤ S. The order continuity of the regular norm implies that there is an U ∈ Lr (E, F ) such that kSn − U kr → 0, thus kSn − U k → 0. Choosing x0 ∈ E with x0 (x0 ) = 1 then we have kyn − U x0 k = kSn x0 − U x0 k ≤ kSn − U kkx0 k → 0 It follows from Theorem 2.4.2 of [6] that the norm on F is order continuous. 0 0 Similarly, for each increasing sequence (x0n )∞ 1 ⊂ [0, x ] ⊂ E , taking y ∈ F+ with kyk = 1 and define T, Tn : E → F by Tn x = x0n (x)y,

T x = x0 (x)y

for

x∈E

then Tn , T ∈ L+ (E, F ) and 0 ≤ Tn ↑ ≤ T . Again there is an V ∈ Lr (E, F ) such that kTn − V kr → 0, thus kTn − V k → 0. Choosing y 0 ∈ F 0 with y 0 (y) = 1, then it is easy to verify that kx0n − V 0 y 0 k = kTn0 y 0 − V 0 y 0 k ≤ kTn0 − V 0 kky 0 k = kTn − V kky 0 k → 0. Theorem 2.4.2 of [6] yields that the norm on E 0 is order continuous. Next result is a characterization of the order continuity of the regular norm on spaces of regular operators. Theorem 2.2. For Banach lattices E and F , the following statements are equivalent. (1) Lr (E, F ) is a vector lattice and the regular norm k · kr on Lr (E, F ) is order continuous. (2) Every positive operator T : E → F is L- and M-weakly compact. Proof. (1) ⇒ (2). For 0 ≤ T : E → F , by Proposition 2.1 it suffices to show that T is Mweakly compact (see Theorem 3.6.17 of [6]). Otherwise, there is a disjoint sequence (xn )∞ 1 ⊂ ball(E) such that kT xn k ≥ δ > 0 for all n ∈ N. Note that 0 ≤ T |xn | → 0 weakly as |xn | → 0 weakly (see Theorem 2.4.14 of [6]). By Corollary 2.3.5 of [6] there exist a sequence (kn ) and a disjoint sequence (yn ) ⊂ F+ such that 0 ≤ yn ≤ T |xkn | and kyn k ≥ c for n ∈ N, where c ∈ (0, δ) is any fixed number from (0, δ). Let Pn : F → {yn }dd be the band projection, hereby {yn }dd denotes the band generated by yn in F . It is easy to verify that Pi ⊥ Pj and Pi ≤ IF − Pj (∀ i 6= j), it follows that P1 + · · · + Pn ↑ ≤ IF , and (P1 + · · · + Pn )T ↑ ≤ T , where IF is the identity operator on F . Now the order continuity of the regular norm implies that ((P1 + · · · + Pn )T )∞ 1 is a k · kr -Cauchy sequence, in particular, kPn T k → 0 as n → ∞. Therefore 0 < c ≤ kyn k = kPn yn k ≤ kPn T |xn |k ≤ kPn T k → 0 this is impossible, so (1) ⇒ (2) holds.

On the order continuity of the regular norm

47

(2) ⇒ (1). For any 0 < y ∈ F+ and 0 < x0 ∈ E+0 , let T : E → F by T x = x0 (x)y. Clearly T ≥ 0 and the L- and M-weak compactness of T yield the relatively weak compactness of both [−y, y] and [−x0 , x0 ]. It follows from Theorem 2.4.2 of [6] that the norms both on E 0 and F are order continuous, Lr (E, F ) is certainly a (Dedekind complete) vector lattice. For any decreasing sequence Tn ∈ L+ (E, F ) with inf{Tn : n ∈ N} = 0, Proposition 3.6.19 of [6] yields that the operator norm, and hence the regular norm, on order interval [0, T1 ] is order continuous, which implies that kTn kr = kTn k → 0. Now the order continuity of the regular norm is following from Theorem 2.4.2 of [6] . It is clear that the identity operator on a Banach lattice E is M-weakly compact if and only if E is finite dimensional. Next result should be not surprise. Corollary 2.3. Let E be a Banach lattice. Then Lr (E) is a vector lattice and the regular norm k · kr on Lr (E) is order continuous if and only if dimE < ∞. It is obvious that if T : E → F is regular then T 0 is also regular, and the converse is false in general. The next results will show that the order continuity of the regular norms on Lr (E, F ), Lr (E, F 00 ) and Lr (F 0 , E 0 ) possess some relations. Theorem 2.4. For Banach lattices E and F , the following assertions are equivalent. (1) The regular norm k · kr on Lr (E, F 00 ) is order continuous. (2) The regular norm k · kr on Lr (F 0 , E 0 ) is order continuous. Proof. Let Φ : L(E, F 00 ) → L(F 0 , E 0 ) by Φ(T ) = T 0 j for T ∈ L(E, F 00 ), where j : F 0 → F 000 the natural embedding. According to Theorem 5.6 of [3] a linear bounded operator T : E → F 00 is regular if and only if Φ(T ) is regular and kΦ(T )kr = kT kr . Moreover Φ is an order continuous isometric lattice isomorphism from (Lr (E, F 00 ), k · kr ) onto (Lr (F 0 , E 0 ), k · kr ). Thus (1) ⇔ (2) is a simple consequence of these facts. Corollary 2.5. Let E and F be Banach lattices such that F is reflexive. Then the following statements are equivalent. (1) The regular norm k · kr on Lr (E, F ) is order continuous. (2) The regular norm k · kr on Lr (F 0 , E 0 ) is order continuous. Theorem 2.6. Assume that E and F are Banach lattices, H ⊂ E and G ⊂ F are closed sublattices. If there is a positive projection P from E onto H, Lr (E, F ) is a vector lattice and the regular norm k · kr on Lr (E, F ) is order continuous, then Lr (H, G) also is a vector lattice and k · kr on Lr (H, G) is order continuous. Proof. Suppose that Lr (E, F ) is a vector lattice and the regular norm k · kr on Lr (E, F ) is order continuous. For 0 ≤ T : H → G, then 0 ≤ T P : E → G ⊂ F , Theorem 2.2 yields that T P is L- and M-weakly compact. For any disjoint sequence (yn )∞ 1 contained in the solid hull is a disjoint sequence in F as G is a sublattice of F , which is of T ball(H) in G, then (yn )∞ 1 contained in the solid hull of (T P )ball(E) as T ball(H) ⊂ (T P )ball(E), so that kyn k → 0, i.e. ∞ T is L-weakly compact. Also for each disjoint sequence (xn )∞ 1 ⊂ ball(H), (xn )1 ⊂ ball(E) is disjoint as H is a sublattice of E, it follows that kT xn k = kT P xn k → 0, which implies that T

48

Z. L. Chen

is M-weakly compact. Again by Theorem 2.2 Lr (H, G) is a vector lattice and the regular norm k · kr on Lr (H, G) is order continuous. Note that each Banach lattice F can be identified with a closed sublattice of F 00 , and so, as a consequence of Theorems 2.4 and 2.6 we have the following result. Corollary 2.7. Let E and F be Banach lattices. If the regular norm k · kr on Lr (F 0 , E 0 ) is order continuous, then so is the regular norm on Lr (E, F ). Remark The regular norm k · kr on Lr (F 0 , E 0 ) may fail to be order continuous even if the regular norm on Lr (E, F ) is order continuous. ¡ ¢0 For example, let E = c0 and F = `1 (`n∞ ) = c0 (`n1 ) . Clearly F is a KB-space and F 0 = `∞ (`n1 ). Define T : `1 → F 0 by ¡ ¢ T (λn ) = (λ1 ), (λ1 , λ2 ), (λ1 , λ2 , λ3 ), · · · , ∀ (λn ) ∈ `1 , it is easy to see that T is an isometric lattice homomorphism into, i.e. F 0 contains a closed sublattice isometrically lattice isomorphic to `1 . Thus Theorem 2.4.14 of [6] implies that the norm on F 00 is not order continuous. Therefore, the regular norm k · kr on Lr (E, F 00 ), and hence on Lr (F 0 , E 0 ), is not order continuous as the norm on F 00 is not order continuous (see the proof of Proposition 2.1). However, it follows from Theorem 3.2 (see below) that the regular norm on Lr (E, F ) is order continuous.

3

Some concrete sufficient conditions

In this section we will present some sufficient conditions on Banach lattices E and F such that the regular norm k · kr on Lr (E, F ) is order continuous. Proposition 3.1. Let E be an AM-space with a strong order unit and F a Banach lattice with an order continuous norm. Then the regular norm k · kr on Lr (E, F ) is order continuous. Proof. We may assume that E is equipped with the strong order unit norm and also the norm on F is order continuous, clearly (Lr (E, F ), k · kr ) is a Banach lattice. For 0 ≤ Tn ↑ ≤ T in Lr (E, F ) then 0 ≤ Tn x ↑ ≤ T x for each x ∈ E+ . It follows from the order continuity of the norm on F that (Tn x)∞ 1 is norm convergent. So there is a S ∈ L+ (E, F ) such that Tn → S with respect to the strong operator topology and obviously Tn ↑ S. In particular kS − Tn kr = kS − Tn k = kSe − Tn ek → 0 where e is a strong order unit of E. Therefore Theorem 2.4.2 of [6] yields that the regular norm k · kr on Lr (E, F ) is order continuous. Remark If E fails to possess a strong order unit the above result is false even if E is an AMspace, E, E 0 and F are atomic with an order continuous norm. For example, let E = F = c0 and then the regular norm k · kr on Lr (E) is not order continuous, compare also Corollary 2.3. Recall that Banach lattice E possesses the positive Schur property if every weakly null sequence in E+ is norm convergent to 0.

On the order continuity of the regular norm

49

Theorem 3.2. Let E be a Banach lattice such that E 0 possesses the positive Schur property, F a KB-space. Then the regular norm k · kr on Lr (E, F ) is order continuous. Proof. By Theorem 2.2 and Theorem 3.6.17 of [6] it suffices to show that each positive operator T : E → F is M-weakly compact. Indeed, if T is not M-weakly compact then there is a disjoint sequence (xn )∞ 1 ⊂ ball(E) with xn ≥ 0 and kT xn k ≥ 2δ > 0 for n ∈ N. Note that T xn → 0 weakly as xn → 0 weakly (see Theorem 2.4.14 of [6]), by Proposition 2.3.4 of [6] there exists 0 a disjoint sequence (yn0 )∞ 1 ⊂ ball(F ), yn ≥ 0, satisfying (T 0 yn0 )(xn ) = yn0 (T xn ) > δ,

for all n.

Also by Theorem 2.5.6 and 3.4.18 of [6], T is weakly compact and so is T 0 by Gantmacher’s theorem (see Theorem 17.2 of [2]), so we may assume that T 0 yn0 is weakly convergent (replacing by a subsequence if necessary), say T 0 yn0 → x0 weakly, then for each x ∈ E x0 (x) = lim T 0 yn0 (x) = lim yn0 (T x) = 0 n→∞

n→∞

as yn0 → 0 in σ(F 0 , F ) (see Corollary 2.4.3 of [6]), i.e. T 0 yn0 → 0 weakly, so the positive Schur property of E 0 implies that kT 0 yn0 k → 0 and it follows that 0 < δ < yn0 (T xn ) = (T 0 yn0 )(xn ) ≤ kT 0 yn0 k → 0. This is impossible, thus T is M-weakly compact. The following result is a dual version of Theorem 3.2 Theorem 3.3. Let F be a Banach lattice with the positive Schur property, E a Banach lattice. Then the regular norm k · kr on Lr (E, F ) is order continuous if and only if E 0 has an order continuous norm. Proof. The part of “only if” easily follows from the proof of Theorem 2.2. If the norm on E 0 is order continuous, for T ∈ L+ (E, F ) and each disjoint sequence (xn )∞ 1 ⊂ ball(E), then |xn | → 0 weakly, and T |xn | → 0 weakly. It follows from the positive Schur property of F that kT xn k → 0 as |T xn | ≤ T |xn |, i.e. T is M-weakly compact, Theorem 2.2 and Theorem 3.6.17 of [6] yield that k · kr on Lr (E, F ) is order continuous. For a Banach lattice E and 1 ≤ p ≤ ∞, recall that E has the strong `p -decomposition property if there exists a constant M such that for all disjoint elements x1 , · · · , xn in E we have P P P 1/p ( ni=1 kxi kp ) ≤ M k ni=1 xi k for p < ∞ and max{kxi k : i = 1, · · · , n} ≤ M k ni=1 xi k in case p = ∞. The number σ(E) = inf{p ≥ 1 : E has the strong `p -decomposition property } is call the upper index of E. Similarly E has the strong `p -composition property if there exists a constant M such that P P 1/p for allP disjoint elements x1 , · · · , xn in E we have k ni=1 xi k ≤ M ( ni=1 kxi kp ) for p < ∞ and k ni=1 xi k ≤ M max{kxi k : i = 1, · · · , n} in case p = ∞. The number s(E) = sup{p ≥ 1 : E has the strong `p -composition property } is called lower index of E. It is known that 1 ≤ s(E) ≤ σ(E) ≤ ∞ for any Banach lattice E. If σ(E) < ∞ then E has an order continuous norm. If s(E) > 1 then the norm on E 0 is order continuous. See [5] for details. Also recall that if the norm on a Banach lattice E is p-superadditive then σ(E) ≤ p; and if E has a p-subadditive norm then s(E) ≥ p, see Proposition 2.8.2 of [6] and [5].

50

Z. L. Chen

Theorem 3.4. Let E and F be Banach lattices. If s(E) > σ(F ) then the regular norm k · kr on Lr (E, F ) is order continuous. Proof. The norm on E 0 clearly is order continuous. Note that if σ(F ) < ∞ then F is a KBspace. Indeed, if F is not a KB-space, then F contains a sublattice H lattice isomorphic to c0 , which implies that σ(F ) = ∞ as σ(c0 ) = ∞. Now the rest is a simple consequence of Theorem 2.2, Theorem 6.7 of [5] and Theorem 3.6.17 of [6]. Corollary 3.5. Let E and F be Banach lattices. If the norm of E is p-subadditive, the norm of F is q-superadditive and 1 ≤ q < p ≤ ∞, then the regular norm k · kr on Lr (E, F ) is order continuous. Remark. It is worth to point out that s(E) > σ(F ) fails to be true in general even if Lr (E, F ) is a KB-space, see [4, example 3.12]. For E and F being Lp - and Lq -spaces respectively we have the following characterization. Theorem 3.6. Let E and F be infinite dimensional Lp -space and Lq -space respectively, then the regular norm k · kr on Lr (E, F ) is order continuous if and only if q < p. Proof. The part of “if” is a simple consequence of Corollary 3.5. To see the part of “only if”, we may first assume that ½ ½ `q if q < ∞ `p if p < ∞ and Gq = Hp = c0 if q = ∞ c0 if p = ∞ are sublattices of E and F respectively. Suppose that p ≤ q then Hp ⊂ Gq . If p < ∞ there is a positive projection P from E onto Hp (the existence of P is following from Theorem 2.7.11 of [6]), then P : E → Hp ⊂ Gq ⊂ F is not M-weakly compact, which, by Theorem 2.2, implies that k · kr on Lr (E, F ) is not order continuous. Also if p = ∞, then q = ∞ and the norm on F is not order continuous, the proof of Theorem 2.2 again yields that k · kr on Lr (E, F ) is not order continuous. Acknowledgments. I would like to thank Professors A. W. Wickstead of and M. R. Weber for useful discussions about the subjects this paper.

References [1] Abramovich, Y. A., When each continuous operator is regular, Funct. Analysis, Optimization and Math. Economics., Oxford Univ. Press, 1990, 133–140. [2] Aliprantis, C.D. and Burkinshaw, O., Positive operators, Academic Press, New York & London, 1985. [3] Chen, Z. L. and A. W. Wickstead, Some applications of Rademacher sequences in Banach lattices, Positivity, 2 (1998), 171–191. [4] Chen, Z. L. and A. W. Wickstead, The order properties of r-compact operators on Banach lattices, submitted.

On the order continuity of the regular norm

51

[5] Dodds, P.G. and Fremlin, D.H., Compact operators on Banach lattices, Israel J. Math., 34 (1979), 287–320. [6] Meyer-Nieberg, P., Banach lattices, Springer-Verlag Berlin Heidelberg New York, 1991. [7] Wickstead, A. W:, AL-spaces and AM-spaces of operators, Positivity, 4 (2000), 303–311.

Z. L. Chen Department of Mathematics Southwest Jiaotong University Chengdu 610031, Peoples Republic of China

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 53-61 (2006)

Positive Asymptotically Regular Operators in L1-spaces and KB-spaces E. Yu. Emel’yanov (Ankara) Abstract. We discuss several recent results on asymptotic periodicity and mean ergodicity of positive operators. Key words: L1 -space, non-commutative L1 -space, KB-space, positive operator, asymptotically periodic operator, mean ergodic operator MSC 2000: 46B42, 47A35, 47C15, 47D07

1 Introduction Lasota, Li and Yorke [13] introduced the notion of constrictiveness for Markov operators in L1 -spaces in order to characterize their asymptotic periodicity. Let L1 (Ω, Σ, µ) be an L1 -space; denote by D = D(Ω, Σ, µ) the set of all densities, i.e., non-negative almost everywhere normone functions in the L1 -space. A linear operator T : L1 → L1 is called a Markov operator if T (D) ⊆ D. Lasota, Li and Yorke called a Markov operator T constrictive, whenever there exists a compact set K ⊆ D satisfying lim dist(T n f , K) = 0 for all f ∈ D. They proved in n→∞ [13] the following remarkable result. Theorem 1 (Lasota – Li – Yorke). Let T be a constrictive Markov operator in an L1 -space. Then there is a Markov operator R of a finite rank such that lim k(T n − Rn ) xk = 0 for all n→∞

x ∈ L1 . In the other words, to describe the asymptotic behavior of T , it is enough to investigate the asymptotic behavior of R, which is given by an appropriate finite-rank matrix. An operator T , for which there exists an operator R of finite rank, satisfying lim k(T n − Rn ) xk = 0

n→∞

for all x, is called asymptotically periodic. Since every asymptotically periodic Markov operator is constrictive, these two notions coincide for Markov operators. In a short time after the paper [13], several significant generalizations of Theorem 1 was obtained by many authors. First of all, the definition of constrictiveness was extended for an arbitrary operator in an arbitrary Banach space. A subset A ⊆ X of a Banach space X is called a constrictor for an operator T in X if lim dist(T n x, A) = 0 (∀x ∈ X kxk ≤ 1). n→∞

An operator T is called constrictive if there is a compact constrictor for T . Due to the uniform boundedness principle, any constrictive operator is power bounded. The notion of constrictor

54

E. Yu. Emel’yanov

depends on the choice of a norm in X. We denote the set of all constrictors for T by Conk·k (T ) or, simple, by Con(T ), if the norm in X is fixed. Remark that the original definition of a constrictive operator of Lasota, Li, and Yorke coincides with the definition above for a Markov operator in an L1 -space. The following theorem, which is an extension of the Lasota – Li – Yorke theorem for the general case of operators in Banach spaces, was obtained by Ph`ong [17] and Sine [23]. Theorem 2 (Ph`ong – Sine). Given an operator T in a Banach space X, the following assertions are equivalent: (i) there exists a compact K ∈ Con(T ) ; (ii) there exists a T -reducing decomposition X = X0 (T ) ⊕ Xr (T ), where X0 (T ) = {x ∈ X : lim kT n xk = 0} and dim(Xr ) < ∞ . n→∞

Theorem 1 shows that constrictive operators can be asymptotically investigated with methods of the linear algebra. This remark asserts the importance of constrictive operators. Unfortunately, with exception of several rather special cases, it is difficult to check whenever an operator T is constrictive or not. Therefore, the problem arises: to relax conditions on an operator T to be constrictive. This problem will be the central problem in this paper.

2 Asymptotic Regularity of Markov Operators in L1-spaces The first important result concerning the problem above was obtained by Komornik [9] and, in more general form, by Komornik and Lasota [10]. Namely, they proved the following theorem. Theorem 3 (Komornik – Lasota). Let T be a Markov operator in an L1 -space. Assume that there are 0 ≤ y ∈ L1 and 0 ≤ η < 1 such that [−y, y] + η BL1 ∈ Con(T ) ,

(1)

where BL1 denotes the closed unit ball of L1 , then T is asymptotically periodic. The following well-known result of Lasota [12] (see also [14, Thm.5.6.2 and Thm.7.4.1]) is an easy consequence of Theorem 3. Theorem 4 (Lasota). Let T be a Markov operator in an L1 -space. Then the following assertions are equivalent: (i) T is asymptotically stable; (ii) there exists a nontrivial lower-bound function for T . Here, we call a Markov operator T asymptotically stable, whenever there exists (always T -invariant) density u such that lim kT n f − uk = 0

n→∞

(∀f ∈ D) .

A function h ∈ L1+ is called a lower-bound function for a Markov operator T if lim k(h − T n f )+ k = 0 (∀f ∈ D) .

n→∞

Positive Asymptotically Regular Operators in L1-spaces and KB-spaces

55

A lower-bound function h is called nontrivial if h 6= 0. In connection with the Lasota – Komornik theorem and the Lasota theorem, the following two questions were investigated. What would be happened if we replace the powers T n of T in Theorems 3 and 4 , by the n−1 P k Ces`aro averages ATn = n1 T of T ? k=0

What would be generalizations of Theorems 3 and 4 for Markov operators in non-commutative L1 -spaces, i.e. preduals of von Neumann algebras? The first question was answered by Wolff and the author in [6] as follows. Theorem 5 (Emel’yanov – Wolff). Let T be a Markov operator in an L1 -space such that there exist y ∈ L1+ and 0 ≤ η < 1 satisfying

T

lim sup (An f − y)+ ≤ η. n→∞

Then T is mean ergodic i.e., the sequence (ATn )∞ n=1 converges strongly. Moreover, the space Fix(T ) of fixed vectors of T is finite-dimensional. This theorem implies the following result in the similar way, as Theorem 3 implies Theorem 4. Theorem 6 (Emel’yanov – Wolff). Let T be a Markov operator in L1 . Then the following assertions are equivalent: (i) there exists a density u such that lim kATn f − uk = 0

n→∞

(∀f ∈ D);

(ii) there exists a non-trivial mean lower-bound function for T . Here we call h ∈ L1+ a mean lower-bound function for a Markov operator T if

 

lim h − ATn f = 0 (∀f ∈ D). n→∞

+

Obviously, any lower-bound function is a mean lower-bound function. The problem of existence of an invariant density is one of the old and central problems in the theory of Markov operators. This problem is closely related to the problem of finding the conditions under which a Makrov operator is mean ergodic since, obviously, every mean ergodic Markov operator possesses an invariant density. In the paper [4] one of such conditions was found. Theorem 7 (Emel’yanov). Given a Markov operator T in an L1 -space. The following assertions are equivalent: (i) T has an invariant density; (ii) lim sup kf − T n f k < 2 for some density f ; n→∞

(iii) lim sup kd − T n gk < 2 for some pair of densities d, g. n→∞

56

E. Yu. Emel’yanov

In the same paper, by using of Theorem 7, it was obtained the following characterization of mean ergodic Frobenius – Perron operators. Theorem 8 (Emel’yanov). Given a Frobenius – Perron operator P in an L1 -space. The following assertions are equivalent: (i) P is mean ergodic; (ii) there exists a density w such that lim sup kP n f − wk < 2 for every density f . n→∞

Remark that the second condition of Theorem 8 is weaker then the condition (1) in the Komornik – Lasota theorem, even it cannot provide that Fix(P ) is finite-dimensional. But it was quite surprise that it guaranties the mean ergodicity of P . For arbitrary Markov operators, obviously, (i) implies (ii), but they are not equivalent. The following problem is open: is it possible to replace in the second condition of Theorem 8 P n on APn ? The second question has a longer history. In several papers in the beginning of 80-th., Sarymsakov and other authors obtained the generalization of Theorem 4 for Markov operators on non-commutative L1 -spaces. This generalization is, mainly, due to Sarymsakov and Grabarnik, and it was announced without a proof by Ayupov and Sarymsakov [3], and by Sarymsakov and Grabarnik [21]. Theorem 9 (Ayupov – Sarymsakov – Grabarnik). Let M be a von Neumann algebra. Then, for any Markov operator T in M∗ , the following assertions are equivalent: (i) T is asymptotically stable; (ii) there exists a non-trivial lower-bound element for T . The definitions of a Markov operator in M∗ , of asymptotic stability, and of lower-bound element are the same as for L1 -spaces, only we have to replace the set D of densities by the set S(M) of all normal states on M (for details, see [19] or [24]). We send to [24] for the definition of completely positive operator in M∗ , that will be used below. Here we only remark that this class of operators in M∗ is natural and important. Preduals of von Neumann algebras are usually called non-commutative L1 -spaces. The following analogue of Theorem 9 for the Ces`aro convergence was obtained in [7]. Theorem 10 (Emel’yanov – Wolff). Let M be a von Neumann algebra and T be a completely positive Markov operator in M∗ . Then the following assertions are equivalent: (i) there exists a T -invariant normal state u such that lim kATn f − uk = 0

n→∞

for any normal state f ∈ M∗ ; (ii) there exists a non-trivial mean lower-bound element for T . Here we call h ∈ M∗+ a mean lower-bound element for a Markov operator T if lim k(h − ATn f )+ k = 0

n→∞

(∀f ∈ S(M)).

Obviously, any lower-bound element is a mean lower-bound element. Indeed, Theorems 9 and 10 can be easily obtained from the following more general result [7], which is, in some sense, an analogue of Theorem 3 for non-commutative L1 -spaces.

Positive Asymptotically Regular Operators in L1-spaces and KB-spaces

57

Theorem 11 (Emel’yanov – Wolff). Let M be a von Neumann algebra, T a Markov operator in M∗ , g ∈ M∗+ , and η ∈ IR, 0 ≤ η < 1, such that lim sup dist(ATn f, [−g, g]) ≤ η n→∞

for any normal state f ∈ M∗ . Then T is mean ergodic. If, moreover, M is atomic and T is a completely positive operator then the space Fix(T ) of all fixed vectors of T is finite-dimensional.

3 Asymptotic Regularity of Positive Operators in KB-spaces Some results of the previous section possess extensions on KB-spaces, which are, by the definition, Banach lattices in which every increasing norm-bounded sequence has a supremum and converges to it in the norm. It is known (see [1] or [16]) that a Banach lattice E is a KB-space if and only if E is a band in its bidual E ∗∗ . The class of KB-spaces is large enough, it contains L1 -spaces as well as reflexive Banach lattices, for instance, all Lp -spaces for 1 < p < ∞. The principal tool in the proofs of the main results of [10] and [6] (Theorems 3, 5, and 6 above) was using the additivity of the norm on the positive part of the L1 -space. The same idea was used in the non-commutative case in [7] (Theorems 10 and 11 above). Since this is no longer true for a general KB-space, to obtain versions of those theorems, we have to use the more delicious technique. The first important result in this direction was obtained by R¨abiger in [18]. Theorem 12 (R¨abiger). Let T be a positive operator in a KB-space E. Moreover, let C := [−z, z] + η · BE be a constrictor for T , where z ∈ E+ and η ∈ IR satisfies 0 ≤ η < 1 and η · supn≥0 kT n k < 1. Then T is asymptotically periodic. In the end of this section we show that the second condition on η in Theorem 12 can be omitted. In the paper [18], R¨abiger proved the following nice result, which is a principal step in the proof of Theorem 12. Theorem 13 (R¨abiger). Let T be a mean ergodic positive contraction in a Banach lattice E. Moreover, let C := [−z, z] + η · BE be a constrictor for T , where z ∈ E+ and 0 ≤ η < 1. Then there exists y ∈ E+ such that T y = y and [−y, y] ∈ Con(T ). In a short time, this result was extended on any positive operator T in a Banach lattice in [8]. Then in [5], it was shown that it is still true for a positive operator in a so-called strongly normal Banach space, in particular, in non-commutative L1 -spaces. Here we formulate it for Banach lattices only (see [8]). Theorem 14 (Gorokhova – Emel’yanov). Let E be a Banach lattice, and T be a positive operator in E. If T has a constrictor [−y, y] + ηBE for some y ∈ E+ , and for some real η, 0 ≤ η < 1, and if the closure of the convex hull of the orbit {T n y}∞ n=1 contains a T -invariant point w, then the order interval 1 [−w, w] 1−η

58

E. Yu. Emel’yanov

is a constrictor for T . Another principal step in the proof of Theorem 12 is to show that the operator T satisfying the condition of this theorem is mean ergodic. This is true even under a weaker assumption. Namely, the following two results, which generalize Theorem 5 for a positive power bounded operators in KB-spaces, were obtained recently in [2]. Theorem 15 (Alpay – Binhadjah – Emel’yanov – Ercan). Let E be a KB-space, T be a positive power bounded operator in E, W be a weakly compact subset of E, and η ∈ IR, 0 ≤ η < 1 be such that lim dist(ATn x, W + ηBE ) = 0 n→∞

for any x ∈ BE . Then T is mean ergodic. Since order intervals in any KB-space are weakly compact, the theorem is true if we replace a weakly compact subset W of E by an order interval [−g, g] for any g ∈ E+ . In this case, we can say even more (see [2]), that the fixed space Fix(T ) of T is finite-dimensional. Theorem 16 (Alpay – Binhadjah – Emel’yanov – Ercan). Let E be a KB-space, T be a positive power bounded operator on E, g ∈ E+ , and η ∈ IR, 0 ≤ η < 1, be such that lim dist(ATn x, [−g, g] + ηBE ) = 0

(2)

n→∞

for any x ∈ BE . Then T is mean ergodic and Fix(T ) is finite-dimensional. Remark that any mean ergodic positive operator T such that dim Fix(T ) < ∞ satisfies the condition (2) for some g ∈ E+ and η ∈ IR, 0 ≤ η < 1. Moreover, η can be taken arbitrary small. There are examples which show that the condition that E is a KB-space cannot be omitted in Theorem 16. Even for Banach lattice with the order continuous norm, this result can fail [2]. It will be interesting to obtain a non-commutative version of Theorem 16, since for Markov operators on non-commutative L1 -spaces this result is true due to Theorem 11. As an analogue for KB-space it seems to be is reasonable to take a non-commutative Lp -space. Now we prove the following result, which generalizes Theorem 12 for arbitrary positive operators in KB-spaces. Indeed, we use the same technique as in the proof of [18, Thm.5.3], with replacing Theorem 13 on Theorem 14, and involve Theorem 16 to show that T is mean ergodic. Theorem 17. Let T be a positive operator in a KB-space E. Moreover, let C := [−z, z]+η·BE be a constrictor for T , where z ∈ E+ and 0 ≤ η < 1. Then T is asymptotically periodic. Proof: Since the operator T satisfies the condition (2), by Theorem 16, T is mean ergodic. By Theorem 14, T has a constrictor [−y, y], where y ∈ E+ , T y = y. Since any order interval in a Banach lattice with order continuous norm is weakly compact, T is weakly almost periodic. Show that the subspace Er (T ) of reversible vectors of T [11, 2.4.] is finite-dimensional, and T |Er (T ) is periodic. The Jacobs – Deleeuw – Glicksberg projection PT : E → Er (T ) [11, 2.4.] is, obviously, positive. By [22, II.11.5], Er (T ) = PT (E) is a Banach lattice with respect to the order induced by E and a suitable equivalent norm. Since [−y, y] ∈ Con(T ) and PT belongs to the weak operator closure of the set {T n : n ∈ IN}, we have BEr (T ) ⊆ PT (BE ) ⊆

Positive Asymptotically Regular Operators in L1-spaces and KB-spaces

59

[−y, y]. Applying PT again it follows that BEr (T ) is order bounded in Er (T ). Thus Er (T ) is lattice isomorphic to an AM-space with a strong unit, and hence by the Kakutani – Krein representation theorem Er (T ) = C(K) for some compact Hausdorff K. On the other hand, Er (T ) is reflexive (since BEr (T ) is weakly compact) and by Grothendieck’s theorem (cf. [22, II.9.9, Cor.2]) dim(Er (T )) < ∞. Thus T |Er (T ) is a positive doubly power bounded operator in a finite-dimensional Banach lattice Er (T ). It is well known (see, for example, [18, Lm.2.4]) that such an operator has a positive inverse and hence T |Er (T ) is a surjective lattice automorphism in Er (T ) and that any lattice automorphism in a finite-dimensional Banach lattice is periodic. Now, to finish the proof, it is enough to show that Ef l (T ) = {x ∈ X : lim kT n xk = 0} . n→∞

Let F be the band in E generated by y. Then F has order continuous norm and is a rang of a positive projection P : E → F . Moreover, there is a strictly positive y ′ ∈ F ∗ (cf. [15, 1.b.15]), and ψ := P ∗ (y ′) is a positive extension of y ′ to E. Let U be a free ultra-filter on IN and let EU be the ultra-power of E with respect to U. Define ˆ T : EU → EU as follows Tˆ yˆ = (T yn )∞ n=1 + c0 (EU ) , Q ∞ where c0 (EU ) = {(xn )∞ n=1 ∈ n=1 E : limU kxn k = 0}. Let yˆ = (y)∞ + c (E ) ∈ EU then S(BE ) ⊆ [−ˆ y , yˆ]. Denote (EU )yˆ the principal ideal 0 U n=1 generated in EU by the element yˆ then Tˆ1 : E → (EU )yˆ is continuous. Thus Tˆ = iyˆ ◦ Tˆ1 admits a factorization through (EU )yˆ, where iyˆ : (EU )yˆ → EU is the canonical injection. By means of hψ ′ , (xn )∞ n=1 + c0 (EU )i := lim hψ , xn i n→∞

the linear functional ψ induces a functional ψ ′ ∈ (EU )∗ . Consider the L1 -space (EU , ψ ′ ) associated to EU , which is a completion of EU with respect to the norm given by ψ ′ , and let jψ′ : EU → (EU , ψ ′ ) be the canonical map. We obtain the following diagrams Tˆ

jψ ′

E → EU → (EU , ψ ′ ) ;

Tˆ

iyˆ

E →1 (EU )yˆ → EU .

Since iyˆ and jψ′ are positive operators and order intervals in the L1 -space (EU , ψ ′ ) are weakly compact, R := jψ′ ◦ iyˆ maps the unit ball [−ˆ y , yˆ] of (EU )yˆ into a weakly compact set, i.e., R is weakly compact. Since (EU )yˆ is lattice isomorphic to a C(K)-space, a result of Grothendieck (cf. [22, II.9.7, II.9.9]) implies that R maps weakly compact sets into norm compact sets. Thus jψ′ ◦ Tˆ = R ◦ Tˆ1 also maps weakly compact sets into norm compact sets.

60

E. Yu. Emel’yanov

Let now x be a flight vector of T [11, 2.4.] and kxk ≤ 1. There is a subsequence (T mk x)∞ k=1 mk ∞ ˆ ′ of (T m x)∞ weakly convergent to zero. Thus (j ◦ T ◦ T x) converges to zero in the ψ m=1 k=1 norm, i.e., lim lim hψ , |T n+mk x|i = lim hψ ′ , |Tˆ ◦ T mk x|i = lim kjψ′ ◦ Tˆ ◦ T mk xk = 0. U k→∞

k→∞

k→∞

In particular, there is a sequence (nk )∞ n=1 of naturals such that lim hψ , |T rk x|i = 0

k→∞

(rk := nk + mk ).

(3)

since [−y, y] ∈ Con(T ) there is a decomposition |T rk x| = ak + bk , k ∈ IN, such that ak ∈ [−y, y] ∩ E+ , bk ∈ E+ and lim kbk k = 0. The sequence (ak )∞ k=1 is order bounded in F and (3) k→∞ implies that lim hy ′ , ak i = lim hP ∗(y ′ ) , ak i = lim hψ , ak i = 0 . k→∞

k→∞

k→∞

By [18, Lm.4.2] we obtain lim kak k = 0, thus k→∞

lim kT rk xk = lim kak + bk k = 0.

k→∞

k→∞

Since T is power bounded, this implies lim kT n xk = 0, and the proof is finished. n→∞

References [1] C.D. Aliprantis; O. Burkinshaw, Positive Operators, Academic Press, Orlando, London, (1985). [2] S. Alpay; A. Binhadjah; E.Yu. Emel’yanov; Z. Ercan, Mean Ergodicity of Positive Operators in KB-spaces, preprint [3] Sh.A. Ayupov; T.A. Sarymsakov, Markov operators on quantum probability spaces, Probability theory and applications, Proc. World Congr. Bernoulli Soc., Tashkent/USSR 1986, Vol.1 (1987), 445–454. [4] E.Yu. Emel’yanov, Invariant densities and mean ergodicity of Markov operators, Israel J. Math. 136 (2003), 373–379. [5] E.Yu. Emel’yanov; M.P.H. Wolff, Positive operators on Banach spaces ordered by strongly normal cone, Positivity 7 (2003), no.1-2, 3–22. [6] E.Yu. Emel’yanov; M.P.H. Wolff, Mean lower bounds for Markov operators, Ann. Pol. Math. 83 (2004), no.1, 11–19. [7] E.Yu. Emel’yanov; M.P.H. Wolff, Asymptotic Behavior of Markov Semigroups on Preduals of von Neumann Algebras, J. Math. Anal. Appl. (to appear) [8] S.G. Gorokhova; E.Yu. Emel’yanov, A sufficient condition for the order boundedness of the attractor of a positive mean ergodic operator acting on a Banach lattice, Siberian Adv. Math. 9 (1999), no.3, 78–85.

Positive Asymptotically Regular Operators in L1-spaces and KB-spaces

61

[9] J. Komornik, Asymptotic periodicity of the iterates of weakly constrictive Markov operators, Tohoku Math. J. (2) 38 no.1 (1986), 15–27. [10] J. Komornik; A. Lasota, Asymptotic decomposition of Markov operators, Bull. Polish Acad. Sci. Math. 35 (1987), no.5-6, 321–327. [11] U. Krengel, Ergodic Theorems, De Gruyter, Berlin – New York (1985). [12] A. Lasota, Statistical stability of deterministic systems, Equadiff 82 (Wurzburg, 1982), 386–419, Lecture Notes in Math., 1017 Springer – Berlin (1983). [13] A. Lasota; T.Y. Li; J.A. Yorke, Asymptotic periodicity of the iterates of Markov operators, Trans. Amer. Math. Soc. 286 (1984), 751–764. [14] A. Lasota; M.C. Mackey, Chaos, fractals, and noise. Stochastic aspects of dynamics, Second edition. Applied Mathematical Sciences, 97. Springer-Verlag, New York, (1994). [15] J. Lindenstrauss; L. Tzafriri, Classical Banach spaces, Vol. I, Ergeb. Math. Grenzgeb., Vol.92, Springer, Berlin 1977. [16] P. Meyer-Nieberg, Banach Lattices, Universitext. Springer-Verlag, Berlin, (1991). [17] Vu Quoc Phong, Asymptotic almost periodicity and compactifying representations of semigroups, Ukrain. Mat. Zh. 38 (1986), 688–692. [18] F. R¨abiger, Attractors and asymptotic periodicity of positive operators on Banach lattices, Forum Math. 7 (1995), 665–683. [19] S. Sakai, C ∗ -algebras and W ∗ -algebras, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 60. Springer-Verlag, New York – Heidelberg (1971). [20] T.A. Sarymsakov, Introduction to quantum probability theory, Fan, Tashkent, (1985). [21] T.A. Sarymsakov; T.Ya. Grabarnik, The regularity of monotone continuous compresions on von Neumann algebras, Doklady AN Uz.SSR 6 (1987), 9-11. [22] H.H. Schaefer, Banach Lattices and Positive Operators, Die Grundlehren der mathematischen Wissenschaften, Band 215. Springer-Verlag, New York – Heidelberg, (1974). [23] R. Sine, Constricted systems, Rocky Mountain J. Math. 21 (1991), 1373–1383. [24] M. Takesaki, Theory of Operator Algebras I, Springer-Verlag, New York – Heidelberg, (1979).

E. Yu. Emel’yanov Department of Mathematics Middle East Technical University 06531 Ankara, Turkey E-mail: [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 63-71 (2006)

Bivariate and marginal function spaces J. J. Grobler1 (Potchefstroom) Abstract. We prove that the continuity of the conditional expectation operator relative to the product marginal measure is a necessary and sufficient condition for the sum of two marginal Banach function spaces Lρ1 and Lρ2 to be closed in the bivariate Banach function space Lρ . We show that the composition of this operator and the conditional expectation operator relative to the measure on the product space is a positive kernel operator and that the closedness of the sum space is also equivalent to the continuity of this kernel operator on the sum space. If the kernel operator maps Lρ into itself it is automatically continuous and criteria for this to happen (for instance being Hilbert-Schmidt) provides necessary conditions for the closedness of the sum space. We apply the results to the Hilbert space case and to the multivariate t-distribution. Key words: Banach function space, bivariate distribution, marginal measure, marginal distribution, conditional expectation, kernel operator, Hilbert-Schmidt operator. MSC 2000: 46E30, 47G10, 62G05

1 Introduction This study is motivated by the problem in Statistics to estimate a probability measure with given marginal probabilities (see [1], [7], [8]). Let (X × Y, Σ ⊗ Λ, P) be a probability space. The marginal probabilities P1 and P2 are defined by P1 (A) := P(A × Y ), A ∈ Σ and P2 (B) := P(X × B), B ∈ Λ. The marginal spaces L2 (X, Σ, P1 ) and L2 (Y, Λ, P2) embed isometrically into L2 (X × Y, Σ ⊗ Λ, P) via the natural embeddings φ : g(x) 7→ g(x)1Y (y) and ψ : h(y) 7→ 1X (x)h(y). The question is to find conditions which will guarantee the closedness of L2 (X, Σ, P1 )+L2 (Y, Λ, P2 ) in L2 (X × Y, Σ ⊗ Λ, P). We studied the problem in general Banach function spaces (for a definition and properties of see A.C. Zaanen [11] and W.A.J. Luxemburg and A.C. Zaanen [5]). In [4] we defined marginal function spaces as follows. Let ρ be a saturated function norm defined on M + (X × Y, Σ ⊗ Λ, P) and denote the function space Lρ (X × Y, Σ ⊗ Λ, P) by Lρ (see [11, Chapter 15]). We assume throughout that the function norm satisfies the Riesz-Fischer property which is equivalent to the 1

The author thanks B de Pagter and J Conradie with whom he discussed the problem and for their helpful remarks. He is also grateful to G Buskes on whose invitation he spent the fall semester 2003 at the University of Mississippi at Oxford, USA.

64

J. J. Grobler

assumption that Lρ is a Banach function space. The marginal function norms ρi are defined by ρ1 (f ) = ρ(φ(f )), ρ2 (g) = ρ(ψ(g)),

f ∈ M + (X, Σ, P1 ) g ∈ M + (Y, Λ, P2 ).

We call the spaces Lρ1 (X, Σ, P1 ) = Lρ1 and Lρ2 (Y, Λ, P2 ) = Lρ2 the marginal spaces of Lρ . We proved in [4] that if Lρ is a Banach function space, then the marginal spaces Lρ1 (P1 ) and Lρ2 (P2 ) are also Banach function spaces and that φ and ψ are isometric lattice isomorphisms of Lρ1 and Lρ2 onto closed Riesz subspaces of Lρ . It is also immediately clear that both ρ1 and ρ2 must be saturated function norms. We shall identify Lρi with its image in Lρ . In order to guarantee that Lρi 6= {0}, we shall tacitly assume throughout the paper that L∞ ⊂ Lρ , i.e., Lρ , and therefore also the marginal spaces Lρ1 and Lρ2 , contain the constant functions. Let (Ω, F, P) be an arbitrary probability space and let G be a sub-σ-algebra of F. For f ∈ L1 (Ω, F, P), we denote the conditional expectation of f with respect to G by EP (f | G) and recall that it is the (P-a.e.) unique G-measurable function with the property that Z Z P E (f | G) dP = f dP for all A ∈ G. A

A

We refer the reader to [2], [10] and [3] for properties of EP (· | G). The conditional expectation EP (· | G) can be extended from a mapping of L1 (Ω, F, P) into itself, to a mapping from M + (Ω, F, P) into itself. Since the extended map needs not map L0 (Ω, F, P)+ into itself (see [3]), we define the domain dom EP (· | G) := {f ∈ L0 (Ω, F, P) : EP (|f | | G) ∈ L0 (Ω, G, P)}. Clearly, dom EP (· | G) is an ideal in L0 (Ω, F, P) which contains L1 (Ω, F, P) and is therefore order dense in L0 (Ω, F, P). For f ∈ dom EP (· | G), we define: EP (f | G) := EP (f + | G) − EP (f − | G). This defines a positive linear operator EP (· | G) : dom EP (· | G) → L0 (Ω, G, P) ⊂ L0 (Ω, F, P). It can be proved that if EP (|f | | G) ∈ L0 (Ω, G, P), then f ∈ L0 (Ω, F, P) (see [3, Proposition 2.3]). e := {A × Y : A ∈ Σ} and Λ e := {X × B : B ∈ Λ} are sub-σ-algebras of We note that Σ e and EP (f | Λ) e by EP (f ) Σ ⊗ Λ. For brevity we denote the conditional expectations EP (f, | Σ) 1 and by EP2 (f ) respectively. We define the associate function seminorm ρ′ of a function norm ρ by Z ′ ρ (g) = sup{ |f g| dP : ρ(f ) ≤ 1}. X×Y

Since we assume ρ to be saturated, ρ′ is a Fatou function norm (see [11]) and gives rise to the Banach function space Lρ′ = L′ρ which is called the associate space of Lρ . In our case we note that ρ′i ≤ (ρ′ )i . We call the function norm ρ fully marginal if ρ′i = (ρ′ )i , i = 1, 2. We proved in [4, Proposition 2.3] that if the function norm ρ is a Fatou norm

Bivariate and marginal function spaces

65

then it is fully marginal if and only if both the conditional expectations EP1 (f ) and EP2 (f ) are contractive projections from Lρ onto the marginal spaces Lρ1 and Lρ2 respectively. To avoid technicalities, we shall assume that the constant functions are also contained in the associate spaces and thus Lρ ⊂ L1 (P) and Lρi ⊂ L1 (Pi ). This may seem to implicate that the extension of the domain of the conditional expectation is superfluous, but Lρ ⊂ L1 (P) does not in general imply Lρ ⊂ L1 (P1 ⊗ P2 ) and so Lρ is not automatically contained in the domain of EPi 1 ⊗P2 . Our main result in [4] was the following. Theorem 1.1. Let (X × Y, Σ ⊗ Λ, P) be a probability space with marginal spaces (X, Σ, P1 ) and (Y, Λ, P2 ) and suppose that the measures P and P1 ⊗P2 are equivalent. Let r be the RadonNikodym derivative of P1 ⊗ P2 with respect to P and let ρ be a fully marginal function norm defined on M + (X × Y, Σ ⊗ Λ, P). If Q(ξ, η, x, y) =

r(x, y)1Y (η) r(ξ, y)

is the kernel of an operator Q which maps Lρ into itself, then Lρ1 + Lρ2 is closed in Lρ (X × Y, Σ ⊗ Λ, P). The same conclusion holds if R(ξ, η, x, y) =

r(x, y)1X (ξ) r(x, η)

is the kernel of an operator R which maps Lρ into itself. Moreover, Q = EP1 E2P1 ⊗P2 and R = EP2 E1P1 ⊗P2 . One can expect a sharper result if one considers the operators Q and R on the closure of Lρ1 + Lρ2 in stead of on the whole space Lρ . We shall prove in this note that indeed, the closedness of the sum space is equivalent to the continuity of the maps EiP1 ⊗P2 from Lρ1 + Lρ2 into Lρi . We also provide some applications.

2 The main results The following general result is widely used in Hilbert space. We need it in Banach spaces. Equivalence (i)–(iii) was shown to the author by B. de Pagter. Let X be a Banach space and suppose that V and W are closed linear subspaces in X. We assume that V ∩W is complemented in X, i.e., there exists a bounded linear projection P : X → X such that P (X) = V ∩ W . Let V0 := {v ∈ V : P v = 0},

W0 := {w ∈ W : P w = 0}.

We note that V0 ∩ W = {0} and V ∩ W0 = {0}. Indeed if z ∈ V0 ∩ W ⊂ V ∩ W, then, since z ∈ V ∩ W, we have P z = z and so 0 = P z = z. Similarly for the other case. In particular this implies that V0 ∩ W0 = {0} and so V0 + W0 = V0 ⊕ W0 algebraically. Theorem 2.1. The following are equivalent (i) V + W is closed in X;

66

J. J. Grobler

(ii) V0 + W0 is closed in X; (iii) there exists a constant C such that max(kvk, kwk) ≤ Ckv + wk for all v ∈ V0 and w ∈ W0 ; (iv) V + W :=

V W X + is closed in X = ; V ∩W V ∩W V ∩W

(v) there exists a constant K such that max(kvk, kwk) ≤ Kkv + wk for all v ∈ V and w ∈ W. Proof. (i) =⇒ (ii): Let (vn ) ⊂ V0 and (wn ) ⊂ W0 be sequences such that vn +wn → x ∈ X. Since V + W is closed, x ∈ V + W, say x = v + w with v ∈ V and w ∈ W. We note that since P vn = 0 = P wn , P x = 0, i.e., P v + P w = 0. Hence, x = (v − P v) + (w − P w) ∈ V0 + W0 . (ii) =⇒ (iii): The space V0 ⊕ W0 with the norm k(v, w)k = max(kvk, kwk) is a Banach space. Define the map v + w 7→ (v, w) from V0 + W0 → V0 ⊕ W0 . This map is easily seen to be closed and hence, since V0 + W0 is also a Banach space, continuous. Hence there exists a constant C > 0 such that max(kvk, kwk) = k(v, w)k ≤ Ckv + wk for all v ∈ V0 and w ∈ W0 . (iii) =⇒ (i): Take (vn ) ⊂ V and (wn ) ⊂ W such that vn + wn → x ∈ X. It follows that (I − P )vn + (I − P )wn → (I − P )x. Noting that (I − P )vn ∈ V0 and (I − P )wn ∈ W0 it follows from the estimate in (ii) that ((I − P )vn ) and ((I − P )wn ) are Cauchy sequences in V0 and W0 both being closed linear subspaces of X. There exists therefore elements v ∈ V0 and w ∈ W0 such that (I − P )vn → v and (I − P )wn → w. Hence, (I − P )x = v + w ∈ V0 + W0 ⊂ V + W. This shows that x = (I − P )x + P x ∈ (V + W ) + V ∩ W ⊂ V + W. (i) =⇒ (iv): Let vn + w n ∈ V + W , v n + wn → u ∈ X. Then (vn + w n )∞ n=1 is a Cauchy sequence in X and we can find a subsequence and a sequence uk ∈ V ∩ W such that zk := (vnk+1 − v1 ) + [(wnk+1 − w1 ) + uk ] ∈ V + W is a Cauchy sequence in X (see [9, proof of Theorem III.5.3]). By assumption it converges to an element v + w ∈ V + W. It follows that u = (v + v1 ) + w + w 1 ∈ V + W . (iv) =⇒ (ii) Let (vn + wn ) be a sequence in V0 + W0 which converges to u ∈ X and note that P u = 0. Then v n +w n converges in X to u and by assumption u = v +w ∈ V +W . Hence, for some z ∈ U ∩ W, we have u = v + w + z = (v − P v) + (w − P w) + (P v + P w + z). But then P v + P w + z = P (P v + P w + z) = P u = 0, and so u = (v −P v) + (w −P w) ∈ V0 + W0 . (iv)⇐⇒ (v): Assume that (iv) holds. Note that V + W is a direct sum. Indeed if u ∈ V ∩ W , then u = v+z1 = w+z2 for appropriate elements z1 and z2 in V ∩W. Writing v0 = v−P v ∈ V0

Bivariate and marginal function spaces

67

and w0 = w − P w ∈ W0 , we see that u = v0 + z1′ = w0 + z2′ , with z1′ , z2′ ∈ V ∩ W. But then zi′ = P zi′ = P u and so z1′ = z2′ resulting in v0 = w0 . This shows that u ∈ V ∩ W, which means that u = 0. We can now repeat the arguments used in the proof of (ii) =⇒ (iii) to conclude that (v) is true. If we assume (v) to be true, then the truth of (iv) follows as in the proof of (iii) =⇒ (i). Remark. The proof that the quotient above is a direct sum also follows by observing that P V = V ∩ W and so, since V = P V + (I − P )V one has that V0 = (I − P )V is algebraically isomorphic to the quotient V , and similarly, W0 is isomorphic to W . We will now specialize our discussion by considering the bivariate function space Lρ (P) with marginal spaces Lρ1 (P1 ) and Lρ2 (P2 ). In order to apply Theorem 2.1, we assume that P and P1 ⊗ P2 are equivalent measures. This implies, by [4, proof of Proposition 3.5] that the intersection of the spaces Lρ1 and Lρ2 consists of the constant functions, which R is then a closed complemented subspace of Lρ1 ∩ Lρ2 . In this case the integral P (f ) = ( X×Y f (x, y) dP) provides the bounded linear projection of Lρ onto Lρ1 ∩ Lρ2 . Theorem 2.2. Let P and P1 ⊗P2 be equivalent measures. The normed subspace Lρ1 +Lρ2 ⊂ Lρ is closed in Lρ if and only if the conditional expectation operators EP1 1 ⊗P2 and EP2 1 ⊗P2 are continuous operators from Lρ1 + Lρ2 into Lρ1 and Lρ2 respectively. Proof. Assume Lρ1 + Lρ2 to be closed. We prove that EP2 1 ⊗P2 is continuous. With P the bounded linear projection of Lρ onto Lρ1 ∩ Lρ2 as explained above, we note that for g + h ∈ Lρ1 + Lρ2 , Z Z P1 ⊗P2 E2 (g + h) = g(x) dP1(x) + h(y) = g dP + h = P g + h, X

X×Y

which shows that EP2 1 ⊗P2 maps Lρ1 + Lρ2 into Lρ2 . We show next that EP2 1 ⊗P2 is closed. To this end, let gn + hn → g + h

(1)

EP2 1 ⊗P2 (gn + hn ) = P gn + hn → f.

(2)

and let

Subtracting (2) from (1) we get that (I − P )gn → g + h − f. Now, from gn + hn → g + h i.e., from (g − gn ) + (h − hn ) → 0, it follows that (I − P )(g − gn ) + (I − P )(h − hn ) → 0 and so from Theorem 2.1 (iii) it follows that (I − P )gn → (I − P )g. Hence, g − P g = g + h − f, and we get the required result that f = P g + h = EP2 1 ⊗P2 (g + h). This shows that EP2 1 ⊗P2 is continuous and the proof that E1P1 ⊗P2 is continuous follows from the symmetry.

68

J. J. Grobler

Conversely, assume that EPi 1 ⊗P2 (i = 1, 2) are continuous maps from Lρ1 + Lρ2 into Lρi . By Theorem 2.1 it is sufficient to prove that Lρ1 ,0 + Lρ2 ,0 is closed. Let therefore gn + hn → f ∈ Lρ with gn + hn ∈ Lρ1 ,0 + Lρ2 ,0 . Then (gn + hn ) is a Cauchy sequence in Lρ1 + Lρ2 and so (EP1 1 ⊗P2 (gn + hn )) = (gn + P hn ) = (gn ) is a Cauchy sequence in Lρ1 ,0 and similarly (hn ) is a Cauchy sequence in Lρ2 ,0 . Let gn → g ∈ Lρ1 ,0 and hn → h ∈ Lρ2 ,0 . Then f = g + h ∈ Lρ1 ,0 + Lρ2 ,0 and so Lρ1 + Lρ2 is closed in Lρ . Theorem 1.1 provided sufficient conditions for the closedness of the sum space Lρ1 + Lρ2 in terms of the kernel operators EPj EiP1 ⊗P2 . The next theorem links this to the present discussion and provides a more general result. Theorem 2.3. Let P and P1 ⊗ P2 be equivalent measures and let ρ be a fully marginal function norm. The subspace Lρ1 + Lρ2 is closed in Lρ if and only if the operators EPj EPi 1 ⊗P2 are continuous operators from Lρ1 + Lρ2 to Lρj (i 6= j, i, j = 1, 2). Proof. Our assumption that ρ is fully marginal implies, as we remarked earlier, that the i = 1, 2 are contractive projections from Lρ into Lρi . Now, if Lρ1 + Lρ2 is closed, then EiP1 ⊗P2 is continuous by the preceding theorem and the restriction of EPj to Lρi is contractive. It follows that EPj EPi 1 ⊗P2 is continuous. EPi ,

Conversely, let fn = gn + hn ∈ Lρ1 ,0 + Lρ2 ,0 converge in Lρ to f. By assumption, EP2 EP1 1 ⊗P2 (gn + hn ) = EP2 (gn + P hn ) = EP2 (gn ) → g := EP2 EP1 1 ⊗P2 (f ). Also, since EP2 is a contractive projection, EP2 (gn + hn ) = EP2 (gn ) + hn → EP2 (f ). It follows that hn = (hn + EP2 (gn )) − EP2 (gn ) → h := EP2 (f ) − g ∈ Lρ2 ,0 , since the latter space is closed. But then, gn = (gn + hn ) − hn → f − h ∈ Lρ1 ,0 . Hence, f = (f − h) + h ∈ Lρ1 ,0 + Lρ2 ,0 . This completes the proof. Remark. Theorem 1.1 follows from Theorem 2.3 since the positive operator EPj EPi 1 ⊗P2 from Lρ into itself is automatically continuous on Lρ and therefore on Lρ1 + Lρ2 . The advantage of applying these operators on the whole space is that they are kernel operators the kernel of which can, in many cases, be found explicitly. Condition can then be formulated so that the operators will map Lρ into itself (see [4, Examples 4.1 and 4.2]). The condition that P and P1 ⊗ P2 are equivalent is a sufficient but not a necessary condition for the theorems of this section to hold. Example 2.4. Let X = Y = [0, 1] and let P be the measure on X × Y defined by Z P(A × B) := δ(x, A) dλ, B

with λ the Lebesgue measure on X and δ(x, A) = 1 if x ∈ A and δ(x, A) = 0 if x ∈ / A. It follows that P is a probability measure concentrated on the diagonal ∆ and that Pi = λ for i =

Bivariate and marginal function spaces

69

R 1, 2. Hence, P ⊥ P1 ⊗ P2 . Note that if f (x, y) is P-integrable, we have that X×Y f (x, y) dP = R R R f (x, y)δ(x, dy) dλ = X f (x, x) dλ. Let ρ be a fully marginal Fatou function norm on X Y M + (X × Y, P). For any f ∈ Lρ , put g(x) = f (x, x) and h(y) = f (y, y) then, Z Z ′′ ρ1 (g) = sup |g(x)u(x)| dλ = sup |f (x, y)u(x)| dP ≤ ρ′′ (f ) < ∞ ρ′1 (u)≤1

X

ρ′ (φ(u))≤1

X×Y

Hence, g ∈ Lρ1 and similarly, h ∈ Lρ2 . It follows that, since f (x, y) = 21 g(x) + 21 h(y) holds P-almost everywhere, f ∈ Lρ1 + Lρ2 . This shows that Lρ1 + Lρ2 = Lρ .

3 Applications 3.1 L2(X × Y, P) Assume as before that P and its product marginal measure P1 ⊗ P2 are equivalent measures. Then L2 (X, P1 ) ∩ L2 (Y, P2 ) = C and in the quotient Hilbert space L2 (X × Y )/C we have that kfk2 = Var f and the inner product in this space is given by hf , gi = Cov(f, g). The correlation coefficient is given by Corr(f, g) = p

Cov(f, g) Var(f ) Var(g)

=

hf , gi kfk kgk

Assume now as in [8] that the correlation coefficient is bounded away from 1 and −1 for all f ∈ L2 (X, P1 ) and g ∈ L2 (Y, P2 ), i.e., there exists a constant 0 ≤ γ < 1 such that | Corr(f, g)| ≤ γ for all f ∈ L2 (X, P1 ) and g ∈ L2 (Y, P2 ).

(A)

Theorem 3.1 (H. Peng and A. Schick). Let P be a probability measure with marginal measures P1 and P2 and suppose that P and P1 ⊗P2 are equivalent measures. If condition (A) above holds then L2 (X, P1 ) + L2 (Y, P2 ) is closed in L2 (X × Y, P). Proof. In the quotient space L2 (P)/C we have kf + gk2 = kfk2 + kgk2 + 2hf, gi ≥ kfk2 + kgk2 − 2|hf, gi| ≥ kfk2 + kgk2 − 2γkfk kgk ≥ (1 − γ)(kf k2 + kgk2 ). Hence, kf + gk ≥ (1 − γ) max(kf k, kgk) and the result follows from Theorem 2.1 (v). In the next corollary we denote by P the measure in R with density function the bivariate normal distribution N with correlation coefficient ρ. Corollary 3.2 (H. Peng and A. Schick). If the the stochastic variables X and Y have joint bivariate normal distribution N with correlation coefficient ρ ∈ (−1, 1) then, for f ∈ L2 (R, B(R), P1 ) and g ∈ L2 (R, B(R), P2 ), | Corr(f (X), g(Y ))| ≤ |ρ|. It follows that L2 (R, B(R), P1 ) + L2 (R, B(R), P2 ) is closed in L2 (R × R, B(R × R), P). Proof This follows from Mehler’s identity, as pointed out in [8] by Peng and Schick. Remark. In [4] we proved, using Theorem 1.1 that this result holds for | ρ | < √15 . This shows that the Peng-Schick result provides a stronger conclusion in this case, as can be expected.

70

J. J. Grobler

3.2 The bivariate t-distribution The bivariate central t-distribution with shape paramete ν (see [6]) has density function −(ν+2)/2  κ x21 − 2ρx1 x2 + x22 p(x1 , x2 ) = p 1+ , |ρ| < 1 ν(1 − ρ2 ) 1 − ρ2 with

κ :=

(2)

Γ((ν + 2)/2) πνΓ(ν/2)

Its marginal densities are −(ν+1)/2  √ x2i pi (xi ) = G(ν)2κ ν 1 + (i = 1, 2) ν R π/2 with G(ν) := 0 cosν u du.

(2)

The kernel Q in Theorem 1.1 satisfies

p1 (x1 )p(ξ, x2 ) p1 (ξ)p(x1 , x2 )  −(ν+1)/2  −(ν+2)/2 ν + x21 ν(1 − ρ2 ) + ξ 2 − 2ρξx2 + x22 = ν + ξ2 ν(1 − ρ2 ) + x21 − 2ρx1 x2 + x22

Q(ξ, x1 , x2 ) =

and to show that its Hilbert-Schmidt norm is finite, we have to integrate p1 (x1 )2 p(ξ, x2 )2 (ν + x21 )−(ν+1) (ν(1 − ρ2 ) + ξ 2 − 2ρξx2 + x22 )−(ν+2) =c p1 (ξ)p(x1 , x2 ) (ν + ξ 2 )−(ν+1)/2 (ν(1 − ρ2 ) + x21 − 2ρx1 x2 + x22 )−(ν+2)/2 over R3 , with the constant c = 2κ2 ν (ν+2)/2 (1 − ρ2 )(ν+1)/2 G(ν). The integrand has no singularities in the origin and is asymptotically equivalent to the function −(ν+2)

ξ −(ν+3) x−ν 1 x2

Therefore, the Hilbert-Schmidt norm of Q is finite if and only if ν > 1. It follows that the sum of the marginal spaces for the bivariate t-distribution is closed if the shape parameter ν satisfies ν > 1.

References [1] P.J. B ICKEL , Y. R ITOV AND J.A. W ELLNER , Efficient estimation of linear functionals of a probability measure P with known marginal distributions Ann. Statist.19 (1991), 1316– 1346. [2] D OOB , J.L., Measure theory, Graduate Texts in Mathematics, Springer-Verlag New York Berlin Heidelberg, 1994.

Bivariate and marginal function spaces

71

[3] G ROBLER , J.J. AND DE PAGTER , B., On operators representable as multiplication conditional expectation operators, J. of Operator theory 48 (2002), 15–40. [4] G ROBLER , J.J., Embedding marginal spaces Research report: FABWI-N-WST: 2005169, 2005, 23p. [5] W.A.J. L UXEMBURG AND A.C. Z AANEN Notes on Banach function spaces Indag. Math 25 (1963), Note I, 135–147; Note II, 148–153; Note III, 239–250; Note IV, 251–263; Note V, 496–504. [6] S. Nadarajah and S Kotz, Sampling distributions associated with the multivariate t distribution, Statistica Neerlandica 59(2) (2005), 214–234. [7] H. P ENG , A. S CHICK , On efficient estimation of linear functionals of a bivariate distribution with known marginals, Statist. and Prob. Letters 59 (2002), 83–91. [8] H. P ENG , A. S CHICK , Efficient estimation of linear functionals of a bivariate distribution with equal but unknown marginals; the least-squares approach. Journal of multivariate analysis 95 (2005), 385–409. [9] S CHECHTER , M., Principles of Functional Analysis, Academic Press, New York, London, 1971. [10] S TROMBERG , K.R., Probability for analysts, Chapman and Hall, New York London, 1994. [11] Z AANEN , A.C., Integration, North-Holland, Amsterdam New York, 1967.

J. J. Grobler Unit for Business Mathematics and Informatics, North-West University, Potchefstroom 2520, South Africa. email address: [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 73-77 (2006)

Positive operators without invariant sublattices A. K. Kitover (Community College of Philadelphia) and A. W. Wickstead (Queens University Belfast) Abstract. Many results are now known which ensure that certain kinds of positive operators on Banach lattices have non-trivial closed invariant sublattices. In particular, this is true for every positive compact operator. We outline several examples of positive operators on Banach lattices which do not have nontrivial closed invariant sublattices. The examples cover cases defined on both AM-spaces and Banach lattices with an order continuous norm and which are and are not atomic. In all these cases we can ensure that the operators do possess non-trivial closed invariant subspaces. Key words: Banach lattices, positive operators, invariant sublattice MSC 2000: 47B65, 47A15

1 Introduction Examples of operators which do not have non-trivial invariant closed subspaces were first produced by Enflo [4] and later examples on ℓ1 were produced by Read [6], [7] and [8]. After a pioneering paper by de Pagter [5], in about 1992 Abramovich, Aliprantis and Burkinshaw commenced a series of papers devoted to the study of the conjecture that every positive operator on a Banach lattice with dimension at least two has a non-trivial closed invariant subspace. An account of their work may be found in Chapter 10 of [1]. There are many positive results to be found there, but we note in particular that every positive compact operator on a Banach lattice has a non-trivial closed invariant sublattice. Indeed either such an operator has strictly positive spectral radius, when the spectral radius is an eigenvalue and the corresponding eigenvector is positive so that the linear span of this eigenvalue is an invariant sublattice, or else the spectral radius is zero and the operator has a non-trivial closed invariant ideal. The question then arises of whether or not every positive operator on a Banach lattice does possess a non-trivial closed invariant sublattice. We were able to produce an example of a positive operator on a Banach lattice which does not possess either a non-trivial closed invariant ideal nor an invariant one-dimensional ideal but which does possess a closed non-trivial invariant sublattice. However after this positive start, we then produced a range of examples where the operator fails to have a non-trivial closed invariant sublattice. These examples are defined on several kinds of Banach lattice, including both AM-spaces and Banach lattices with an order continuous norm and, in each case, which are or are not atomic. On each of these four sorts of Banach lattice, we are able to produce positive operators which do have non-trivial closed invariant subspaces but which do not have non-trivial closed invariant sublattices.

74

A. K. Kitover, A. W. Wickstead

The examples outlined here seem to put paid to any positive result that involves only a condition on the Banach lattice, apart from the obvious finite dimensional case and the almost trivial case when the Banach lattice is non-separable, which is proved in [2], which also contains complete proofs of the examples outlined here. The authors would like to thank Mahesh Nerurkahr for useful discussions connected with Examples 2.4 and 2.5.

2 The examples The first example shows that, for positive operators, non-trivial closed invariant sublattices may exist when neither non-trivial closed invariant ideals nor invariant one-dimensional sublattices do. In this example, we actually have a complete description of the non-trivial closed invariant sublattices. The notation ℓp (Z) is used to denote the usual space of p-summable bilateral sequences indexed by all integers, in contrast to ℓp which is indexed by the non-negative integers. Example 2.1. Let X = ℓp (Z), (1 ≤ p < ∞) or c0 (Z) and T be the operator on X defined by T x = (xn−1 + xn+1 ), then 1. T has no positive eigenvector. 2. T has no non-trivial closed invariant ideals. 3. The T -invariant closed sublattices of X are precisely the sublattices Hq = {x ∈ X : xm = xn if m + n = 2q} for either q ∈ Z or q −

1 2

∈ Z.

A slight modification of this example gives us our first example of a positive operator which does not have a non-trivial closed invariant sublattice. Example 2.2. Let X = ℓp , (1 ≤ p < ∞) or c0 and T be the operator on X defined by ( xn−1 + xn+1 if n > 0 (T x)n = x1 if n = 0. then there is no non-trivial closed T -invariant sublattice of X. Proof. The proof starts by that there are no positive eigenvectors. If x = (xn ) ∈ c we Pshowing ∞ n use the function f (z) = n=0 x)nz when T x corresponds to
f (z) − f (0) . zf (z) + z We then solve for f and show that the sequence of coefficients in its power series expansion does not lie in X. If H were a proper closed T -invariant sublattice we start by showing that there is no m ∈ N such that xm = 0 for all x ∈ H. On the other hand there must be non-negative integers m > n and 0 < α ∈ R such that xm = αxn for all x ∈ H, and we show that the restriction map of H onto the first m coordinates is one-to-one so that H is finite dimensional and hence T has a positive eigenvector corresponding to the spectral radius of T|H .

Positive operators without invariant sublattices

75

Note that when p = 2 the operator T is self-adjoint so certainly has a plentiful supply of non-trivial closed invariant subspaces. The proof needed in Example 2.2 breaks down if we take X = c or X = ℓ∞ as c0 is a non-trivial invariant closed ideal in that case. This is of special interest as a theorem of M.G. Krein, [1] Corollary 9.46, asserts that for any positive operator T on a C(K)-space the adjoint T ∗ has a positive eigenvector, from which it is immediate that T has a non-trivial closed invariant subspace, namely the kernel of such an eigenvector. Thus it might have been conjectured that positive operators on C(K)-spaces had non-trivial closed invariant sublattices. Here is an example to show that this is not so. Like our previous example, the next one has many atoms. Example 2.3. Let X = c and T be the operator on X defined by ( xn−1 + xn+1 + x0 if n > 0 (T x)n = x1 + x0 if n = 0. then there is no non-trivial closed T -invariant sublattice of X. Proof. Again the proof starts by showing there are no one-dimensional invariant sublattices using a function viewpoint to show first that the corresponding eigenvalue is at most 2, followed by elementary estimates. Again, if H is a non-trivial closed T -invariant sublattice we start by showing that there is no m with xm = 0 for all x ∈ H. If there are non-negative integers m > n and 0 < α ∈ R with xm = αxn for all x ∈ H the proof proceeds as in the preceding example. The only remaining alternative is that there exists precisely one m ∈ N and 0 < α ∈ R such that xm = α limn→∞ xn for all x ∈ H. This defines H completely and it is then easily seen not to be T -invariant. Note that this operator is also defined on ℓ∞ where it certainly has an invariant closed sublattice namely c. We can also produce examples where the Banach lattice has no atoms. Γ denotes the unit circle and C(Γ) the space of all continuous real-valued functions on Γ. We will identify C(Γ) with the set {f ∈ C[0, 2π] : f (0) = f (2π)}. We use the symbol ∔ to mean addition modulo 2π. Example 2.4. Let α ∈ [0, 2π) be such that α/π is an irrational number. There is f ∈ C(Γ) such that if we set w = exp(f ) and let T be the positive operator on C(Γ) defined as (T h)(x) = w(x)h(x ∔ α), then the operator T does not have a non-trivial closed invariant sublattice in C(Γ). Proof. The function f is chosen so that there is no g ∈ C(Γ) with f (θ) = g(θ ∔ α) − g(θ) for all θ ∈ [0, 2π), which is precisely the condition needed to ensure that T does not have a one-dimensional T -invariant sublattice. If H were a non-trivial T -invariant closed sublattice then there are x, y ∈ [0, 2π), with x 6= y, and λ ≥ 0 such that g(y) = λg(x) for all x ∈ H. We first show that λ > 0 and that H contains a strictly positive function g. If (x − y)/π is irrational then any strictly positive function in H must be proportional to g from which it follows that H is one-dimensional, whilst if (x − y)/π = p/q is rational then the proof depends on showing that 1 wn (θ) ≤ ≤C C wn (θ ∔ pq π)

76

A. K. Kitover, A. W. Wickstead

for all θ ∈ [0, 2π) and all n ∈ N where C = kgk2 kg −1k2 and wn (θ) = w(θ)w(θ ∔ α) . . . w(θ ∔ (n + 1)α).

Of course, Krein’s theorem again shows that T has a non-trivial invariant subspace. We conclude our examples with one to show that it is possible for a purely non-atomic Banach lattice with an order continuous norm to support a positive operator which has no non-trivial invariant sublattices. Example 2.5. If v is a real analytic function in L∞ (Γ) which is not a trigonometric polynomial and w = exp(v) then there is a subset V of the second category in [0, 2π) such that for any α α ∈ V the number 2π is irrational and the operator T = wTα has no non-trivial closed invariant sublattices in Lp , 1 ≤ p < ∞. Proof. A crucial part of the proof might be of independent interest. We show that if X is a norm-closed, T -invariant vector sublattice of Lp (Γ) and M(X) = {f ∈ L∞ (Γ) : f X ⊆ X} then M(X) is a σ(L∞ , L1 )-closed subalgebra and sublattice of L∞ (Γ), which is rotation invariant and is the σ(L∞ , L1 )-closed linear span of the functions exp(ınθ) that it contains. The remainder of the proof is rather technical and we refer the interested reader to [2] for details. A similar example can be presented on any symmetric ideal in L0 (Γ) with an order continuous norm. The number α in this example is chosen from a second category subset of Γ. The Liouville numbers form a set of first category in Γ so that we could also ensure that α is not a Liouville number. It would then follow from [3] that the operator T will have a closed non-trivial invariant subspace even though it does not have a closed non-trivial invariant sublattice.

References [1] Y.A. Abramovich and C.D. Aliprantis, An Invitation to Operator Theory, AMS Graduate Studies in Mathematics 50, Providence Rhode Island (2002). [2] A.K. Kitover and A.W. Wickstead, Invariant sublattices for positive operators, submitted. [3] A.M. Davie, Invariant subspaces for Bishop’s operators, Bull. London Math. Soc. 6 (1974), 343–348. [4] P. Enflo, On the invariant subspace problem for Banach spaces, Seminaire Maurey-Schwarz (1975–1976); Acta Math. 158 (1987), 213–313. [5] B. de Pagter, Irreducible compact operators, Math. Z. 192 (1986), 149–153. [6] C.J. Read, A solution to the invariant subspace problem on the space ℓ1 , J. London Math. Soc. 17 (1985), 305–317. [7] C.J. Read, A short proof concerning the invariant subspace problem, J. London Math. Soc. 34 (1986), 335–348.

Positive operators without invariant sublattices

77

[8] C.J. Read, Quasinilpotent operators and the invariant subspace problem. J. London Math. Soc. 56 (1997), 595–606.

A.K. Kitover Department of Mathematics Community College of Philadelphia USA. [email protected] A.W. Wickstead Department of Pure Mathematics Queens University Belfast Northern Ireland [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 79-92 (2006)

Stochastic Processes on the Basis of New Measure Theory Heinz K¨onig (Saarbr¨ucken) Abstract. The present article describes the reformation of certain basic structures, first in measure and integration as in the previous work of the author, and on this basis then in stochastic processes. Both times the aim is to overcome certain well-known substantial difficulties. Key words: Traditional and new stochastic processes, their canonical and maximal measures, their essential subsets, the Wiener and Poisson processes, inner premeasures and their maximal inner extensions, projective limit theorems MSC 2000: 28A05, 28A12, 28C20, 60A10, 60G05, 60G07, 60G17

0 Introduction In the present article the author wants to describe the reformation of certain basic structures, first in measure and integration in his 1997 book [12] and in subsequent papers summarized in his 2002 survey article [14], and on this basis then in stochastic processes [15][16][17][18]. The reasons were certain substantial difficulties with the traditional theories, which we start to recall and to which we shall come back. We also refer to the treatises listed in the references below. The traditional abstract theory of measure and integration which emerged from the achievements of Borel and Lebesgue in the first two decades of the 20th century is burdened with its total limitation to sequential procedures and its neglect of regularity. The alternative concept of Bourbaki [2] which arose in the middle of the century was able to relieve this burden but produced new ones, first of all its Procrustean bed in topology. In particular there is a methodical point which also reappeares in the later sequential and nonsequential abstract variants: In spite of the deliberate and innovative turn from (often unnoticed) outer to explicit inner regularity, based on the profound rˆole of compactness, one went on to produce the basic entities, now intended to be of pronounced inner character, with the weapons of the outer arsenal - a procedure which soon afterwards had to be repaired with that unfortunate construction named the essential one. All this has been made clear in [14]. Since the famous 1933 work of Kolmogorov [11] the mathematical theory of probability is a part of abstract measure theory, and hence exposed to its imperfections as well. The central notion ever since is that of stochastic processes, its first systematic treatment is due 1953 to Doob [6]. A stochastic process amounts to a probability measure (prob measure for short), called its canonical measure, on a certain σ algebra in the path space, composed of time domain Extended version of lectures presented at the S´eminaire d’Initiation a` l’Analyse, Universit´e Pierre et Marie Curie (Paris VI), 1 April 2004, and at the conference Positivity IV, Technische Universit¨at Dresden, 29 July 2005

80

H. König

and state space. The probabilistic context requires that a stochastic process be rooted in the finite subsets of the time domain. Combined with the above limitation to sequential procedures this forces the σ algebra in question to consist of so-called countably determined subsets of the path space. In the all-important case of an uncountable time domain the class of these subsets is much too narrow, a fact which is the obvious reason for well-known serious difficulties with stochastic processes. Thus it enforces the ad hoc formation of an unforeseeable multitude of measure extensions of the canonical measure, as a rule in the guise of so-called versions of the stochastic process under consideration. A vast crowd of them turned out to be pathological, so that one has to find out the substantial ones. A related issue is to detect those subsets of the path space which could be named the essential ones for the stochastic process, that is those subsets which support the essential features of the process, and which a priori can be far from obvious. The most prominent example of such an essential subset is the set of continuous paths for the traditional Wiener measure, the canonical measure of one-dimensional Brownian motion. Note that this particular example comes from experimental observations outside of mathematics! In its more than fifty years the traditional theory of stochastic processes has not been able to produce an adequate notion of an essential subset. The usual attempt due to Doob defines these subsets to be the sets of outer canonical measure one; equivalent for a subset is that the canonical measure has a measure extension which lives on the set. But the class of these subsets turned out to be pathological as well. Both times it appears to be a natural idea that the collection of measure extensions of the canonical measure needs a drastic and clever reduction. One could even think of a unique measure extension, provided that it has a wide domain in order to expose the full breadth of the process under consideration. But the traditional theory of stochastic processes did not contribute to this idea. However, the new structure in measure and integration quoted at the outset [12][14] was able to achieve the aim in question and to resolve the connected problems with stochastic processes [15]-[18], after that its innovative force had already been confirmed through other applications. The decisive step is a new projective limit theorem of Kolmogorov type in terms of inner premeasures and in its τ (:=nonsequential) version, which in final form is in [16] theorem 3. This τ theorem inspires an immediate variant of the concept of stochastic processes: after due modification of the structure in the state space one defines the new stochastic processes in terms of inner τ premeasures instead of measures. Then the fundamental extension procedure, which is the heart of the new structure, provides a new stochastic process with a unique and highly distinguished prob measure on the path space, called its maximal measure, defined on an immense domain which in particular reaches far beyond the class of countably determined subsets. After this it is natural to define the essential subsets for a new process to be those subsets of the path space on which the maximal measure of the process lives. For all that the new stochastic processes remain, in view of the above projective limit theorem, rooted in the finite subsets of the time domain like the traditional ones. Thus the new concept is able to unite two aspects which seemed to be incompatible under the former ones. After this the main question is of course how the new stochastic processes are related to the traditional ones. The answer is that the two concepts are in one-to-one correspondence whenever the state space is a Polish topological space, on the traditional side equipped with its Borel σ algebra and on the new side with the lattice of its compact subsets. The correspondence is kind of a restriction and is as simple and natural as it could be, so that in practice the two

Stochastic Processes on the Basis of New Measure Theory

81

kinds of stochastic processes can be identified. In particular, the maximal measure for the new process is an extension of the canonical measure for the traditional process. The next question is on the notion of essential subsets: it is whether in case of a Polish state space the above well-defined essential subsets for a new stochastic process are in reasonable connection with the dreamt-of essential subsets for the related traditional process. In [15]-[18] we considered the two typical examples of the Wiener and Poisson processes with state space R. Both times it turned out that in essence the new maximal measure lives on those subsets of the path space which are the classical examples of essential subsets in the traditional intuitive sense. This appears to be a pleasant confirmation, even though some contrast remains, for example with the set of c`adl`ag paths in the Poisson process. In conclusion we want to refer to some work of predecessors. In the particular frame of compactness the classical Kolmogorov projective limit theorem has a variant for Radon measures which first appeared 1943 in Kakutani [9]. In this frame then the foundational problems for stochastic processes have been attacked in the 1959 paper of Nelson [19] and in the 1972 and 1980 books of Tjur [21][22]. In particular [22] chapter 10 contains a number of results on the above notion of essential subsets. But it becomes visible that beyond compact Polish state spaces an adequate treatment requires the new measure-theoretic foundations laid down in [12][14]. For time-honoured evidence we invoke the discussions in [5] and in the historical note of [2] chapter IX. In this connection we also note that the work of Nelson [19] and Tjur [21][22] did not at all find due attention in the subsequent literature on stochastic processes. To be sure, the whole enterprise requires some trace of compactness. Yet the present work makes clear that this is not topological compactness, but rather the different and more flexible notion of set theoretical τ compactness, manifested in the formation of the lattice S in section 3 below. Thus the usual projective limit theorem for Radon measures on Hausdorff spaces is not nearly as good as claimed in [4] p.65, because it does not even cover the simple example 3.5 below. Rather it seems that the true adequate projective limit theorem appears first in the present 3.1 = [16] theorem 4 with its extension [16] theorem 3, of course with their roots in [15] section 4. In view of all this it is plain that in the present context of stochastic processes, after detachment from the traditional abstract theory of measure and integration, the need is not at all for foundation into topological concepts, but rather for a kind of measure-theoretic foundation which comprises the notions of regularity and τ continuity - in short for a conception as developed in [12][14] and sketched in section 2 below. For one more evidence, Bourbaki has still not fulfilled his promise of 1952 to develop probability in his Treatise.

1 The Traditional Stochastic Processes We fix an infinite index set T called the time domain, and a measurable space (Y, B), that is a nonvoid set Y equipped with a σ algebra B in Y , called the state space. One forms the T -fold product set X := Y T , called the path space, the members of which are the paths x = (xt )t∈T : T → Y . For t ∈ T let Ht : X → Y be the canonical projection x 7→ xt . In X = Y T one forms the finite-based product set system B[T ] := { Π Bt : Bt ∈ B ∀ t ∈ T with Bt = Y ∀∀ t ∈ T }, t∈T

82

H. König

where ∀∀ means for all except for finitely many, and the generated σ algebra A := Aσ(B[T ] ), which is the smallest σ algebra A in X such that the Ht : X → Y for all t ∈ T are measurable A − B. It is well known that for uncountable T the formation A is quite narrow, because its members A ∈ A are countably determined in the sense that A = {x ∈ X : (xt )t∈D ∈ E} for some nonvoid countable D ⊂ T and some E ⊂ Y D . It is this situation where the traditional notion of stochastic processes comes into existence: A stochastic process with time domain T and state space (Y, B), for T and (Y, B) for short, amounts to be a prob measure α : A → [0, ∞[ on A. In view of the size of the measurable space (X, A) it is a nontrivial problem how to produce such stochastic processes. The standard method is via projective limits. Let I consist of the nonvoid finite subsets p, q, · · · of T . For p ∈ I one forms the product set Y p , with Hp : X → Y p the canonical projection x 7→ (xt )t∈p , and also the canonical projections Hpq : Y q → Y p for the pairs p ⊂ q in I. In Y p one forms the usual product set system Bp := B × · · · × B and the generated σ algebra Bp := Aσ(Bp ). Besides the prob measures α : A → [0, ∞[, that is the stochastic processes for T and (Y, B), one considers the families (βp )p∈I of prob measures βp : Bp → [0, ∞[ which are projective in the sense that −1 βp = βq (Hpq (·))|Bp for all pairs p ⊂ q in I (which makes sense because Hpq is measurable Bq − Bp ). Each prob measure α : A → [0, ∞[ produces such a projective family (βp )p∈I via βp = α(Hp−1(·))|Bp (which as before makes sense because Hp is measurable A − Bp ). One notes that the correspondence α 7→ (βp )p∈I is injective, but it need not be surjective. The projective family (βp )p∈I is called solvable iff it comes from some and hence from a unique prob measure α : A → [0, ∞[, called the projective limit of the family (βp )p∈I . Thus a stochastic process for T and (Y, B) can also be defined as a solvable projective family (βp )p∈I , called the family of finite-dimensional distributions of the process. There is a famous particular situation (Y, B) where all projective families (βp )p∈I for all T are solvable: it is the situation that Y is a Polish topological space and B = Bor(Y ) its Borel σ algebra. This is the projective limit theorem due to Kolmogorov [11] chapter III section 4. The fundamental fact behind it is that in a Polish space Y all finite (and all locally finite) measures on Bor(Y ) are inner regular with respect to the lattice Comp(Y ) of its compact subsets. The situation will be contained in the development of section 3 as a basic special case. 1.1 E XAMPLES. Let T = [0, ∞[ and Y = R with B = Bor(R). We fix a family (ϑt )t∈T of prob measures ϑt : B → [0, ∞[ with ϑ0 = δ0 |B which under convolution fulfils ϑs ⋆ ϑt = ϑs+t for all s, t ∈ T . One proves that it produces a projective family (βp )p∈I of prob measures βp : Bp → [0, ∞[, defined to be β{t} = ϑt for t ∈ T , and via induction for q = {t(0), t(1), · · · , t(n)} and p = {t(1), · · · , t(n)} with 0 ≦ t(0) < t(1) < · · · < t(n) to be
βq B(0)×B(1) × · · · × B(n) Z   βp−t(0) (B(1) − u) × · · · × (B(n) − u) dϑt(0) (u) = B(0)

with B(0), B(1), · · · , B(n) ∈ B. The Kolmogorov theorem then furnishes the stochastic process α : A → [0, ∞[. The most prominent examples are R −x2 /2t 1 e dx, and the Wiener process α for the ϑt : ϑt (B) = √2πt B

Stochastic Processes on the Basis of New Measure Theory

83

∞

the Poisson process α for the ϑt : ϑt (B) = e−t Σ (tl /l!)δl (B), l=0

with B ∈ B and t > 0. The Wiener process α : A → [0, ∞[ has the physical interpretation of the one-dimensional Brownian motion. The experimental observations quoted above say that ”all paths are continuous”. In mathematical context this statement has the intuitive sense that ”the subset C(T, R) of the path space X = RT is essential for the process α”. This cannot mean that C(T, R) has full measure for α, for C(T, R) is not countably determined and hence not in A. The traditional theory of stochastic processes proves that C(T, R) has outer α measure one, with the implications which result from the specialities of C(T, R). However, this result is clouded by the obvious fact that the complement X \ C(T, R) has outer α measure one as well. Thus it is reasonable to make a problem out of the true significance of full outer α measure. We start to describe the condition in a few lines in terms of some suitable equivalences, in independent notations up to the end of 1.2 below. Let K : Ω → X be a map between nonvoid sets Ω and X. For a σ algebra P in Ω one →

defines the direct image KP := {A ⊂ X : K −1 (A) ∈ P}, which is a σ algebra in X. For →

→

a measure P : P → [0, ∞] on P one defines the direct image KP : KP → [0, ∞] to be →

→

→

KP (A) = P (K −1 (A)) for A ∈ KP, which is a measure on KP, and a prob measure when P is one. If P lives on the subset T ⊂ Ω, that is if all N ⊂ Ω \ T are in P with P (N) = 0, then →

one verifies that KP lives on the image set K(T ) ⊂ K(Ω) ⊂ X. →

Next if A is a σ algebra in X, then A ⊂ KP means that K : Ω → X is measurable P − A →

in the usual sense. If α : A → [0, ∞] is a measure on A, then α = KP |A means that α is the image measure of P on A under K in the usual sense. In this case one also says that K : (Ω, P, P ) → X is a version of α. In these terms one has the equivalences [16] proposition 1 which follow. 1.2 P ROPOSITION. Let α : A → [0, ∞[ be a prob measure on the measurable space (X, A), and define its outer envelope α⋆ : P(X) → [0, ∞[ to be α⋆ (M) = inf{α(A) : A ∈ A with A ⊃ M}

for M ⊂ X.

For a subset C ⊂ X then the following are equivalent. 0) α⋆ (C) = 1 (in which case C is called thick for α). 1) α has a version K : (Ω, P, P ) → X with image K(Ω) ⊂ C. 1’) α has a version K : (Ω, P, P ) → X with image K(Ω) = C. 2) α has a measure extension ρ : R → [0, ∞[ with C ∈ R and ρ(C) = 1. 2’) α has a measure extension ρ : R → [0, ∞[ which lives on C. In this case α has a unique minimal measure extension ρ : R → [0, ∞[ which lives on C (minimal with respect to the inclusion ⊂ of domains). This is ρ : ρ(R) = α⋆ (R ∩ C) on R := {R ⊂ X : R ∩ C = A ∩ C for some A ∈ A}. After this we return to the situation of the present section. We repeat [16] theorem 1 which demonstrates in all sharpness that the condition of outer α measure one has no connection with reasonable notions of essentialness.

84

H. König

1.3 T HEOREM. Fix an arbitrary path a = (at )t∈T ∈ X and form C(a) := {x ∈ X : xt = at for all t ∈ T except countably many ones}. Then C(a) has α⋆ (C(a)) = 1 for all stochastic processes α for T and (Y, B). Thus after 1.2 each such α has versions K : (Ω, P, P ) → X with K(Ω) ⊂ C(a) and measure extensions ρ : R → [0, ∞[ which live on C(a). Note that C(a) is = X when T is countable, but is of obvious smallness when T is uncountable, and then X is the disjoint union of myriads of such C(a). Proof. Fix A ∈ A with A ⊃ C(a). We prove that A′ = ∅ and hence A = X. Let A′ = {x ∈ X : (xt )t∈D ∈ E} for some nonvoid countable D ⊂ T and E ⊂ Y D , and assume that A′ 6= ∅. Take u = (ut )t∈T ∈ A′ , and define x = (xt )t∈T to be xt = ut for t ∈ D and and xt = at for t ∈ T \ D. Then x ∈ A′ ⊂ (C(a))′ , whereas x ∈ C(a) by definition. Thus we obtain a contradiction. Under the impression of this absurd collection of thick subsets, and hence of measure extensions and of versions for each traditional stochastic process α : A → [0, ∞[ with uncountable time domain T , we proceed to our new structure in measure and integration, in the hope to find clarification and simplification. The basic step in this new structure are parallel outer and inner extension procedures for certain set functions. For historical reasons the outer versions look more familiar, but in recent years the inner versions became more and more authoritative. Thus the basis in the present context will be the inner τ version.

2 The Inner Extension Theories Let X be a nonvoid set. We start to recall the fundamental ideas 1914 of Carath´eodory [3] on the extension of set functions. On the one hand he defines for a set function Θ : P(X) → [0, ∞] with Θ(∅) = 0 the set system C(Θ) := {A ⊂ X : Θ(M) = Θ(M ∩ A) + Θ(M ∩ A′ ) ∀ M ⊂ X}, the members of which are called measurable Θ. It turns out that Θ|C(Θ) is a content on an algebra in X. On the other hand he defines for a set function ϕ : S → [0, ∞] on a set system S in X with ∅ ∈ S and ϕ(∅) = 0 the so-called outer measure ϕ◦ : P(X) → [0, ∞] to be ∞

∞

ϕ◦ (A) = inf{ Σ ϕ(Sl ) : (Sl )l in S with ∪ Sl ⊃ A}. l=0

l=0

His main theorem then reads as follows. If ϕ : S → [0, ∞] is a content on a ring and upward σ continuous, then ϕ◦ |C(ϕ◦ ) is a measure on a σ algebra in X and an extension of ϕ. In the traditional theory this theorem is the most fundamental tool in order to produce nontrivial measures. However, it has been under quite some criticism. In the traditional frame the attacks are towards the formation C(·), as an unmotivated and artificial one, while as a rule no doubt falls upon the outer measure formation ϕ 7→ ϕ◦ . But the new structure to be described below will disclose that the opposite is true: There are in fact serious deficiencies around the Carath´eodory theorem, but it is the particular form of his outer measure which must be blamed

Stochastic Processes on the Basis of New Measure Theory

85

for them, whereas the formation C(·) remains the decisive methodical idea and even improves when put into the adequate context. The main defects of the theorem are as follows. 1) The measure extension it produces is of an obvious outer regular character, like ϕ◦ itself. It is mysterious how an inner regular counterpart could look - while inner regular aspects become more and more important. 2) The measure extension it produces is of an obvious sequential character. It is mysterious how a nonsequential counterpart could look - while nonsequential aspects become more and more important. Both times the sum in the definition of ϕ◦ is a crucial obstacle. 3) The proof of the theorem suffers a complete breakdown as soon as one attempts to pass from rings S to less restrictive set systems like lattices - while lattices of subsets become more and more important. All these defects will disappear under the new structure in measure and integration, to which we proceed now. We shall be concerned with the inner theories, with likewise an obvious contrast to Bourbaki [2] from the start. Our ancestors are Kisy´nski [10] of 1968 and Topsøe [23] of 1970. Let as before X be a nonvoid set. We adopt a kind of shorthand notation, in that • = ⋆στ marks three parallel theories, where ⋆ stands for finite, σ for sequential or countable, and τ for nonsequential or arbitrary. As an example, for a nonvoid set system S in X let S• and S• denote the systems of the intersections and the unions of the nonvoid • subsystems of S. In the sequel we assume that S is a lattice in X with ∅ ∈ S and that ϕ : S → [0, ∞[ is an isotone set function with ϕ(∅) = 0. The basic definitions are as follows. We define an inner • extension of ϕ to be an extension α : A → [0, ∞] of ϕ which is a content on a ring, and such that moreover S• ⊂ A and α|S• is downward • continuous (note that α|S• < ∞), and α is inner regular S• . We define ϕ to be an inner • premeasure iff it admits inner • extensions. The subsequent inner extension theorem characterizes those ϕ which are inner • premeasures, and then describes all inner • extensions of ϕ. The theorem is in terms of the inner • envelopes ϕ• : P(X) → [0, ∞] of ϕ, defined to be ϕ• (A) = sup{ inf ϕ(M) : M ⊂ S nonvoid • with M ↓⊂ A}, M ∈M

where M ↓⊂ A means that M is downward directed with intersection contained in A. We also need their satellites ϕB • : P(X) → [0, ∞] with B ⊂ X, defined to be ϕB • (A) = sup{ inf ϕ(M) : M ⊂ S nonvoid • with M ∈M

M ↓⊂ A and M ⊂ B ∀M ∈ M}. We note that ϕ• is inner regular S• . Moreover ϕ = ϕ• |S iff ϕ is downward • continuous, and ϕ• (∅) = 0 iff ϕ is downward • continuous at ∅. 2.1 I NNER E XTENSION T HEOREM. Assume that ϕ : S → [0, ∞[ is isotone with ϕ(∅) = 0. Then ϕ is an inner • premeasure iff ϕ is supermodular and downward • continuous, and ϕ(B) ≦ ϕ(A) + ϕ• (B \ A) for all A ⊂ B in S.

86

H. König

Equivalent is ϕ is supermodular and downward • continuous at ∅, and ϕ(B) ≦ ϕ(A) + ϕB • (B \ A) for all A ⊂ B in S. In this case Φ := ϕ• |C(ϕ• ) is an inner • extension of ϕ, and a measure on a σ algebra when • = στ ; also Φ is complete. All inner • extensions of ϕ are restrictions of Φ. Moreover we have the localization principle which reads for A ⊂ X: S ∩ A ∈ C(ϕ• ) for all S ∈ S =⇒ A ∈ C(ϕ• ). Thus we have S ⊂ S• ⊂ C(ϕ• ). It is plain that the members of S• are the most basic measurable subsets. The prominent rˆole of ϕ• |C(ϕ• ) as the unique maximal inner • extension of ϕ emphasizes the fundamental nature of Carath´eodory’s formation C(·). There is no such fact in the traditional context: If ϕ : S → [0, ∞] is an upward σ continuous content on a ring S in X then ϕ◦ |C(ϕ◦ ) need not be a maximal measure extension of ϕ (for example for S = {∅, X} and ϕ 6= 0 one has ϕ◦ |C(ϕ◦ ) = ϕ). We also note a special case of particular importance: S is called • compact iff each nonvoid • subsystem M ⊂ S fulfils M ↓ ∅ ⇒ ∅ ∈ M. It is obvious that in this case the above functions ϕ are all downward • continuous at ∅. Thus the second equivalent condition in 2.1 becomes much simpler. The most natural example is that X is a Hausdorff topological space with S = Comp(X). For an isotone set function ϕ : S → [0, ∞[ with ϕ(∅) = 0 then the three conditions • = ⋆στ in 2.1 turn out to be identical, and if fulfilled produce the same ϕ• and hence the same Φ = ϕ• |C(ϕ• ). In this case ϕ is called a Radon premeasure and Φ the maximal Radon measure which results from ϕ. The localization principle implies that C(ϕ• ) ⊃ Bor(X).

3 The New Stochastic Processes We fix as before an infinite set T called the time domain. But this time we assume the state space (Y, K) to consist of a nonvoid set Y and of a lattice K in Y which contains the finite subsets of Y and is • compact. We retain the path space X := Y T and the projections Ht : X → Y for t ∈ T . In X = Y T we form the finite-based product set system (K ∪ {Y })[T ] := { Π St : St ∈ K ∪ {Y } ∀ t ∈ T with St = Y ∀∀ t ∈ T }, t∈T

and S := ((K ∪ {Y })[T ] )⋆ . Thus S is a lattice in X with ∅, X ∈ S and is • compact after [13] 2.6. This formation is the decisive step in the new enterprise. Next we let as before I consist of the nonvoid finite subsets p, q, · · · of T . We retain for p ∈ I the product set Y p and the projection Hp : X → Y p , and for the pairs p ⊂ q in I the projections Hpq : Y q → Y p . In Y p we form the usual product set system Kp := K × · · · × K −1 and the generated lattice Kp = (Kp )⋆ . Note that Hp−1 (Kp ) ⊂ S, but as a rule Hpq (Kp ) 6⊂ Kq for p ⊂ q in I. We turn to the relevant set functions. These are on the one hand on X = Y T the inner • premeasures ϕ : S → [0, ∞[ with ϕ(X) = 1 (the inner • prob premeasures for short) with their maximal inner • extensions Φ = ϕ• |C(ϕ• ) (thus with Φ(X) = 1). On the other hand we

Stochastic Processes on the Basis of New Measure Theory

87

consider the families (ϕp )p∈I of inner • prob premeasures ϕp : Kp → [0, ∞[ with their Φp (thus −1 with Φp (Y p ) = 1), which are projective in the sense that ϕp = (ϕq )• (Hpq (·))|Kp for all p ⊂ q in I. These entities are connected via the present main result [16] theorem 4 which follows. It is a comprehensive counterpart of the classical Kolmogorov projective limit theorem invoked in section 1. 3.1 T HEOREM. The family of the maps ϕ 7→ ϕp := ϕ(Hp−1 (·))|Kp

for p ∈ I

defines a one-to-one correspondence between the inner • prob premeasures ϕ : S → [0, ∞[ and the projective families (ϕp )p∈I of inner • prob premeasures ϕp : Kp → [0, ∞[. It fulfils Φ(A) = inf Φp (Hp (A)) for A ∈ S• . p∈I

→

Moreover (ϕp )• = ϕ• (Hp−1 (·)) on P(Y p ) and Φp = H p Φ for all p ∈ I. We want to note that this projective limit theorem appears in [16] theorem 3 in an even more comprehensive version: instead of the fixed state space (Y, K) one admits a family of individual pairs (Yt , Kt ) for the t ∈ T . But for the present context the above specialization will be adequate. The present result appears to be much more favourable than the traditional one: This time all projective families (ϕp )p∈I deserve to be called solvable. Also the relations between these families (ϕp )p∈I and their projective limits ϕ look deeper than before. But the main benefit compared with the traditional situation is that in case • = τ the resultant prob measure Φ = ϕ• |C(ϕ• ) on X has an immense domain: In fact, even the most prominent subclass Sτ ⊂ C(ϕτ ) contains for example all A ⊂ X of the form A = Π Kt with Kt ∈ K ∪ {Y } ∀ t ∈ T , and hence t∈T

reaches far beyond the class of countably determined subsets. On the other side the result preserves the traditional one in that one admits all inner τ prob premeasures ϕ : S → [0, ∞[, and the projective limit theorem 3.1 asserts that all of them remain rooted in the finite subsets of T . Thus we feel entitled to define a stochastic process with time domain T and state space (Y, K), for short for T and (Y, K), to be an inner τ prob premeasure ϕ : S → [0, ∞[. The maximal inner τ extension Φ = ϕτ |C(ϕτ ) of ϕ will be called its maximal measure. We proceed to the comparison with the traditional situation in the most fundamental particular case. The result is [16] theorem 5. Its proof combines the above theorems 2.1 and 3.1 with the basic properties of Polish spaces. 3.2 T HEOREM. Assume that Y is a Polish space with B = Bor(Y ) and K = Comp(Y ). There is a one-to-one correspondence between the traditional stochastic processes α : A → [0, ∞[ for T and (Y, B), and the new stochastic processes ϕ : S → [0, ∞[ for T and (Y, K). The correspondence rests upon S ⊂ A ⊂ C(ϕτ ) and reads ϕ = α|S and α = Φ|A. Moreover ϕτ = (α⋆ |Sτ )⋆ ≦ α⋆ . Proof of the final assertion. We have ϕ⋆ ≧ α⋆ ≧ Φ⋆ and hence ϕ⋆ |Sτ ≧ α⋆ |Sτ ≧ Φ⋆ |Sτ = Φ|Sτ = ϕτ |Sτ . Now ϕ⋆ |Sτ = ϕτ |Sτ because ϕτ |Sτ is downward τ continuous. Therefore ϕτ |Sτ = α⋆ |Sτ , and hence ϕτ = (ϕτ |Sτ )⋆ = (α⋆ |Sτ )⋆ since ϕτ is inner regular Sτ .

88

H. König

Thus in the present particular case the situation is as claimed in the introduction: we have a universal extension procedure which assigns to each traditional stochastic process α : A → [0, ∞[ the maximal measure Φ = ϕτ |C(ϕτ ) of its counterpart ϕ = α|S. These are in fact simple and natural formulae. We complete the comparison with an addendum on the new essential subsets C ∈ C(ϕτ ) with Φ(C) = 1. 3.3 A DDENDUM. Assume that C ∈ C(ϕτ ) with Φ(C) = 1. Then α⋆ (C) = 1. Moreover the unique minimal measure extension ρ : R → [0, ∞[ of α obtained in 1.2 fulfils R ⊂ C(ϕτ ) and ρ = Φ|R. Proof. The final assertion in 3.2 implies that α⋆ (C) = 1. Next for R ∈ R we have on the one hand R ∩ C = A ∩ C for some A ∈ A and hence R ∩ C ∈ C(ϕτ ), and on the other hand R ∩ C ′ ∈ C(ϕτ ), because C ′ ∈ C(ϕτ ) with Φ(C ′ ) = 0 and Φ is complete. Thus R ∈ C(ϕτ ). For R ∈ R now Φ(R) = Φ(R ∩ C) ≦ α⋆ (R ∩ C) = ρ(R), and hence Φ = ρ on R. In the final section 4 we shall invoke the two typical examples with state space R defined above, in order to convince ourselves that the measure extension Φ has adequate behaviour with respect to its essential subsets. In the remainder of the present section we continue to assume a Polish state space Y . We equip X = Y T with the product topology and want to describe the partial connection of the new stochastic processes with the topological species of Radon premeasures. The result is [16] corollary 1. 3.4 T HEOREM. Let as before Y be a Polish space with B = Bor(Y ) and K = Comp(Y ), and let X = Y T be equipped with the product topology. 0) We have Sτ ⊂ Cl(X) (:= the closed subsets of X), and Comp(X) = {S ∈ Sτ : S ⊂ some F ∈ KT } ⊂ Sτ , with KT the usual product set system. In particular Comp(X) = Sτ iff Y is compact. 1) Let ϕ : S → [0, ∞[ be an inner τ prob premeasure, and assume that sup{Φ(S) : S ∈ Comp(X)} = 1. Then φ := ϕτ |Comp(X) is a Radon premeasure with φτ = ϕτ . Hence Φ = φτ |C(φτ ) is maximal Radon. We shall see that the assumption in 1) is fulfilled for the two examples in section 4. But there are natural situations where this assumption is violated. We conclude with a simple example (which makes sense in the full frame of the present section). 3.5 E XAMPLE. Let the ϑt : K → [0, ∞[ for t ∈ T be inner τ prob premeasures, and the ϕp = Π ϑt for p ∈ I be their products in the sense of [13] section 1. Thus the ϕp : Kp → [0, ∞[ t∈p

are inner τ prob premeasures with
(ϕp )τ Π At = Π (ϑt )τ (At ) t∈p

t∈p

for At ⊂ Y ∀t ∈ p,

and hence form a projective family (ϕp )p∈I . Let ϕ : S → [0, ∞[ with Φ = ϕτ |C(ϕτ ) be the resultant stochastic process for T and (Y, K). We claim that if T is uncountable and ϑt < 1 on K for all t ∈ T then Φ|Comp(X) = 0, so that the assumption in 3.4.1) is violated. In fact, for

Stochastic Processes on the Basis of New Measure Theory

89

S ∈ Comp(X) we have S ⊂ some F ∈ KT , that is F = Π Kt with Kt ∈ K ∀ t ∈ T . For t∈T

p ∈ I thus

Φ(S) ≦ Φ(F ) ≦ Φ Π Kt × Y T \p = ϕp Π Kt = Π ϑt (Kt ). t∈p

t∈p

t∈p

Now there exists an uncountable M ⊂ T such that ϑt (Kt ) ≦ some c < 1 for all t ∈ M. It follows that Φ(S) ≦ ccard(p) for all p ⊂ M and hence that Φ(S) = 0.

4 Specializations and Examples The present section assumes T = [0, ∞[, and for the initial part a Polish state space Y with B = Bor(Y ) and K = Comp(Y ) as before. We consider in the path space X = Y T a few subsets of particular importance C = C(T, R) ⊂ D ⊂ E ⊂ F ⊂ X = Y T , defined as follows: F consists of the paths x : T → Y which possess all one-sided limits − x± t ∈ Y for t ∈ T , with the convention x0 := x0 . Then E consists of the paths x ∈ F which at each t ∈ T are either left or right continuous, and D of the paths x ∈ F which are right continuous at all t ∈ T , the so-called c`adl`ag ones. Note that all these subsets are not countably determined, and hence are not in A. However, the first main result in [17] theorem 1.1 asserts that for each pair of stochastic processes α : A → [0, ∞[ and ϕ : S → [0, ∞[ as in 3.2 one has the remarkable fact which follows. The proof combines ideas from Nelson [19] and Tjur [21][22] with the Choquet capacitability theorem. 4.1 T HEOREM. Assume that Y fulfils condition COMP: There exists a sequence of compact subsets K(n) ⊂ Y ∀n ∈ N such that each compact K ⊂ Y is contained in some K(n). For each pair of stochastic processes α and ϕ then C = C(T, R) and E, F are members of C(ϕτ ) and fulfil α⋆ (·) = Φ(·). After this we specialize to Y = R and turn to the one-dimensional Wiener and Poisson processes as defined in section 1. We want to obtain some basic examples of essential subsets. We start with the Wiener process α. The basic point for the sequel is the well-known relation Z |Ht − Hs |a dα = (t − s)a/2 M(a) for 0 ≦ s < t, with some constant M(a) < ∞ for all a > 0. The main result [15] theorem 6.1 which follows needs but a weakened form of this relation. 4.2 T HEOREM (G ENERALIZED W IENER P ROCESS ). Assume that the stochastic process α fulfils Z |Ht − Hs |a dα ≦ c(t − s)1+b for 0 ≦ s < t,

90

H. König

with some real constants a, b, c > 0. Fix 0 < γ ≦ 1 with γ < b/a, and for m ∈ N define Em (γ) := {x ∈ X :|x0 | ≦ m and |xv − xu | ≦ m2(u∨v)(1−γ) |v − u|γ ∀ u, v ∈ T }. σ Then Em (γ) ∈
Comp(X) ⊂ Sτ . For m → ∞ we have Em (γ) ↑ some E(γ) ∈ (Sτ ) ⊂ C(ϕτ ) with Φ E(γ) = 1. Thus the maximal measure Φ lives on E(γ). E(γ) is a certain class of locally H¨older continuous functions with exponent γ on T . In particular E(γ) ⊂ C(T, R), so that Φ likewise lives on C(T, R). After 3.4.1) also Φ is maximal Radon. The Wiener process α itself fulfils all this with the exponents 0 < γ < 1/2. We pass to the Poisson process α. We start to recall the main result [16] theorem 6. For t ∈ R define [t] := the largest integer ≦ t and {t} := the smallest integer ≧ t. 4.3 T HEOREM (P OISSON P ROCESS ). For m ∈ N define Zm ⊂ X (called Em (T ) in [16]) to consist of the x ∈ X such that x is integer valued with x0 = 0 and increasing, and
xv − xu ≦ {(1/2) {2n v} − [2n u] } ≦ {2n (v − u)} for all 0 ≦ u < v ≦ n and m ≦ n ∈ N. − Then Zm ∈ Comp(X) ⊂ Sτ , and the x ∈ Zm fulfil x+ t − xt ≦ 1 for all t ∈ T . For m → ∞ we have Zm ↑ some Z ∈ (Sτ )σ ⊂ C(ϕτ ) with Φ(Z) = 1 (called E(T ) in [16]). Thus the maximal measure Φ lives on Z. In view of 3.4.1) also Φ is maximal Radon. The properties of Z furnish at once that Z ⊂ E. It follows that Φ(E) = 1. It is a remarkable difference between our treatment of the Poisson process and the traditional one that the previous approach started from the discrete state space Y = Z instead of Y = R. After this we turn to the subset D ⊂ E which is more delicate. As in [17] section 4 we define X ◦ ⊂ X to consist of the paths x : T → R which are integer − valued with x0 = 0 and increasing, so that x ∈ F , which fulfil x+ t − xt ≦ 1 for all t ∈ T , so ◦ that x ∈ E, and are unbounded xt ↑ ∞ for t ↑ ∞. Then Z ∩ X consists of the unbounded members of Z. It follows from 4.3 combined with [16] proposition 7.2) that Z ∩ X ◦ ∈ C(ϕτ ) with Φ(Z ∩ X ◦ ) = 1. This implies that X ◦ ∈ C(ϕτ ) with Φ(X ◦ ) = 1. Next note that each path x ∈ X ◦ ⊂ E has an infinite sequence of jump points, each of height =1, which increases ↑ ∞. For each fixed n ∈ N define L(n) and R(n) to consist of those paths x ∈ X ◦ which are left/right continuous at the nth jump point. Thus X ◦ = L(n) ∪ R(n) and L(n) ∩ R(n) = ∅. Then the second main result in [17] theorem 5.2 reads as follows. Its last assertion (but not the quantitative first one) has a certain precedent in Tjur [22] 10.1.2 and 10.9.4. 4.4 T HEOREM. We have ϕτ (L(n)) = ϕτ (R(n)) = 0 for all n ∈ N. Thus L(n) and R(n) are nonmeasurable C(ϕτ ). T Now D ∩ X ◦ = R(n) from the definition. Therefore ϕτ (D ∩ X ◦ ) = 0, and combined n∈N

with X ◦ ∈ C(ϕτ ) and Φ(X ◦ ) = 1 we obtain

ϕτ (D) = ϕτ (D ∩ X ◦ ) + ϕτ (D ∩ (X ◦ )′ ) = 0.

Stochastic Processes on the Basis of New Measure Theory

91

It follows that the subsets D and D ∩ X ◦ are either nonmeasurable C(ϕτ ), or are in C(ϕτ ) with Φ(·) = 0. At the moment we do not know the answer. All these observations are in sharp contrast to the traditional result that α⋆ (D) = 1 and even α⋆ (D ∩ X ◦ ) = 1, which in the present frame results from [16] remark 8 and proposition 7.2). The traditional treatment of the Poisson process is in terms of D or rather of D ∩ X ◦ , that means in terms of measure extensions of the canonical measure α which live on these sets. We see that D and D ∩ X ◦ cannot maintain this rˆole when the treatment is based on the present ϕ and its maximal measure Φ. For the moment the final resolution of this problem, to be based on a definitive accord on the rˆole of the subsets D and D ∩ X ◦ , must remain open. The present author thinks that his subsequent article [18] can be an appropriate answer. Of course much will depend on when the traditional school of stochastic processes will start to face the present new approach. At last we want to note that in both of the above examples the pathological thick subsets C(a) for the a ∈ X in theorem 1.3 are measurable C(ϕτ ) with Φ(C(a)) = 0.

References [1] H. Bauer, Wahrscheinlichkeitstheorie. 4th ed. de Gruyter 1991, English translation 1996. [2] N. Bourbaki, Int´egration. Chap.1–4, 2i`eme ed. Hermann 1965, Chap.5, 2`eme ed. Hermann 1967, Chap.IX, Hermann 1969, English translation Springer 2004. ¨ [3] C. Carath´eodory, Uber das lineare Mass von Punktmengen - eine Verallgemeinerung des L¨angenbegriffs. Nachr. K. Ges. Wiss. G¨ottingen, Math.-Nat. Kl. 1914, pp. 404–426. Reprinted in: Gesammelte Mathematische Schriften, Vol. IV, pp. 249–275. C. H. Beck 1956. [4] C. Dellacherie and P. A. Meyer, Probability and Potential. North-Holland 1978. [5] J. L. Doob, Probability in function spaces. Bull. Amer. Math. Soc. 53(1947), 15–30. [6] J. L. Doob, Stochastic Processes. Wiley 1953. [7] D. H. Fremlin, Measure Theory. Vol.1–4 Torres Fremlin 2000–2003 (in the references the first digit of an item indicates its volume). http://www.essex.ac.uk/maths/staff/ fremlin/mt.htm. [8] W. Hackenbroch and A. Thalmaier, Stochastische Analysis. Teubner 1994. [9] S. Kakutani, Notes on infinite product measure spaces II. Proc. Imp. Acad. Tokyo 19(1943), 184–188. [10] J. Kisy´nski, On the generation of tight measures. Studia Math. 30(1968), 141–151. [11] A. Kolmogorov (= Kolmogoroff), Grundbegriffe der Wahrscheinlichkeitsrechnung. Springer 1933, Reprint 1973.

92

H. König

[12] H. K¨onig, Measure and Integration: An Advanced Course in Basic Procedures and Applications. Springer 1997. [13] H. K¨onig, The product theory for inner premeasures. Note di Mat. 17(1997), 235–249. [14] H. K¨onig, Measure and Integration: An attempt at unified systematization. Rend. Istit. Mat. Univ. Trieste 34(2002), 155–214. Preprint under http://www.math.uni-sb.de/ PREPRINTS/preprint42.pdf. [15] H. K¨onig, Projective limits via inner premeasures and the true Wiener measure. Mediterranean J. Math. 1(2004), 3–42. Preprint under http://www.math.uni-sb.de/PREPRINTS/ preprint83.pdf. [16] H. K¨onig, Stochastic processes in terms of inner premeasures. Note di Mat. Preprint under http://www.math.uni-sb.de/PREPRINTS/preprint105.pdf. [17] H. K¨onig, The new maximal measures for stochastic processes. Z. Analysis Anwendungen. Preprint under http://www.math.uni-sb.de/PREPRINTS/preprint 117.pdf. [18] H. K¨onig, Essential sets and support sets for stochastic processes. Preprint under http://www.math.uni-sb.de/PREPRINTS/preprint118.pdf. [19] E. Nelson, Regular probability measures on function spaces. Ann. Math. 69(1959), 630– 643. [20] K. R. Stromberg, Probability for Analysts. Chapman & Hall 1994. [21] T. Tjur, On the Mathematical Foundations of Probability. Lecture Notes 1, Inst. Math. Statist. Univ. Copenhagen 1972. [22] T. Tjur, Probability based on Radon Measures. Wiley 1980. [23] F. Topsøe, Topology and Measure. Lect. Notes Math. 133, Springer 1970.

Heinz K¨onig Universit¨at des Saarlandes Fakult¨at f¨ur Mathematik und Informatik 66041 Saarbr¨ucken, Germany [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 93-106 (2006)

Boolean Valued Analysis and Positivity A. G. Kusraev and S. S. Kutateladze (Russia) Abstract. This is an overview of the recent results of interaction of Boolean valued analysis and vector lattice theory. Key words: Boolean valued analysis, vector lattice, positive operator MSC 2000: 46 A 40

Boolean valued analysis is a general mathematical method that rests on a special modeltheoretic technique. This technique consists generally in comparison between the representations of arbitrary mathematical objects and theorems in two different set-theoretic models whose constructions start with principally distinct Boolean algebras. We usually take as these models the cosiest Cantorian paradise, the von Neumann universe of Zermelo–Fraenkel set theory, and a special universe of Boolean valued “variable” sets trimmed and chosen so that the traditional concepts and facts of mathematics acquire completely unexpected and bizarre interpretations. The use of two models, one of which is formally nonstandard, is a family feature of nonstandard analysis. For this reason, Boolean valued analysis means an instance of nonstandard analysis in common parlance. By the way, the term Boolean valued analysis was minted by G. Takeuti. Proliferation of Boolean valued models is due to P. Cohen’s final breakthrough in Hilbert’s Problem Number One. His method of forcing was rather intricate and the inevitable attempts at simplification gave rise to the Boolean valued models by D. Scott, R. Solovay, and P. Vopˇenka. Professor M. Weber had invited us to the Positivity Conference at the end of 2004 when we were completing our book “Introduction to Boolean Valued Analysis.” The book was recently published in Russian and so this article is a kind of presentation. Another recent event of relevance to this article is grievous. Saunders Mac Lane, a cofather of category theory, passed away in San Francisco on April 14, 2005. The power of mathematics rests heavily on the trick of socializing the objects and problems under consideration. The understanding of the social medium of set-theoretic models belongs to category theory. Topos theory provides a profusion of categories of which classical set theory is an ordinary member. Mathematics has thus acquired infinitely many new degrees of freedom. All these achievements rest on category theory. “There remains to us, then, the pursuit of truth, by way of proof, the concatenation of those ideas which fit, and the beauty which results when they do fit.” So wrote Saunders Mac Lane, a great genius, creator, master, and servant of mathematics. We reverently dedicate this article to the memory of this eternal and tragicomical mathematical Knight of the Sorrowful Figure.

94

A. G. Kusraev, S. S. Kutateladze

1 Boolean Requisites We start with recalling some auxiliary facts about the construction and treatment of Boolean valued models. 1.1. Let B be a complete Boolean algebra. Given an ordinal α, put

Vα(B) := {x : x is a function ∧ (∃β)(β < α (B) ⊂ Vβ ∧ im(x) ⊂ B)}.

∧ dom(x)

After this recursive definition the Boolean valued universe V(B) or, in other words, the class of B-sets is introduced by [ V(B) := Vα(B), α∈On

with On standing for the class of all ordinals. In case of the two element Boolean algebra 2 := {0, 1} this procedure yields a version of the classical von Neumann universe V (see 2.1 (2)). Let ϕ be an arbitrary formula of ZFC, Zermelo–Fraenkel set theory with choice. The Boolean truth value [ ϕ]] ∈ B is introduced by induction on the length of a formula ϕ by naturally interpreting the propositional connectives and quantifiers in the Boolean algebra B and taking into consideration the way in which this formula is built up from atomic formulas. The Boolean truth values of the atomic formulas x ∈ y and x = y, with x, y ∈ V(B) , are defined by means of the following recursion schema: _ [ x ∈ y]] = y(t) ∧ [ t = x]], t∈dom(y)

[ x = y]] =

_

t∈dom(x)

x(t) ⇒ [ t ∈ y]] ∧

_

y(t) ⇒ [ t ∈ x]].

t∈dom(y)

The sign ⇒ symbolizes the implication in B; i.e., a ⇒ b := a∗ ∨ b where a∗ is as usual the complement of a. The universe V(B) with the Boolean truth value of a formula is a model of set theory in the sense that the following statement is fulfilled. 1.2. Transfer Principle. For every theorem ϕ of ZFC, we have [ ϕ]] = 1; i.e., ϕ is true inside V(B) . Enter into the next agreement: If x is an element of V(B) and ϕ(·) is a formula of ZFC, then the phrase “x satisfies ϕ inside V(B) ” or, briefly, “ϕ(x) is true inside V(B) ” means that [[ϕ(x)]] = 1. This is sometimes written as V(B) |= ϕ(x). Given x ∈ V(B) and b ∈ B, define the function bx : z 7→ bx(z) (z ∈ dom(x)). Here we presume that b∅ := ∅ for all b ∈ B. 1.3. Mixing Principle. Let (bξ )ξ∈Ξ be a partition of unity in B, i.e. supξ∈Ξ bξ = sup B = 1 and ξ 6= η → bξ ∧ bη = 0. To each family (xξ )ξ∈Ξ in V(B) there exists a unique element x in the separated universe such that [ x = xξ ] ≥ bξ (ξ ∈ Ξ). P This element is called the mixing of (xξ )ξ∈Ξ by (bξ )ξ∈Ξ and is denoted by ξ∈Ξ bξ xξ . 1.4. Maximum Principle. If ϕ is a formula of ZFC then there is a B-valued set x0 satisfying [[(∃x)ϕ(x)]] = [ ϕ(x0 )]].

Boolean Valued Analysis and Positivity

95

2 The Escher Rules Boolean valued analysis consists primarily in comparison of the instances of a mathematical object or idea in two Boolean valued models. This is impossible to achieve without some dialog between the universes V and V(B) . In other words, we need a smooth mathematical toolkit for revealing interplay between the interpretations of one and the same fact in the two models V and V(B) . The relevant ascending-and-descending technique rests on the functors of canonical embedding, descent, and ascent. 2.1. We start with the canonical embedding of the von Neumann universe V. Given x ∈ V, we denote by x∧ the standard name of x in V(B) ; i.e., the element defined by the following recursion schema: ∅∧ := ∅, dom(x∧ ) := {y ∧ : y ∈ x}, im(x∧ ) := {1}. Observe some properties of the mapping x 7→ x∧ we need in the sequel. (1) For an arbitrary x ∈ V and a formula ϕ of ZFC we have _ [ (∃y ∈ x∧ ) ϕ(y)]] = [ ϕ(z ∧ )]], z∈x

[ (∀y ∈ x ) ϕ(y)]] = ∧

^

[ ϕ(z ∧ )]].

z∈x

(2) If x and y are elements of V then, by transfinite induction, we establish x ∈ y ↔ V(B) |= x ∈ y ∧ , x = y ↔ V(B) |= x∧ = y ∧ . In other words, the standard name can be considered as an embedding of V into V(B) . Moreover, it is beyond a doubt that the standard name sends V onto V(2) , which fact is demonstrated by the next proposition: (3) The following holds: (∀u ∈ V(2) ) (∃!x ∈ V) V(B) |= u = x∧ . A formula is called bounded or restricted if each bound variable in it is restricted by a bounded quantifier; i.e., a quantifier ranging over a particular set. The latter means that each bound variable x is restricted by a quantifier of the form (∀x ∈ y) or (∃x ∈ y) for some y. 2.2. Restricted Transfer Principle. For each bounded formula ϕ of ZFC and every collection x1 , . . . , xn ∈ V the following holds: ϕ(x1 , . . . , xn ) ↔ V(B) |= ϕ(x∧1 , . . . , x∧n ). Henceforth, (B) working in the separated universe V , we agree to preserve the symbol x∧ for the distinguished element of the class corresponding to x. Observe for example that the restricted transfer principle yields: ∧

“Φ is a correspondence from x to y”

↔ V(B) |= “Φ∧ is a correspondence from x∧ to y ∧ ”; “f : x → y” ↔ V(B) |= “f ∧ : x∧ → y ∧ ”

(moreover, f (a)∧ = f ∧ (a∧ ) for all a ∈ x). Thus, the standard name can be considered as a covariant functor of the category of sets (or correspondences) inside V to an appropriate subcategory of V(2) in the separated universe V(B) . 2.3. A set X is finite if X coincides with the image of a function on a finite ordinal. In symbols, this is expressed as fin(X); hence, fin(X) := (∃ n)(∃ f )(n ∈ ω ∧ f is a function ∧ dom(f ) = n ∧ im(f ) = X)

96

A. G. Kusraev, S. S. Kutateladze

(as usual ω := {0, 1, 2, . . . }). Obviously, the above formula is not bounded. Nevertheless there is a simple transformation rule for the class of finite sets under the canonical embedding. Denote by Pfin (X) the class of all finite subsets of X; i.e., Pfin (X) := {Y ∈ P(X) : fin(Y )}. For an arbitrary set X the following holds: V(B) |= Pfin (X)∧ = Pfin (X ∧ ). 2.4. Given an arbitrary element x of the (separated) Boolean valued universe V(B) , we define the descent x↓ of x as x↓ := {y ∈ V(B) : [ y ∈ x]] = 1}. We list the simplest properties of descending: (1) The class x↓ is a set, i.e., x↓ ∈ V for all x ∈ V(B) . If [ x 6= ∅]] = 1 then x↓ is a nonempty set. (2) Let z ∈ V(B) and [ z 6= ∅]] = 1. Then for every formula ϕ of ZFC we have ^ [ (∀x ∈ z) ϕ(x)]] = [ ϕ(x)]], x∈z↓

[ (∃x ∈ z) ϕ(x)]] =

_

[ ϕ(x)]].

x∈z↓

Moreover, there exists x0 ∈ z↓ such that [ ϕ(x0 )]] = [ (∃x ∈ z) ϕ(x)]]. (3) Let Φ be a correspondence from X to Y in V(B) . Thus, Φ, X, and Y are elements of V(B) and, moreover, [ Φ ⊂ X × Y ] = 1. There is a unique correspondence Φ↓ from X↓ to Y ↓ such that Φ↓(A↓) = Φ(A)↓ for every nonempty subset A of X inside V(B) . The correspondence Φ↓ from X↓ to Y ↓ of the above proposition is called the descent of the correspondence Φ from X to Y inside V(B) . (4) The descent of the composite of correspondences inside V(B) is the composite of their descents: (Ψ ◦ Φ)↓ = Ψ↓ ◦ Φ↓. (5) If Φ is a correspondence inside V(B) then (Φ−1 )↓ = (Φ↓)−1 . (6) Let IdX be the identity mapping inside V(B) of a set X ∈ V(B) . Then (IdX )↓ = IdX↓ . (7) Suppose that X, Y, f ∈ V(B) are such that [ f : X → Y ] = 1, i.e., f is a mapping from X to Y inside V(B) . Then f ↓ is a unique mapping from X↓ to Y ↓ satisfying [ f ↓(x) = f (x)]] = 1 for all x ∈ X↓. By virtue of (1)–(7), we can consider the descent operation as a functor from the category of B-valued sets and mappings (correspondences) to the category of the usual sets and mappings (correspondences) (i.e., in the sense of V). (8) Given x1 , . . . , xn ∈ V(B) , denote by (x1 , . . . , xn )B the corresponding ordered n-tuple inside V(B) . Assume that P is an n-ary relation on X inside V(B) ; i.e., X, P ∈ V(B) and ∧ [[P ⊂ X n ] = 1, where n ∈ ω. Then there exists an n-ary relation P ′ on X↓ such that (x1 , . . . , xn ) ∈ P ′ ↔ [ (x1 , . . . , xn )B ∈ P ] = 1. Slightly abusing notation, we denote the relation P ′ by the same symbol P ↓ and call it the descent of P . 2.5. Let x ∈ V and x ⊂ V(B) ; i.e., let x be some set composed of B-valued sets or, in other words, x ∈ P(V(B) ). Put ∅↑ := ∅ and dom(x↑) := x, im(x↑) := {1} if x 6= ∅. The element x↑ (of the separated universe V(B) , i.e., the distinguished representative of the class {y ∈ V(B) : [ y = x↑]] = 1}) is called the ascent of x.

Boolean Valued Analysis and Positivity

97

(1) For all x ∈ P(V(B) ) and every formula ϕ we have the following: ^ [ (∀z ∈ x↑) ϕ(z)]] = [ ϕ(y)]], y∈x

[ (∃z ∈ x↑) ϕ(z)]] =

_

[ ϕ(y)]].

y∈x

Introducing the ascent of a correspondence Φ ⊂ X × Y , we have to bear in mind a possible distinction between the domain of departure X and the domain dom(Φ) := {x ∈ X : Φ(x) 6= ∅}. This circumstance is immaterial for the sequel; therefore, speaking of ascents, we always imply total correspondences; i.e., dom(Φ) = X. (2) Let X, Y, Φ ∈ V(B) , and let Φ be a correspondence from X to Y . There exists a unique correspondence Φ↑ from X↑ to Y ↑ inside V(B) such that Φ↑(A↑) = Φ(A)↑ is valid for every subset A of dom(Φ) if and only if Φ is extensional; i.e., satisfies the condition y1 ∈ Φ(x1 ) → W [[x1 = x2 ]] ≤ y2 ∈Φ(x2 ) [ y1 = y2 ] for x1 , x2 ∈ dom(Φ). In this event, Φ↑ = Φ′ ↑, where Φ′ := {(x, y)B : (x, y) ∈ Φ}. The element Φ↑ is called the ascent of the initial correspondence Φ. (3) The composite of extensional correspondences is extensional. Moreover, the ascent of a composite is equal to the composite of the ascents inside V(B) : On assuming that dom(Ψ) ⊃ im(Φ) we have V(B)  (Ψ ◦ Φ)↑ = Ψ↑ ◦ Φ↑. Note that if Φ and Φ−1 are extensional then (Φ↑)−1 = (Φ−1 )↑. However, in general, the extensionality of Φ in no way guarantees the extensionality of Φ−1 . (4) It is worth mentioning that if an extensional correspondence f is a function from X to Y then the ascent f ↑ of f is a function from X↑ to Y ↑. Moreover, the extensionality property can be stated as follows: [ x1 = x2 ] ≤ [ f (x1 ) = f (x2 )]] for all x1 , x2 ∈ X. 2.6. Given a set X ⊂ V(B) , we denote by the symbol mix(X) the set of all mixings of the form mix(bξ xξ ), where (xξ ) ⊂ X and (bξ ) is an arbitrary partition of unity. The following propositions are referred to as the arrow cancellation rules or ascending-and-descending rules. There are many good reasons to call them simply the Escher rules. (1) Let X and X ′ be subsets of V(B) and let f : X → X ′ be an extensional mapping. Suppose that Y, Y ′ , g ∈ V(B) are such that [ Y 6= ∅]] = [ g : Y → Y ′ ] = 1. Then X↑↓ = mix(X), Y ↓↑ = Y, f ↑↓ = f, and g↓↑ = g. (2) From 2.3 (8) we easily infer the useful relation: Pfin (X↑) = {θ↑ : θ ∈ Pfin (X)}↑. Suppose that X ∈ V, X 6= ∅; i.e., X is a nonempty set. Let the letter ι denote the standard name embedding x 7→ x∧ (x ∈ X). Then ι(X)↑ = X ∧ and X = ι−1 (X ∧ ↓). Using the above relations, we may extend the descent and ascent operations to the case in which Φ is a correspondence from X to Y ↓ and [ Ψ is a correspondence from X ∧ to Y ] = 1, where Y ∈ V(B) . Namely, we put Φ↑ := (Φ ◦ ι)↑ and Ψ↓ := Ψ↓ ◦ ι. In this case, Φ↑ is called the modified ascent of Φ and Ψ↓ is called the modified descent of Ψ. (If the context excludes ambiguity then we briefly speak of ascents and descents using simple arrows.) It is easy to see that Ψ↑ is a unique correspondence inside V(B) satisfying the relation [ Φ↑(x∧ ) = Φ(x)↑]] = 1 (x ∈ X). Similarly, Ψ↓ is a unique correspondence from X to Y ↓ satisfying the equality Ψ↓(x) = Ψ(x∧ )↓ (x ∈ X). If Φ := f and Ψ := g are functions then these relations take the form [[f↑(x∧ ) = f (x)]] = 1 and g↓(x) = g(x∧ ) for all x ∈ X. 2.7. Various function spaces reside in functional analysis, and so the problem is natural of replacing an abstract Boolean valued system by some function-space analog, a model whose ele-

98

A. G. Kusraev, S. S. Kutateladze

ments are functions and in which the basic logical operations are calculated “pointwise.” An example of such a model is given by the class VQ of all functions defined on a fixed nonempty set Q and acting into V. The truth values on VQ are various subsets of Q: The truth value [[ϕ(u1 , . . . , un )]] of ϕ(t1 , . . . , tn ) at functions u1 , . . . , un ∈ VQ is calculated as follows: 
[ ϕ(u1 , . . . , un )]] = q ∈ Q : ϕ u1 (q), . . . , un (q) . A. G. Gutman and G. A. Losenkov solved the above problem by the concept of continuous polyverse which is a continuous bundle of models of set theory. It is shown that the class of continuous sections of a continuous polyverse is a Boolean valued system satisfying all basic principles of Boolean valued analysis and, conversely, each Boolean valued algebraic system can be represented as the class of sections of a suitable continuous polyverse. More details are collected in [15, Chapter 6]. 2.8. Every Boolean valued universe has the collection of mathematical objects in full supply: available in plenty are all sets with extra structure: groups, rings, algebras, normed spaces, etc. An abstract Boolean set or set with B-structure is a pair (X, d), where X ∈ V, X 6= ∅, and d is a mapping from X × X to B such that d(x, y) = 0 ↔ x = y; d(x, y) = d(y, x); d(x, y) ≤ d(x, z) ∨ d(z, y) all x, y, z ∈ X. To obtain an easy example of an abstract B-set, given ∅ 6= X ⊂ V(B) put d(x, y) := [ x 6= y]] = ¬[[x = y]] for x, y ∈ X. Another easy example is a nonempty X with the discrete B-metric d; i.e., d(x, y) = 1 if x 6= y and d(x, y) = 0 if x = y. Let (X, d) be some abstract B-set. There exist an element X ∈ V(B) and an injection ι : X → X ′ := X ↓ such that d(x, y) = [ ιx 6= ιy]] for all x, y ∈ X and every element x′ ∈ X ′ admits the representation x′ = mixξ∈Ξ (bξ ιxξ ), where (xξ )ξ∈Ξ ⊂ X and (bξ )ξ∈Ξ is a partition of unity in B. The element X ∈ V(B) is referred to as the Boolean valued realization of X. If X is a discrete abstract B-set then X = X ∧ and ιx = x∧ for all x ∈ X. If X ⊂ V(B) then ι↑ is an injection from X↑ to X (inside V(B) ). A mapping f from a B-set (X, d) to a B-set (X ′ , d′ ) is said to be contractive if d(x, y) ≥ d′ (f (x), f (y)) for all x, y ∈ X. We see that an abstract B-set X embeds in the Boolean valued universe V(B) so that the Boolean distance between the members of X becomes the Boolean truth value of the negation of their equality. The corresponding element of V(B) is, by definition, the Boolean valued representation of X. In case a B-set X has some a priori structure we may try to furnish the Boolean valued representation of X with an analogous structure, so as to apply the technique of ascending and descending to the study of the original structure of X. Consequently, the above questions may be treated as instances of the unique problem of searching a well-qualified Boolean valued representation of a B-set with some additional structure. We call these objects algebraic B-systems. Located at the epicenter of exposition, the notion of an algebraic B-system refers to a nonempty B-set endowed with a few contractive operations and B-predicates, the latter meaning B-valued contractive mappings. The Boolean valued representation of an algebraic B-system appears to be a conventional two valued algebraic system of the same type. This means that an appropriate completion of

Boolean Valued Analysis and Positivity

99

each algebraic B-system coincides with the descent of some two valued algebraic system inside V(B). On the other hand, each two valued algebraic system may be transformed into an algebraic B-system on distinguishing a complete Boolean algebra of congruences of the original system. In this event, the task is in order of finding the formulas holding true in direct or reverse transition from a B-system to a two valued system. In other words, we have to seek here for some versions of the transfer or identity preservation principle of long standing in some branches of mathematics.

3 Boolean Valued Numbers Boolean valued analysis stems from the fact that each internal field of reals of a Boolean valued model descends into a universally complete Kantorovich space. Thus, a remarkable opportunity opens up to expand and enrich the treasure-trove of mathematical knowledge by translating information about the reals to the language of other noble families of functional analysis. We will elaborate upon the matter in this section. 3.1. Recall a few definitions. Two elements x and y of a vector lattice E are called disjoint (in symbols x ⊥ y) if |x| ∧ |y| = 0. A band of E is defined as the disjoint complement M ⊥ := {x ∈ E : (∀y ∈ M) x ⊥ y} of a nonempty set M ⊂ E. The inclusion-ordered set B(E) of all bands in E is a complete Boolean algebra with the Boolean operations: L ∧ K = L ∩ K,

L ∨ K = (L ∪ K)⊥⊥ ,

L∗ = L⊥

(L, K ∈ B(E)).

The Boolean algebra B(E) is often referred as to the base of E. A band projection in E is a linear idempotent operator in π : E → E satisfying the inequalities 0 ≤ πx ≤ x for all 0 ≤ x ∈ E. The set P(E) of all band projections ordered by π ≤ ρ ⇐⇒ π ◦ ρ = π is a Boolean algebra with the Boolean operations: π ∧ ρ = π ◦ ρ,

π ∨ ρ = π + ρ − π ◦ ρ,

π ∗ = IE − π

(π, ρ ∈ (E)).

Let u ∈ E+ and e ∧ (u − e) = 0 for some 0 ≤ e ∈ E. Then e is a fragment or component of u. The set E(u) of all fragments of u with the order induced by E is a Boolean algebra where the lattice operations are taken from E and the Boolean complement has the form e∗ := u − e. 3.2. A Dedekind complete vector lattice is also called a Kantorovich space or K-space, for short. A K-space E is universally complete if every family of pairwise disjoint elements of E is order bounded. (1) Theorem. Let E be an arbitrary K -space. Then the correspondence π 7→ π(E) determines an isomorphism of the Boolean algebras P(E) and B(E). If there is an order unity 1 in E then the mappings π 7→ π 1 from P(E) into E(E) and e 7→ {e}⊥⊥ from E(E) into B(E) are isomorphisms of Boolean algebras too. (2) Theorem. Each universally complete K -space E with order unity 1 can be uniquely endowed by multiplication so as to make E into a faithful f -algebra and 1 into a ring unity. In this f -algebra each band projection π ∈ P(E) is the operator of multiplication by π(1).

100

A. G. Kusraev, S. S. Kutateladze

3.3. By a field of reals we mean every algebraic system that satisfies the axioms of an Archimedean ordered field (with distinct zero and unity) and enjoys the axiom of completeness. The same object can be defined as a one-dimensional K-space. Recall the well-known assertion of ZFC: There exists a field of reals R that is unique up to isomorphism. Successively applying the transfer and maximum principles, we find an element R ∈ V(B) for which [[ R is a field of reals ] = 1. Moreover, if an arbitrary R ′ ∈ V(B) satisfies the condition [[R ′ is a field of reals ] = 1 then [ the ordered fields R and R ′ are isomorphic ] = 1. In other words, there exists an internal field of reals R ∈ V(B) which is unique up to isomorphism. By the same reasons there exists an internal field of complex numbers C ∈ V(B) which is unique up to isomorphism. Moreover, V(B) |= C = R ⊕ iR. We call R and C the internal reals and internal complexes in V(B) . 3.4. Consider another well-known assertion of ZFC: If P is an Archimedean ordered field then there is an isomorphic embedding h of the field P into R such that the image h(P) is a subfield of R containing the subfield of rational numbers. In particular, h(P) is dense in R. Note also that ϕ(x), presenting the conjunction of the axioms of an Archimedean ordered field x, is bounded; therefore, [ ϕ(R∧ ) ] = 1, i.e., [ R∧ is an Archimedean ordered field ] = 1. “Pulling” 3.2 (2) through the transfer principle, we conclude that [ R∧ is isomorphic to a dense subfield of R ] = 1. We further assume that R∧ is a dense subfield of R and C∧ is a dense subfield of C . It is easy to note that the elements 0∧ and 1∧ are the zero and unity of R. Observe that the equalities R = R∧ and C = C∧ are not valid in general. Indeed, the axiom of completeness for R is not a bounded formula and so it may thus fail for R∧ inside V(B) . 3.5. Look now at the descent R↓ of the algebraic system R. In other words, consider the descent of the underlying set of the system R together with descended operations and order. For simplicity, we denote the operations and order in R and R↓ by the same symbols +, · , and ≤. In more detail, we introduce addition, multiplication, and order in R↓ by the formulas z = x + y ↔ [ z = x + y ] = 1, z = x · y ↔ [ z = x · y ] = 1, x ≤ y ↔ [ x ≤ y ] = 1 (x, y, z ∈ R↓). Also, we may introduce multiplication by the usual reals in R↓ by the rule y = λx ↔ [ λ∧ x = y ] = 1 (λ ∈ R, x, y ∈ R↓). The fundamental result of Boolean valued analysis is Gordon’s Theorem which reads as follows: Each universally complete Kantorovich space is an interpretation of the reals in an appropriate Boolean valued model. Formally, we have the following 3.6. Gordon Theorem. Let R be the reals inside V(B) . Then R↓, (with the descended operations and order, is a universally complete K -space with order unity 1. Moreover, there exists an isomorphism χ of B onto P(R↓) such that χ(b)x = χ(b)y ↔ b ≤ [ x = y ] ,

for all x, y ∈ R↓ and b ∈ B.

χ(b)x ≤ χ(b)y ↔ b ≤ [ x ≤ y ]

Boolean Valued Analysis and Positivity

101

The converse is also true: Each Archimedean vector lattice embeds in a Boolean valued model, becoming a vector sublattice of the reals (viewed as such over some dense subfield of the reals). 3.7. Theorem. Let E be an Archimedean vector lattice, let R be the reals inside V(B) , and let  be an isomorphism of B onto B(E). Then there is E ∈ V(B) such that (1) E is a vector sublattice of R over R∧ inside V(B) ; (2) E ′ := E ↓ is a vector sublattice of R↓ invariant under every band projection χ(b) (b ∈ B) and such that each set of positive pairwise disjoint sets in it has a supremum; (3) there is an o-continuous lattice isomorphism ι : E → E ′ such that ι(E) is a coinitial sublattice of R↓; (4) for every b ∈ B the band projection in R↓ onto {ι((b))}⊥⊥ coincides with χ(b). Note also that E and R coincide if and only if E is Dedekind complete. Thus, each theorem about the reals within Zermelo–Fraenkel set theory has an analog in an arbitary Kantorovich space. Translation of theorems is carried out by appropriate general functors of Boolean valued analysis. In particular, the most important structural properties of vector lattices such as the functional representation, spectral theorem, etc. are the ghosts of some properties of the reals in an appropriate Boolean valued model. More details and references are collected in [15]. 3.8. The theory of vector lattices with a vast field of applications is thoroughly covered in many monographs (for instance, see [19, 23]). The credit for finding the most important instance among ordered vector spaces, an order complete vector lattice or K-space, is due to L. V. Kantorovich. This notion appeared in Kantorovich’s first article on this topic, where he wrote: “In this note, I define a new type of space that I call a semiordered linear space. The introduction of such a space allows us to study linear operations of one abstract class (those with values in such a space) as linear functionals.” Thus the heuristic transfer principle was stated for K-spaces which becomes the Ariadna thread of many subsequent studies. The depth and universality of Kantorovich’s principle are explained within Boolean valued analysis. 3.9. Applications of Boolean valued models to functional analysis stem from the works by E. I. Gordon and G. Takeuti. If B in 3.6 is the algebra of µ-measurable sets modulo µ-negligible sets then R↓ is isomorphic to the universally complete K-space L0 (µ) of measurable functions. This fact (for the Lebesgue measure on an interval) was already known to D. Scott and R. Solovay (see [15]). If B is a complete Boolean algebra of projections in a Hilbert space then R↓ is isomorphic to the space of selfadjoint operators A(B). These two particular cases of Gordon’s Theorem were intensively and fruitfully exploited by G. Takeuti (see the bibliography in [15]). The object R↓ for general Boolean algebras was also studied by T. Jech [8]. Theorem 3.7 was obtained by A. G. Kusraev [10]. A close result (in other terms) is presented in T. Jech’s article [8] where some Boolean valued interpretation is revealed of the theory of linearly ordered sets. More details can be found in [15].

4 Band Preserving Operators This section deals with the class of band preserving operators. Simplicity of these operators notwithstanding, the question about their order boundedness is far from trivial.

102

A. G. Kusraev, S. S. Kutateladze

4.1. Recall that a complex K-space is the complexification GC := G ⊕ iG of a real K-space G. A linear operator T : GC → GC is band preserving or contractive or a stabilizer if, for all f, g ∈ GC , from f ⊥ g it follows that T f ⊥ g. Disjointness in GC is defined just as in G (see 3.1), whereas |z| := sup{Re(eiθ z) : 0 ≤ θ ≤ π} for z ∈ GC . (1) Let EndN (GC ) stand for the set of all band preserving linear operators in GC , with G := R↓. Clearly, EndN (GC ) is a complex vector space. Moreover, EndN (GC ) becomes a faithful unitary module over the ring GC if we define gT as gT : x 7→ g · T x for all x ∈ G. This follows from the fact that multiplication by a member of GC is a band preserving operator and the composite of band preserving operators is band preserving too. (2) Denote by EndC∧ (C ) the element of V(B) representing the space of all C∧ -linear mappings from C to C . Then EndC∧ (C ) is a vector space over C∧ inside V(B) , and EndC∧ (C )↓ is a faithful unitary module over GC . 4.2. Following [11] it is easy to prove that a linear operator T in the K-space GC is band preserving if and only if T is extensional. Since each extensional mapping has an ascent, T ∈ EndN (GC ) has the ascent τ := T ↑ which is a unique internal functional from C to C such that [[τ (x) = T x]] = 1 (x ∈ GC ). We thus arrive at the following assertion: The modules EndN (GC ) and EndC∧ (C )↓ are isomorphic by sending each band preserving operator to its ascent. By Gordon’s Theorem this assertion means that the problem of finding a band preserving operator in G amounts to solving (for τ : C → C ) inside V(B) the Cauchy functional equation: τ (x + y) = τ (x) + τ (y) (x, y ∈ C ) under the subsidiary condition τ (λx) = λτ (x) (x ∈ C , λ ∈ C∧ ). As another subsidiary condition we may consider the Leibniz rule τ (xy) = τ (x)y+xτ (y)(in which case τ is called a C∧ -derivation) or multiplicativity τ (xy) = τ (x)τ (y). These situations are addressed in 4.5. W 4.3. An element g ∈ G+ is locally constant with respect to f ∈ G+ if g = ξ∈Ξ λξ πξ f for some numeric family (λξ )ξ∈Ξ and a family (πξ )ξ∈Ξ of pairwise disjoint band projections. A universally complete K-space GC is called locally one-dimensional if all elements of G+ are locally constant with respect to some order unity of G (and hence each of them). P Clearly, a Kspace GC is locally one-dimensional if each g ∈ GC may be presented as g = o- ξ∈Ξ λξ πξ 1f with some family (λξ )ξ∈Ξ ⊂ C and partition of unity (πξ )ξ∈Ξ ⊂ P(G). 4.4. A σ-complete Boolean algebra B is called σ-distributive if _ ^ ^ _ bn,m = bn,ϕ(n) . n∈N m∈N ϕ∈NN n∈N for every double sequence (bn,m )n,m∈N in B. As an example of a σ-distributive Boolean algebra we may take a complete atomic Boolean algebra, i.e., the boolean of a nonempty set. It is worth observing that there are nonatomic σ-distributive complete Boolean algebras (see [12, 5.1.8]). We now address the problem which is often referred to in the literature as Wickstead’s problem: Characterize the universally complete vector lattices spaces in which every band preserving linear operator is order bounded. According to 4.2, Boolean valued analysis reduces Wickstead’s problem to that of order boundedness of the endomorphisms of the field C viewed as a vector space and algebra over the field C∧ .

Boolean Valued Analysis and Positivity

103

4.5. Theorem. Let P be an algebraically closed and topologically dense subfield of the field of complexes C. The following are equivalent: (1) P = C; (2) every P-linear function on C is order bounded; (3) there are no nontrivial P-derivations on C; (4) each P-linear endomorphism on C is the zero or identity function; (5) there is no P-linear automorphism on C other than the identity. The equivalence (1) ↔ (2) is checked by using a Hamel basis of the vector space C over P. The remaining equivalences rest on replacing a Hamel basis with a transcendence basis (for details see [13]). Recall that a linear operator D : GC → GC is a C-derivation if D(f g) = D(f )g + f D(g) for all f, g ∈ GC . It can be easily checked that every C-derivation is band preserving. Interpreting Theorem 4.5 in V(B) , we arrive at 4.6. Theorem. If B is a complete Boolean algebra then the following are equivalent: (1) C = C∧ inside V(B) ; (2) every band preserving linear operator is order bounded in the complex vector lattice C ↓; (3) there is no nontrivial C-derivation in the complex f -algebra C ↓; (4) each band preserving endomorphism is a band projection in C ↓; (5) there is no band preserving automorphism other than the identity in C ↓. (6) the K -space R↓ is locally one-dimensional; (7) B is σ -distributive. 4.7. The question was raised by A. W. Wickstead in [22] whether every band preserving linear operator in a universally complete vector lattice is automatically order bounded. The first example of an unbounded band preserving linear operator was suggested by Yu. A. Abramovich, A. I. Veksler, and A. V. Koldunov in [1, 2]. The equivalence (1) ↔ (6) is trivial, whereas (2) ↔ (6) combines a result of Yu. A. Abramovich, A. I. Veksler, and A. V. Koldunov [1, Theorem 2.1] and that of P. T. N. McPolin and A. W. Wickstead [21, Theorem 3.2]. The equivalence (6) ↔ (7) was obtained by A. E. Gutman who also found an example of a purely nonatomic locally one-dimensional Dedekind complete vector lattice (see [7]). The equivalences (1) ↔ (3) ↔ (4) ↔ (5) belong to A. G. Kusraev [13].

5 Boolean Valued Positive Functionals A linear functional on a vector space is determined up to a scalar from its zero hyperplane. In contrast, a linear operator is recovered from its kernel up to a simple multiplier on a rather special occasion. Fortunately, Boolean valued analysis prompts us that some operator analog of the functional case is valid for each operator with target a Kantorovich space, a Dedekind complete vector lattice. We now proceed along the lines of this rather promising approach. 5.1. Let E be a vector lattice, and let F be a K-space with base a complete Boolean algebra B. By 3.2, we may assume that F is a nonzero space embedded as an order dense ideal in the universally complete Kantorovich space R↓ which is the descent of the reals R inside the separated Boolean valued universe V(B) over B. An operator T is F -discrete if [0, T ] = [0, IF ] ◦ T ; i.e., for all 0 ≤ S ≤ T there is some ∼ 0 ≤ α ≤ IF satisfying S = α ◦ T . Let L∼ a (E, F ) be the band in L (E, F ) spanned by F -

104

A. G. Kusraev, S. S. Kutateladze

∼ ⊥ ∧∼ discrete operators and L∼ )a and (E ∧∼ )d . d (E, F ) := La (E, F ) . By analogy we define (E The members of L∼ d (E, F ) are usually called F -diffuse. 5.2. As usual, we let E ∧ stand for the standard name of E in V(B) . Clearly, E ∧ is a vector lattice over R∧ inside V(B) . Denote by τ := T ↑ the ascent of T to V(B) . Clearly, τ acts from E ∧ to the ascent F ↑ = R of F inside the Boolean valued universe V(B) . Therefore, τ (x∧ ) = T x inside V(B) for all x ∈ E, which means in terms of truth values that [ τ : E ∧ → R]] = 1 and (∀x ∈ E) [[τ (x∧ ) = T x]] = 1. Let E ∧∼ stand for the space of all order bounded R∧ -linear functionals from E ∧ to R. Clearly, E ∧∼ := L∼ (E ∧ , R) is a K-space inside V(B) . The descent E ∧∼ ↓ of E ∧∼ is a K-space. Given S, T ∈ L∼ (E, F ), put τ := T ↑ and σ := S↑. 5.3. Theorem. For each T ∈ L∼ (E, F ) the ascent T ↑ of T is an order bounded R∧ -linear functional on E ∧ inside V(B) ; i.e., [ T ↑ ∈ E ∧∼ ] = 1. The mapping T 7→ T ↑ is a lattice isomorphism of L∼ (E, F ) and E ∧∼ ↓. Moreover, the following hold: (1) T ≥ 0 ↔ [ τ ≥ 0 ] = 1; (2) S is a fragment of T ↔ [ σ is a fragment of τ ] = 1; (3) T is a lattice homomorphism if and only if so is τ inside V(B) ; (4) T is F -diffuse ↔ [ τ is diffuse ] = 1; ∧∼ (5) T ∈ L∼ )a ] = 1; a (E, F ) ↔ [ τ ∈ (E ∼ ∧∼ (6) T ∈ Ld (E, F ) ↔ [ τ ∈ (E )d ] = 1. Since τ , the ascent of an order bounded operator T , is defined up to a scalar from ker(τ ), we infer the following analog of the Sard Theorem. 5.4. Theorem. Let S and T be linear operators from E to F . Then ker(bS) ⊃ ker(bT ) for all b ∈ B if and only if there is an orthomorphism α of F such that S = αT . We see that a linear operator T is, in a sense, determined up to an orthomorphism from the family of the kernels of the strata bT of T . This remark opens a possibility of studying some properties of T in terms of the kernels of the strata of T . 5.5. Theorem. An order bounded operator T from E to F may be presented as the difference of some lattice homomorphisms if and only if the kernel of each stratum bT of T is a vector sublattice of E for all b ∈ B. Straightforward calculations of truth values show that T+ ↑ = τ+ and T− ↑ = τ− inside V(B) . Moreover, [[ker(τ ) is a vector sublattice of E ∧ ] = 1 whenever so are ker(bT ) for all b ∈ B. Since the ascent of a sum is the sum of the ascents of the summands, we reduce the proof of Theorem 5.5 to the case of the functionals on using 5.3 (3). 5.6. Recall that a subspace H of a vector lattice E is a G-space or Grothendieck subspace (cp. [6, 18]) provided that H enjoys the following property:

(∀x, y ∈ H) (x ∨ y ∨ 0 + x ∧ y ∧ 0 ∈ H). By simple calculations of truth values we infer that [ ker(τ ) is a Grothendieck subspace of E ∧ ]] = 1 if and only if the kernel of each stratum bT is a Grothendieck subspace of E. We may now assert that the following appears as a result of “descending” its scalar analog. 5.7. Theorem. The modulus of an order bounded operator T : E → F is the sum of some pair of lattice homomorphisms if and only if the kernel of each stratum bT of T with b ∈ B is a Grothendieck subspace of the ambient vector lattice E .

Boolean Valued Analysis and Positivity

105

To prove the relevant scalar versions of Theorems 5.5 and 5.7, we use one of the formulas of subdifferential calculus (cp. [14]): 5.8. Theorems 5.5 and 5.7 were obtained by S. S. Kutateladze in [16, 17]. Note that the sums of lattice homomorphisms were first described by S. J. Bernau, C. B. Huijsmans, and B. de Pagter in terms of n-disjoint operators in [3]. A survey of some conceptually close results on n-disjoint operators is given in [12].

References [1] Abramovich Yu. A., Veksler A. I., and Koldunov A. V., “On disjointness preserving operators,” Dokl. Akad. Nauk SSSR, 289, No. 5, 1033–1036 (1979). [2] Abramovich Yu. A., Veksler A. I., and Koldunov A. V., “Disjointness preserving operators, their continuity, and multiplicative representation,” in: Linear Operators and Their Applications [in Russian], Leningrad: Leningrad Ped. Inst., 1981. [3] Bernau S. J., Huijsmans C. B., and de Pagter B., “Sums of lattice homomorphisms,” Proc. Amer. Math. Soc., 115, No. 1, 51–156 (1992). [4] Gordon E. I., “Real numbers in Boolean valued models of set theory and K-spaces,” Dokl. Akad. Nauk SSSR, 237, No. 4, 773–775 (1977). [5] Gordon E. I., “K-spaces in Boolean valued models of set theory,” Dokl. Akad. Nauk SSSR, 258, No. 4, 777–780 (1981). [6] Grothendieck A., “Une caract´erisation vectorielle-m´etrique des espaces L1 ,” Canad. J. Math., 4, 552–561 (1955). [7] Gutman A. E., “Locally one-dimensional K-spaces and σ-distributive Boolean algebras,” Siberian Adv. Math., 5, No. 2, 99–121 (1995). [8] Jech T. J., “Boolean-linear spaces,” Adv. in Math., 81, No. 2, 117–197 (1990). [9] Jech T., Lectures in Set Theory with Particular Emphasis on the Method of Forcing, Berlin: Springer-Verlag, 1971. [10] Kusraev A. G., “Numeric systems in Boolean valued models of set theory,” in: Proceedings of the VIII All-Union Conference in Mathematical Logic (Moscow), Moscow, 1986, p. 99. [11] Kusraev A. G., “On band preserving operators,” Vladikavkaz Math. J., 6, No. 3, 48–58 (2004). [12] Kusraev A. G., Dominated Operators, Dordrecht: Kluwer Academic Publishers, 2000. [13] Kusraev A. G., “Automorphisms and derivations in extended complex f -algebras,” Siberian Math. J., 46, No. 6 (2005).

106

A. G. Kusraev, S. S. Kutateladze

[14] Kusraev A. G. and Kutateladze S. S., Subdifferentials: Theory and Applications, Novosibirk: Nauka, 1992; Dordrecht: Kluwer Academic Publishers, 1995. [15] Kusraev A. G. and Kutateladze S. S., Introduction to Boolean Valued Analysis [in Russian], Moscow: Nauka, 2005. [16] Kutateladze S. S., “On differences of Riesz homomorphisms,” Siberian Math. J. 46, No. 2, 305–307 (2005). [17] Kutateladze S. S., “On Grothendieck subspaces,” Siberian Math. J., 46, No. 3, 489–493 (2005). [18] Lindenstrauss J. and Wulbert D. E., “On the classification of the Banach spaces whose duals are L1 -spaces,” J. Funct. Anal., 4, No. 3, 322–249 (1969). [19] Luxemburg W. A. J. and Zaanen A. C., Riesz Spaces. Vol. 1, Amsterdam; London: NorthHolland, 1971. [20] Maharam D., “On positive operators,” Contemporary Math., 26, 263–277 (1984). [21] McPolin P. T. N. and Wickstead A. W., “The order boundedness of band preserving operators on uniformly complete vector lattices,” Math. Proc. Cambridge Philos. Soc., 97, No. 3, 481–487 (1985). [22] Wickstead A. W., “Representation and duality of multiplication operators on Archimedean Riesz spaces,” Compositio Math., 35, No. 3, 225–238 (1977). [23] Zaanen A. C., Riesz Spaces. Vol. 2, Amsterdam etc.: North-Holland, 1983.

Anatoly Kusraev Institute of Applied Mathematics and Informatics Vladikavkaz, Russia Semen Kutateladze Sobolev Institute Novosibirsk, Russia

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 107-113 (2006)

The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions Giorgio Metafune (Lecce) and Abdelaziz Rhandi (Marrakesh) Abstract. We present in this note a direct PDE proof of the characterization of the dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions. If we assume in addition that the coefficients of the second order derivatives are C 1 , which means that the operator can be written in divergence form, then an alternative proof based on the theory of irreducible semigroups is given. Key words: Elliptic operators in divergence and non divergence form, first eigenvalue, Dirichlet boundary conditions, irreducible semigroups MSC 2000: Primary 35P05, 35J25; Secondary 47A75, 47D06, 47B60, 46N20

1

Introduction

In this short note we consider a uniformly elliptic operator A in the nondivergence form Au :=

N X i,j=1

aij Dij u +

N X

bi Di u − cu,

i=1

where we suppose that the coefficients are real-valued and aij = aji ∈ C α (Ω), bi ∈ C α (Ω), 0 ≤ c ∈ C α (Ω), and N X aij (x)ξi ξj ≥ ν|ξ|2 , ∀x ∈ Ω, ξ ∈ RN i,j=1

for a constant ν > 0. Here we assume also that Ω is an open connected bounded domain of RN such that ∂Ω ∈ C 2,α . It is known that T the operator A with domain Dp := W 2,p (Ω) ∩ W01,p (Ω), 1 < p < ∞, (resp. D∞ := {u ∈ p p, since u ∈ C 2,α (Ω) (see [3, Theorem 18, Chap.3, Sect.8]). Consequently, the eigenfunctions (and hence σ(A)) do not depend on p.

108

G. Metafune, A. Rhandi

Our aim is to characterise the dominant (or the leading) eigenvalue for A with Dirichlet boundary conditions (see e.g. [2, 6.5.2], [4, Theorem 8.38]). More precisely, we propose to prove the following result. Theorem 1.1. The spectral bound s(A) = sup{Re λ : λ ∈ σ(A)} satisfies (i) s(A) is the dominant eigenvalue of A. (ii) There exists a corresponding eigenfunction, which is positive within Ω. (iii) s(A) is simple; that is, dim ker(s(A) − A) = 1 In Section 2 we present a PDE proof based on parabolic and elliptic maximum principles and Hopf’s Lemma. Assuming more regularity on the coefficients aij (·) we give in Section 3 an alternative proof using the theory of irreducible semigroups on Banach lattices. For this proof we do not use Hopf’s lemma.

2

A PDE proof

Since σ(A) does not depend on p, let us fix p = 2. To show (i) we have to prove first that σ(A) 6= ∅. In fact, suppose that σ(A) = ∅. Then for every ω ∈ R there is Mω ≥ 1 such that kT (t)k ≤ Mω eωt ,

∀t ≥ 0,

(2.1)

since s(A) = ω0 (A), the growth bound of A, for positive semigroups on L2 -spaces (cf. [6, C-IV, Theorem 1.1]). Let λ > 0, 0 ≤ h ∈ Cc∞ (Ω), h 6= 0, and w ∈ C 2,α (Ω) such that ½ λw − Aw = h on Ω, w|∂Ω = 0. Using the maximum principle we get w ≥ 0 on Ω, and by the strong maximum principle we obtain w > 0 on Ω (cf. [4, Theorem 3.5]). Since h ∈ Cc∞ (Ω), there exists a > 0 such that h ≤ aw. Hence, 0 ≤ λw − Aw ≤ aw. On the other hand, the function u(t, x) := (T (t)w)(x) is the solution of  t > 0, x ∈ Ω,  ∂t u(t, x) − Au(t, x) = 0, u(t, x) = 0, x ∈ ∂Ω, t > 0,  u(0, x) = w(x), x ∈ Ω. Let us consider the function v(t, x) := e(λ−a)t w(x). Then v satisfies   ∂t v − Av = e(λ−a)t [(λ − a)w − Aw] ≤ 0, v(t, x) = 0, x ∈ ∂Ω,  v(0, x) = w(x).

The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions

109

So, by the parabolic maximum principle (cf. [9, Theorem 4.26]), we obtain 0 ≤ e(λ−a)t w(x) ≤ u(t, x) for t ≥ 0 and x ∈ Ω. Hence, kT (t)wk ≥ e(λ−a)t kwk,

∀t ≥ 0.

This contradicts (2.1). Therefore, σ(A) 6= ∅. This means that s(A) > −∞. Now, s(A) = ω0 (A)), since for every ε > 0 there is Mε ≥ 1 with kT (t)k ≤ Mε e(s(A)+ε)t , t ≥ 0 (here we use s(A) = ω0 (A)), we deduce that {Re λ > s(A)} ⊂ ρ(A) and for Re λ > s(A) and f ∈ X2 Z ∞ R(λ, A)f = e−λt T (t)f dt. 0

From the positivity of T (·) we get Z ∞ |R(λ, A)f | ≤ et Re λ T (t)|f | dt = R(Re λ, A)|f |

(2.2)

0

for f ∈ X2 . This implies that s(A) ∈ σ(A) (see [6, C-III, Corollary 1.4]). To prove (ii) let us consider (λn ) in (s(A),R∞), λn → s(A) such that limn→∞ kR(λn , A)|f |k = ∞ ∞. Take un := R(λn , A)|f |. Since un = 0 eλn t T (t)|f | dt, we have un ≥ 0. By assumptions we know that un ∈ H 2 (Ω) ∩ H01 (Ω), λn un − Aun = |f | and limn→∞ kun k = ∞. So, by taking vn := kuunn k , it follows that vn ∈ H 2 (Ω) ∩ H02 (Ω) and λn vn − Avn =

|f | . kun k

(2.3)

Hence the graph norm of vn is bounded. Thus, vn is bounded in H 2 (Ω) and therefore there is a subsequence (vnk ) of (vn ) converging weakly to v in H 2 (Ω) and hence strongly in H 1 (Ω), by Rellich-Kondrachov’s theorem. This implies that 0 ≤ v ∈ H01 (Ω), kvk = 1 and limk→∞ Avnk = Av weakly in L2 (Ω). Take now the limit in (2.3) we get s(A)v − Av = 0. Thus, A has a non-negative eigenfunction corresponding to s(A). Let us prove now that v(x) > 0 for all x ∈ Ω. From the elliptic regularity we know that v ∈ C 2,α (Ω). Suppose that there is x0 ∈ Ω such that v(x0 ) = 0. This means that x0 is an interior minimum with values zero. This contradicts the strong maximum principle. For (iii) we will use Hopf’s lemma (cf. [2, 6.4.2]). Since v(x) > 0 for all x ∈ Ω and v|∂Ω = 0, it ∂v follows from Hopf’s lemma that ∂ν (x) < 0 for all x ∈ ∂Ω, where ν is the outer unit normal. Let 0 6= u ∈ ker(s(A) − A). Then, u ∈ C 2,α (Ω) and u|∂Ω = 0. Since s(A) ∈ R, and by considering the real an imaginary part of u, one can assume that u is real. Changing u to −u, if necessary, ∂ (v − εu) < 0 on one can suppose that {u > 0} 6= ∅. For ε > 0 sufficiently small we have ∂ν ∂Ω. Hence, v − εu ≥ 0 on a neighbourhood of ∂Ω. In fact, assume that there is (xn ) ⊆ Ω such that dist(xn , ∂Ω) < n1 and (v − εu)(xn ) < 0. Since Ω is a compact, there is a subsequence of (xn ), which will be denoted also by (xn ), converging to x0 ∈ ∂Ω. Take yn = P∂Ω xn , where P∂Ω is a projection into the boundary ∂Ω of Ω. Using Taylor’s formula, we get (v − εu)(xn ) = ∇(v − εu)(yn ) · (xn − yn ) + o(|xn − yn |) ∂ = − (v − εu)(yn )|xn − yn | + o(|xn − yn |). ∂ν

110

G. Metafune, A. Rhandi

Hence,

∂ o(|xn − yn |) (v − εu)(yn ) > → 0 ( as n → ∞). ∂ν |xn − yn |

∂ This contradicts the fact that ∂ν (v − εu) < 0 on ∂Ω. On the other hand, since v − ε0 u ≥ 0 outside of a neighbourhood of ∂Ω for some ε0 sufficiently small, it follows that v − ε00 u ≥ 0 on Ω for some ε00 > 0. Therefore,

γ := sup{α > 0 : v − αu ≥ 0 on Ω} > 0. Moreover, γ < ∞, since {u > 0} 6= ∅. By setting w := v − γu, one can see that w ≥ 0 and satisfies ½ s(A)w − Aw = 0, w|∂Ω = 0. If w 6= 0 then, by the strong maximum principle and Hopf’s lemma, we obtain w > 0 on Ω and ∂ w < 0 on ∂Ω. Now, we repeat the same arguments as above and we get w − εu ≥ 0. Thus, ∂ν v − (γ + ε)u ≥ 0 for some ε > 0, which contradicts the definition of γ. This ends the proof of the theorem.

3

Proof using the theory of positive semigroups

In this section we assume furthermore that aij ∈ C 1 (Ω) for all i, j = 1, · · · N . Before giving an alternative proof based on the theory of positive semigroups, let recall first the definition of irreducible semigroups on Lp (Ω), 1 ≤ p < ∞. A positive semigroup T (·) generated by A on X = Lp (Ω) is called irreducible if there is no closed ideal other than {0} and X that is invariant under T (t) for all t > 0. A measurable subset Y of Ω is called T (t)-invariant if f = f χY implies T (t)f = χY T (t)f . On Lp (Ω) closed ideals are characterised as follows (see [11]): A subspace I of Lp (Ω) is a closed ideal if and only if there exists a measurable subset Y of Ω such that I = {ϕ ∈ X; ϕ(x) = 0 a.e. x ∈ Y }. It follows that T (·) is irreducible on X if and only if Ω and ∅ are the only invariant sets, modulo null sets. The following properties of irreducible semigroups are very useful for the proof of Theorem 1.1 (cf. [10, Proposition 2.5.3]). For the proof we need the following simple lemma (cf. [10, Lemma 1.1.14]). For completeness we include the proofs. Lemma 3.1. Let E be a totally ordered (this means x ∈ E ⇒ 0 ≤ x or x ≤ 0) real Banach lattice. Then dim E ≤ 1. P ROOF. Let 0 6= e ∈ E+ and x ∈ E. We consider the non-empty closed subsets C+ := {α ∈ R : αe ≥ x} and C− := {α ∈ R : αe ≤ x} of R. It is obvious that C+ ∪ C− = R. Since R is connected, it follows that C+ ∩ C− 6= ∅. Hence there is α ∈ R such that x = αe. Theorem 3.2. Assume that A is the generator of an irreducible positive semigroup T (·) on a complex Banach lattice E. Then the following assertions hold.

The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions

111

(a) Every positive eigenvector of A is a quasi-interior point. (b) Every positive eigenvector of A∗ is strictly positive. (c) If ker(s(A) − A∗ ) contains a positive element, then dim ker(s(A) − A) ≤ 1. P ROOF. (a) Let x be a positive eigenvector of A and Ex := lin[−x, x] the ideal generated by x. If λ is such that Ax = λx, then λ ∈ R. This follows from 1 x ≥ 0 and Ax = lim+ (T (t)x − x). t→0 t Then, T (t)x = eλt x for t ≥ 0. For y ∈ Ex there exists c ≥ 0 with |y| ≤ cx, and therefore |T (t)y| ≤ T (t)|y| ≤ ceλt x,

t ≥ 0.

Consequently, T (t)Ex ⊆ Ex holds for all t ≥ 0. Since 0 6= x ∈ Ex and T (·) is irreducible, it follows that Ex = E. (b) Let x∗ be a positive eigenvector of A∗ and λ its corresponding eigenvalue. By the same argument as above we have λ ∈ R and T (t)∗ x∗ = eλt x∗ for t ≥ 0. Hence, h|T (t)u|, x∗ i ≤ hT (t)|u|, x∗ i = h|u|, eλt x∗ i,

u ∈ E, t ≥ 0.

Thus, I := {u ∈ E : h|u|, x∗ i = 0} is a T (t)–invariant closed ideal for all t ≥ 0. Since x∗ = 6 0 we have I $ E and so by the irreducibility we obtain I = {0}. Therefore, x∗ À 0. (c) Let 0  x∗ ∈ ker(s(A) − A∗ ). It follows from (b) that x∗ is strictly positive. For x ∈ ker(s(A) − A) we have T−s(A) (t)x := e−s(A)t T (t)x = x and hence, |x| = |T−s(A) (t)x| ≤ T−s(A) (t)|x|,

t ≥ 0.

Thus, for t ≥ 0, h|x|, x∗ i ≤ hT−s(A) (t)|x|, x∗ i = h|x|, x∗ i. This implies that hT−s(A) (t)|x| − |x|, x∗ i = 0, and since x∗ À 0, we obtain T−s(A) (t)|x| = |x| for t ≥ 0. Therefore, |x| ∈ ker(s(A) − A). ¡ ¢+ Since T−s(A) (t)x ≤ T−s(A) (t)x+ , one can see by the same arguments as above that x+ ∈ ker(s(A) − A) and x− ∈ ker(s(A) − A). This implies that F := ER ∩ ker(s(A) − A) is a real sublattice of E. For x ∈ F we consider the ideal Ex+ (resp. Ex− ) generated by x+ (resp. x− ). Then, Ex+ and Ex− are T−s(A) (t)–invariant for all t ≥ 0. Since Ex+ and Ex− are orthogonal, it follows from the irreducibility of T−s(A) (·) that x+ = 0 or x− = 0. Consequently, F is totally ordered. So by Lemma 3.1, we have dim F = dim ker(s(A) − A) ≤ 1.

112

G. Metafune, A. Rhandi

Remark 3.3. Quasi-interior points of Lp (Ω)+ , 1 ≤ p < ∞, are exactly functions f in Lp (Ω) satisfying f (x) > 0 for a.e. x ∈ Ω (see [11]). P ROOF OF T HEOREM 1.1. Since aij ∈ C 1 (Ω) for all i, j = 1, . . . , N, it follows that A can be written in divergence form. Hence, it is proved, by abstract methods using positivity, that the semigroup T (·) generated by A with Dirichlet boundary conditions is irreducible on L2 (Ω) (see [8, Theorem 4.5]). (i) Let us prove first that σ(A) 6= ∅. Since σ(A) does not depend on p, we take p = 2. It follows from the compactness of the resolvent of A and de Pagter’s theorem (see [7], [5, Theorem 4.2.2]) that r(R(λ, A)) > 0 for λ > 0. So by the spectral mapping theorem for the resolvent, σ(A) 6= ∅. Since T (·) is a positive C0 -semigroup on L2 (Ω), it follows from [6, Theorem 1.1.CIII] that s(A) ∈ σ(A). (ii) Using Theorem 3.2.(a) and Remark 3.3, it suffices to prove that there is a non-negative eigenfunction corresponding to s(A). This was proved before (see (2.3)). (iii) From Theorem 3.2.(c) one has to show only the existence of 0 ≤ u∗ ∈ ker(s(A) − A∗ ). Since σ(A) = σ(A∗ ), and the resolvent of A is compact, we get σp (A) = σp (A∗ ). Hence, ker(s(A) − A∗ ) 6= {0}. Let 0 6= v ∗ ∈ ker(s(A) − A∗ ). Then es(A)t |v ∗ | = |T (t)∗ v ∗ | ≤ T (t)∗ |v ∗ |

(3.1)

for t ≥ 0. Choose 0 < v ∈ ker(s(A) − A), by (ii). Then, hT (t)∗ |v ∗ | − es(A)t |v ∗ |, vi = h|v ∗ |, T (t)v − es(A)t vi = 0. So, by (3.1), we obtain T (t)∗ |v ∗ | = es(A)t |v ∗ |,

t ≥ 0.

This implies 0 ≤ |v ∗ | ∈ ker(s(A) − A∗ ) and Theorem 1.1 is proved. Acknowledgement. The second author wishes to express his gratitude to the Alexander von Humboldt foundation for the financial support.

References [1] S. Agmon: On the eigenfunctions and the eigenvalues of general elliptic boundary value problems, Comm. Pure Appl. Math. 15 (1962), 119-147. [2] L.C. Evans: Partial Differential Equations, Graduate Studies in Mathematics v. 19, AMS 1998. [3] A. Friedman: Partial Differential Equations of Parabolic Type, Prentice Hall, New-Jersey, 1964. [4] D. Gilbarg and N.S. Trudinger: Elliptic Partial Differential Equations of Second Order, Springer-Verlag 2001. [5] P. Meyer-Nieberg: Banach Lattices, Springer-Verlag 1991. [6] R. Nagel (Ed.): One-Parameter Semigroups of Positive Operators, Lecture Notes in Math. 1184 Springer-Verlag 1986.

The dominant eigenvalue of nonsymmetric elliptic operators with Dirichlet boundary conditions

113

[7] B. de Pagter: Irreducible compact operators, Math. Z. 192 (1986), 149-153. [8] E.M. Ouhabaz: Analysis of Heat Equations on Domains, Princeton University Press 2005. [9] M. Renardy and R.C. Rogers: An Introduction to Partial Differential Equations, SpringerVerlag, Second Edition, 2004. [10] A. Rhandi: Spectral Theory for Positive Semigroups and Applications, Quaderno del’Universita di Lecce, Q1 (2002), 51 pages. [11] H.H. Schaefer: Banach Lattices and Positive Operators, Springer-Verlag 1974. [12] H.B. Stewart: Generation of analytic semigroups by strongly elliptic operators, Trans. Amer. Math. Soc. 199 (1974), 141-162. [13] H.B. Stewart: Generation of analytic semigroups by strongly elliptic operators under general boundary conditions, Trans. Amer. Math. Soc. 259 (1980), 299-310.

Giorgio Metafune Dipartimento di Matematica ”Ennio De Giorgi” Universit`a di Lecce C.P. 193, 73100 Lecce, Italy [email protected] Abdelaziz Rhandi Department of Mathematics University of Marrakesh B.P. 2390, 40000 Marrakesh, Morocco [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 115-129 (2006)

R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry B. dePagter (Delft) and W. J. Ricker (Eichstätt) Abstract. When is an R-bounded representation Φ : C(K) → L (X), with X a Banach space and K a compact Hausdorff space, induced via spectral integration against an R-bounded, L (X)-valued spectral measure defined on the Borel σ-algebra B(K)? Similarly, which representations Ψ : L1 (G) → L (X), b with G a lca group, arise as the spectral integral of an R-bounded spectral measure defined on B(G), b is the dual group? These questions, answered in [20] are discussed in further detail. Particular where G emphasis is given to examples and to formulating some open questions. The role of special classes of spaces X, especially Banach lattices, is also considered. Key words: R-boundedness, representations, spectral measure, Banach lattice MSC 2000: Primary 22D12, 46B20; Secondary 43A25, 46B42, 47B40, 47B60

The notion of R-boundedness for families of linear operators was formally introduced by E. Berkson and T.A. Gillespie in [4] (where it is called the R-property), although it was already implicit in earlier work of J. Bourgain, [3]. Since its conception in the mid–1990’s, R-boundedness has played an increasingly important role in various branches of functional analysis, operator theory, harmonic analysis and partial differential equations; see for example [1], [5], [6], [7], [13], [15], [16], [19], [20], [26], [27], and the references therein. Let K be a compact Hausdorff space and C(K) be the Banach algebra (relative to pointwise operations) of all C-valued, continuous functions on K equipped with the usual sup-norm k·k∞ ; its unit is the constant function 1. By L (X) we denote the Banach algebra of all bounded linear operators from the (complex) Banach space X into itself, equipped with the operator norm topology; its unit is the identity operator I on X. Given a continuous representation Φ : C(K) → L (X), there always exists a finitely additive spectral measure Q : B(K) → L (X ∗ ), where X ∗ is the dual Banach space of X and B(K) is the σ-algebra of all Borel subsets of K, such that each scalar measure hx, Qx∗ i : A 7→ hx, Q(A)x∗ i is regular and σ-additive on B(K) for each x ∈ X, x∗ ∈ X ∗ , and satisfies Z ∗ hΦ(f )x, x i = f dhx, Qx∗ i, f ∈ C(K). K

Under certain conditions on Φ, e.g. relative compactness of W1 (Φ) := {Φ(f ) : f ∈ C(K), kf k∞ ≤ 1}

(1)

for the weak operator topology, or geometric properties of X (e.g. X does not contain a copy of c0 ), it is known that each projection Q(A) is the dual operator P (A)∗ of some projection

116

B. dePagter, W. J. Ricker

P (A) ∈ L (X). So, Z Φ(f ) = f dP,

f ∈ C(K),

(2)

K

is then given via integration against a regular, σ-additive (in the strong operator topology; briefly sot) spectral measure P : B(K) → L (X); see [9], [10], [24], for example. In the recent paper [20], a detailed investigation is made of the interaction between Rboundedness of the representation Φ (always assumed to be continuous) and the existence of an R-bounded spectral measure P satisfying (2). These results are then used in [20] to characterize those representations Ψ : L1 (G) → L (X) which are R-bounded; applications are also presented. Here G is a locally compact abelian (briefly, lca) group and L1 (G) is the Banach algebra of all integrable functions (for Haar measure λ) equipped with convolution as its multiplication. The purpose of this note is to summarize certain salient features of [20] and to elaborate further on some of the results of [20]. Particular emphasis is given to presenting some relevant and non-trivial examples and to formulating some open questions. The role of some special classes of spaces X (especially Banach lattices) will also be discussed. So, let us begin. Given a Banach space X, a non-empty collection T ⊆ L (X) is called R-bounded if there exists M ≥ 0 such that  

2 1/2

2 1/2 Z 1 Z 1 n n

X

X



 rj (t)xj dt (3) rj (t)Tj xj dt ≤ M 



0 0 j=1

j=1

for all {Tj }nj=1 ⊆ T , all {xj }nj=1 ⊆ X and all n ∈ N \ {0}. Here {rj }∞ j=1 is the sequence of Rademacher functions on [0, 1]. Every R-bounded collection is obviously uniformly bounded. For more details on R-boundedness we refer to [5], [27]. In particular, the sot-closed absolute convex hull of any R-bounded collection is also R-bounded. The following result is from [20, Proposition 2.17 & Remark 2.18]. The proof is based on the methods of Banach lattices via a careful analysis of certain cyclic subspaces associated with Φ. Proposition 1. Let Φ : C(K) → L (X) be an R-bounded representation, that is, the set W1 (Φ) as given by (1) is R-bounded in L (X). Then there exists a regular, sot σ-additive spectral measure P : B(K) → L (X) satisfying (2). Moreover, P is necessarily R-bounded, that is, Ran(P ) := {P (A) : A ∈ B(K)} is R-bounded in L (X). Conversely, if P : B(K) → L (X) is any regular, sot σ-additive spectral measure which is R-bounded, then the representation ΦP : C(K) → L (X) defined by spectral integration via R f 7→ K f dP is necessarily R-bounded. Moreover, W1 (ΦP ) is contained in the operator norm closed set 4aco (Ran(P )), where “aco” denotes absolutely convex hull. A Banach space X has property (α), [21, Definition 2.1], if

2 Z 1Z 1 m X n

X

ε r (s)r (t)x

jk j k jk ds dt

0 0 j=1 k=1

2 Z 1Z 1 m X n

X

≤β rj (s)rk (t)xjk ds dt

0 0 j=1 k=1

R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry

117

for some constant β > 0 and every choice of xjk ∈ X, εjk ∈ {−1, 1} and all m, n ∈ N \ {0}. Every Banach lattice with finite cotype has property (α). The following consequence of Proposition 1 (see [20, Corollary 2.19]) illustrates how the geometry of X can influence the nature of C(K)-representations. Proposition 2. Let X have property (α). Then every representation Φ : C(K) → L (X) is necessarily R-bounded. In particular, there always exists a regular, R-bounded, sot σ-additive spectral measure P : B(K) → L (X) satisfying (2). Recall that a commuting family of projections M ⊆ L (X) containing 0, I and such that P Q ∈ M and (I − P ) ∈ M whenever P, Q ∈ M , is called a Boolean algebra (briefly, B.a.) of projections in X. Observe that M ∗∗ := {P ∗∗ : P ∈ M } is then a B.a. in the bidual X ∗∗ . Lemma 3. Let the Banach space X be either a Grothendieck space with the Dunford-Pettis property (briefly, GDP) or a hereditarily indecomposable space (briefly, H.I.). Then every Rbounded B.a. of projections in X is finite. Proof. Let M ⊆ L (X) be any R-bounded B.a. of projections. By [19, Theorem 6.6] the sot closure M s is Bade complete (in the sense of [10, p.2195]). If X is a GDP space apply [22], and if X is H.I. apply [23, Appendix], to conclude that M s and hence, also M , is finite. Examples of GDP-spaces include ℓ∞ , L∞ -spaces and H ∞ (D). Examples of H.I. spaces are due to W.T. Gowers and B. Maurey, [14]; also reflexive, separable, and even super-reflexive spaces of this kind are known, [2], [11]. For such kinds of spaces (i.e. GDP or H.I.), what is the connection between X and X ∗∗ ? If X ∗ is a H.I. space, then so is X (take a projection in X and consider its dual operator). Hence, if X ∗∗ is H.I., then so is X. What about the converse? The Banach space X given in Theorem 0.7 of [2] is H.I. but, X ∗ is not. The same is true of the space X in Theorem 0.1 of [2]. On the other hand, the space X in Theorem 0.3 of [2] has the property that both it and X ∗ are H.I. spaces, but X ∗∗ is not. Combining Lemma 3 above with Lemma 2.4 and Remark 2.26 of [20] yields the following Proposition 4. Let X be a H.I. space and Φ : C(K) → L (X) be any R-bounded representation. Then there exist finitely many points w1 , . . . , wn ∈ K and non-zero, pairwise disjoint projections P1 , . . . , Pn ∈ L (X) with P1 + . . . + Pn = I such that Φ(f ) =

n X

f (wj )Pj ,

f ∈ C(K).

(4)

j=1

Because of Lemma 3, one would expect a similar result to Proposition 4 for GPD-spaces. This is true, but there is a (potential) difference. Unlike for H.I. spaces, there are known examples of Banach spaces X which are not themselves GDP but, X ∗∗ is a GDP space (e.g. c0 ). Question 1. Does there exist X which is GDP, but X ∗∗ not? This seems to be unknown. Some comments are in order. Let X have the Schur property. Then X also has the DP property, [8, p.177]. An example of C. Stegall shows that there exist Schur spaces X such that X ∗ lacks the DP property, [8, p.178]. According to [8, Corollary, p.177] the space X ∗∗ then also lacks the DP property. So, the DP-property may be lost when

118

B. dePagter, W. J. Ricker

passing to the bidual. However, ℓ1 shows that Schur spaces need not be Grothendieck spaces (in general). Of course, Stegall’s space may be a Grothendieck space but, this seems to be difficult to decide. So, in view of Question 1, we have the following version of Proposition 4 for GDPspaces, [20, Proposition 2.25]; it relies on the fact that a subset T ⊆ L (X) is R-bounded if and only if T ∗∗ ⊆ L (X ∗∗ ) is R-bounded, [15, Lemma 2.4], [20, Lemma 2.20]. Proposition 5. Let X be a Banach space such that either X or X ∗∗ is a GDP space. Then every R-bounded representation Φ : C(K) → L (X) has the form (4). We now turn our attention to applying C(K)-representations to representations of the Bab The Fourier nach algebra L1 (G) under convolution, where G is a lca group with dual group G. transform of f ∈ L1 (G) is denoted by fb. A representation Ψ : L1 (G) → L (X), always S assumed to be continuous, is called essential if f ∈L1 (G) Ψ(f )X is dense in X. The following result (see [20, Proposition 3.2]) follows by applying Proposition 1 to the dense subalgebra b and by passing to C(G b∞ ), where G b∞ denotes the L1 (G)∧ := {fb : f ∈ L1 (G)} of C0 (G) b 1-point compactification of G. Proposition 6. Let Ψ : L1 (G) → L (X) be an essential representation which is Fourier Rbounded, that is, the set b ∞ ≤ 1} ⊆ L (X) V1 (Ψ) := {Ψ(f ) : f ∈ L1 (G), kfk

(5)

is R-bounded. Then there exists an R-bounded, regular, sot σ-additive spectral measure P : b → L (X) such that B(G) Z Ψ(f ) = fb dP, f ∈ L1 (G). (6) b G

b → Conversely, given any R-bounded, regular, sot σ-additive spectral measure P : B(G) L (X), the map Ψ defined via the spectral integrals (6) is an essential, Fourier R-bounded representation of L1 (G). Every essential representation Ψ : L1 (G) → L (X) has the form Z Ψ(f ) = f (g)U(g) dλ(g), f ∈ L1 (G),

(7)

G

for some unique homomorphism U : G → L (X) which is bounded (i.e. supg∈G kU(g)k < ∞) and sot continuous, [20, Remark 3.5]. Define the absolutely convex set Λ1 (Ψ) := {Ψ(f ) : f ∈ L1 (G), kf k1 ≤ 1} ⊆ L (X). Lemma 7. Let Ψ : L1 (G) → L (X) be an essential representation and U : G → L (X) satisfy (7). Then Λ1 (Ψ) = aco{U(g) : g ∈ G},

(8)

where the bars denote sot-closure in L (X). In particular, Ψ is R-bounded (meaning Λ1 (Ψ) is R-bounded) if and only if U is R-bounded (meaning {U(g) : g ∈ G} is R-bounded).

R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry

119

P Proof. Let ξ = nj=1 xj ⊗ x∗j denote an element of L (X)∗ . Suppose that |hU(g), ξi| ≤ 1 for all g ∈ G. Then, for each f ∈ L1 (G) satisfying kf k1 ≤ 1 we have n X |hΨ(f ), ξi| = hΨ(f )xj , x∗j i j=1 Z n X = f (g) hU(g)xj , x∗j i dλ(g) G j=1 Z ≤ |f (g)| · |hU(g), ξi| dλ(g) ≤ 1. G

This implies that Λ1 (Ψ) ⊆ aco{U(g) : g ∈ G}. Conversely, suppose that ξ now satisfies |hΨ(f ), ξi| ≤ 1 for all f ∈ L1 (G) with kf k1 ≤ 1. That is, Z n X ∗ hU(g)xj , xj i dλ(g) ≤ 1, kf k1 ≤ 1. f (g) G j=1

P Then h(g) := nj=1 hU(g)xj , x∗j i belongs to the unit ball of L∞ (G). But, h in continuous and so |h(g)| ≤ 1 for every g ∈ G, that is, |hU(g), ξi| = |h(g)| ≤ 1,

g ∈ G.

Accordingly, for all g ∈ G we have U(g) ∈ aco(Λ1 (Ψ)) = Λ1 (Ψ), as Λ1 (Ψ) is already absolutely convex. This establishes equality in (8). The final claim follows from the facts that the absolutely convex hull of an R-bounded set is R-bounded and the sot-closure of an R-bounded set is R-bounded. b ∞ ≤ kf k1 for all f ∈ L1 (G) it is clear that Since kfk Λ1 (Ψ) ⊆ V1 (Ψ),

(9)

with closures taken in the sot, after noting that both Λ1 (Ψ) and V1 (Ψ) are absolutely convex sets. b → L (X) as If the essential representation Ψ is given via a regular spectral measure P : B(G) in (6), then necessarily b ⊆ V1 (Ψ); Ran(P ) := {P (A) : A ∈ B(G)}

see Remark 2.14 and the proof S of Proposition 3.2 in [20]. The following example shows that it can happen that Ran(P ) * r>0 rΛ1(Ψ) !

Example 8. Let G = Z be the additive group of integers, in which case L1 (G) = ℓ1 (Z). Then b = T ≃ [0, 2π) is the circle group. Let X = Lp ([0, 2π)), where p ∈ [1, ∞) is arbitrary. For G each A ∈ B([0, 2π)), define P (A) ∈ L (X) to be the operator MχA of multiplication by χA . Then P : B([0, 2π)) → L (X) is a regular, sot σ-additive spectral measure and U : G → L (X) given by Z Z 2π n U(n) = z dP (z) = exp(−int) dP (t), n ∈ Z, T

0

120

B. dePagter, W. J. Ricker

is precisely the bounded homomorphism n 7→ T n , for n ∈ Z, where T ∈ L (X) is the operator Mϕ of multiplication by ϕ(t) := exp(it) for t ∈ [0, 2π). The map ψ 7→ Mψ is an isometric isomorphism of the Banach algebra L∞ ([0, 2π)) onto the (operator norm) closed subalgebra M∞ := {Mψ : ψ ∈ L∞ ([0, 2π))} of L (X). Via this identification, the weak operator topology restricted to M∞ coincides with the weak-∗ ∞ 1 ∞ 1 topology σ(L , L ) on L ([0, 2π)). Moreover, the representation Ψ : ℓ (Z) → L (X), given R by f 7→ T fb dP = Mfb, for f ∈ ℓ1 (Z), takes its values in M∞ . Accordingly, the set Λ1 (Ψ) can be identified with the subalgebra A ⊆ L∞ ([0, 2π)) consisting of the σ(L∞ , L1 )-closure of nX o X int an e : |an | ≤ 1 . n∈Z

n∈Z

1 If b ∞ ≤ 1, then a bipolar argument shows that h ∈ RA and w ∈ L ([0, 2π)) satisfies kwk −1 2π h(t)w(t) dt ≤ 1. Since the Dirichlet kernel {wN }∞ (2π) N =1 , given by 0

wN (t) :=

X

eikt ,

t ∈ [0, 2π),

|k|≤N

(and which satisfies w bN ∞ ≤ 1 for N ∈ N) is unbounded in L1 ([0, 2π)), we can choose N large enough such that kwN k1 ≥ 20, say. So, there exists A ∈ B([0, 2π)) satisfying R (2π)−1 A |wN (t)| dt ≥ 5 and hence, χA ∈ / A. Accordingly, P (A) ∈ / Λ1 (Ψ). Actually, given any r > 0, a similar argument (using kwN k1 → ∞ as N → ∞) shows that there exists Ar ∈ B([0, 2π)) such that P (Ar ) ∈ / rΛ1 (Ψ). In view of Example 8, some further comments concerning (9) and Lemma 7 are in order. For f ∈ L1 (G), define fe(g) := f (−g) for g ∈ G, in which case f 7→ fe is a continuous involution in L1 (G). Let Ψ : L1 (G) → L (X) be any essential representation and Ran(Ψ)u denote the b : L1 (G)∧ → Ran(Ψ) denote the operator norm closure in L (X) of the range of Ψ. Let Ψ u b (well defined) linear map fb 7→ Ψ(f ) for f ∈ L1 (G), where we consider L1 (G)∧ ⊆ C0 (G). Suppose that Ran(Ψ)u is Banach algebra isomorphic to a closed subalgebra of C(M), for some compact Hausdorff space M, so that b fb)k = kΨ(f )k ≤ K sup |Ψ(f )(m)|, kΨ(

f ∈ L1 (G),

(10)

m∈M

for some constant K > 0. Then, for each m ∈ M, the map f 7→ Ψ(f )(m) is a linear, multiplicative functional on the Banach algebra L1 (G) and hence, is automatically bounded. b such that Ψ(f )(m) = According to [25, Theorem 1.2.2], for each m ∈ M there exists γm ∈ G fb(γm ) and hence, by (10), we have b fb)k ≤ K sup |fb(γm )| ≤ K kfbk∞ . kΨ( m

b and (10), there exists α > 0 such that By density of L1 (G)∧ in C0 (G) kΨ(f )k ≤ α,

f ∈ L1 (G), kfbk∞ ≤ 1.

So, we have established the following

R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry

121

Lemma 9. Let Ψ : L1 (G) → L (X) be an essential representation such that Ran(Ψ)u is Banach algebra isomorphic to a closed subalgebra of C(M) for some compact Hausdorff space M. Then the set V1 (Ψ) as given by (5) is uniformly bounded in L (X). Suppose now that X is a Hilbert space and U : G → L (X) is the unique homomorphism satisfying (7). According to a result of B. Sz.-Nagy, [10, XV Lemma 6.1], there exists a selfadjoint isomorphism T ∈ L (X) such that W (g) := T −1 U(g)T,

g ∈ G,

is a unitary homomorphism. Then Z −1 ΨW (f ) := T Ψ(f )T = f (g)W (g) dλ(g),

f ∈ L1 (G),

G

is a ∗-representation in L (X). Accordingly, Ran(ΨW )u is an abelian C ∗ -algebra and hence, is isomorphic to a C(M)-space. From Ran(Ψ) = T Ran(ΨW )T −1 it follows that Ran(Ψ)u is also isomorphic to C(M). According to Lemma 9 we see that V1 (Ψ) is uniformly bounded in L (X). Since R-bounded subsets and uniformly bounded subsets of L (X) coincide (recall X is a Hilbert space) we have established the following fact. Proposition 10. Let X be a Hilbert space and Ψ : L1 (G) → L (X) be any essential representation. Then (i) V1 (Ψ) is R-bounded in L (X), and (ii) Λ1 (Ψ) is R-bounded in L (X). The following consequence of Proposition 6 is known, [20, Proposition 3.6]. Proposition 11. Let U : G → L (X) be a bounded and sot continuous homomorphism. The following two statements are equivalent. (i) the set Z

f (g)U(g) dλ(g) : f ∈ L (G), kfbk∞ ≤ 1 1

G



(11)

is R-bounded in L (X).

b → L (X) (ii) there exists an R-bounded, regular, sot σ-additive spectral measure P : B(G) such that Z U(g) = (g, u) dP (u), g ∈ G, b G

is the Fourier-Stieltjes transform of P . Moreover, if either of these conditions is satisfied, then U is R-bounded.

122

B. dePagter, W. J. Ricker

For a Hilbert space X,we see from (9) and Proposition 10 that it is possible to replace (11) in Proposition 11 with the smaller set Z  1 f (g)U(g) dλ(g) : f ∈ L (G), kf k1 ≤ 1 . (12) G

Keeping in mind Example 8 we pose the following Question 2. Let X be a Banach space and U : G → L (X) be a bounded, sot continuous homomorphism. With (12) in place of (11), are statements (i) and (ii) in Proposition 11 still equivalent? We now describe some techniques for deciding about the R-boundedness or not of particular representations. For the definition of a (complex) Banach space satisfying a lower p-estimate, with 1 ≤ p < ∞, or an upper q-estimate, with 1 < q ≤ ∞, we refer to [18, p.82]. Any Lp space, with 1 ≤ p < ∞, satisfies a lower p-estimate and, for 1 < p ≤ ∞, an upper p-estimate. The following result is crucial for examples, [20, Proposition 3.8]. Proposition 12. Let E be a (complex) Banach lattice and T ⊆ L (E). (i) Suppose that E satisfies a lower p-estimate for some 1 ≤ p < 2. Assume that there exists a constant c > 0 with the property that for every n ∈ N there exist T1 , . . . , Tn ∈ T and x 6= 0 in E such that {T1 x, . . . , Tn x} is a disjoint system in E and kTj xk ≥ ckxk for 1 ≤ j ≤ n. Then T is not R-bounded. (ii) Suppose that E satisfies an upper q-estimate for some 2 < q ≤ ∞. Assume there exists a constant c > 0 with the property that for every n ∈ N, there exist R1 , . . . , Rn ∈ T and a disjoint system {y1, . . . , yn } in E such that Rj yj = x 6= 0 for all 1 ≤ j ≤ n and kyj k ≤ ckxk . Then T is not R-bounded. Every homomorphism U : Z → L (X) is necessarily of the form m 7→ T m , for m ∈ Z, where T = U(1). In particular, U is bounded iff sup{kT m k : m ∈ Z} < ∞. In this case the isomorphism T ∈ L (X) is called power bounded. For such homomorphisms the following consequence of Proposition 12 (see [20, Corollary 3.10]) is useful. Proposition 13. Let E be a Banach lattice which satisfies either a lower p-estimate for some 1 ≤ p < 2 or an upper q-estimate for some 2 < q ≤ ∞. Let T ∈ L (E) be a power bounded operator such that, for every n ∈ N, there exist k1 , . . . , kn ∈ Z and x 6= 0 in E with the property that {T k1 x, . . . , T kn x} is a disjoint system in E. Then the homomorphism m 7→ T m , for m ∈ Z, fails to be R-bounded. Let E = ℓ∞ (Z) and U : Z → L (E) be the bounded representation m 7→ T m , where T ∈ L (E) is the unit right-shift operator. Then {U(m) : m ∈ Z} is the group of all translation operators in E. The Banach lattice ℓ∞ (Z) satisfies an upper ∞-estimate. Moreover, for each n ∈ N, the non-zero element x := χ{0} ∈ ℓ∞ (Z) has the property that {T x, . . . , T n x} is a disjoint family in ℓ∞ (Z). By Proposition 13 the translation group in ℓ∞ (Z) fails to be Rbounded. Note that c0 (Z)∗∗ = ℓ∞ (Z). By the comments just prior to Proposition 5, stating that Rbounded subsets of L (c0 (Z)) are also R-bounded in L (c0 (Z)∗∗ ), it follows that the translation

R-boundedness of C(K)-representations, group homomorphisms, and Banach space geometry

123

group in c0 (Z) is not R-bounded either. For further examples related to discrete groups of operators in Lp (G)-spaces (generated by a single translation operator) which fail to be R-bounded, we refer to Proposition 3.11 and Corollary 3.12 of [20]. On a positive note we have the following Proposition 14. Let X be a Banach space with property (α) and T ∈ L (X) be an isomorphism which is a power bounded, scalar-type spectral operator. Then the homomorphism m 7→ T m , for m ∈ Z, is R-bounded. Proof. By the hypotheses, the spectrum R σ(T ) ⊆ T and there is a sot σ-additive spectral measure P : B(T) → L (X) such that T = T λ dP (λ); see Chapter XV of [10], for example. Then Ran(P ) is a bounded B.a. of projections in L (X) and hence, is R-bounded, [19, Theorem 3.3]. Since each function λ 7→Rλm , for m ∈ Z, has sup-norm 1 on T and by the properties of spectral integrals we have T m = T λm dP (λ), it follows that Z  m {T : m ∈ Z} ⊆ f dP : f ∈ C(T), kf k∞ ≤ 1 ⊆ 4aco Ran(P ). T

Accordingly {T m : m ∈ Z} is R-bounded in L (X). We conclude with two examples which do not follow from the results of [20]. The proof of the first example was suggested by Tuomas Hytönen. Example 15. Let G = T and X = c0 (Z). For each ϕ ∈ ℓ∞ (Z), let Mϕ ∈ L (X) be the operator of (coordinate-wise) multiplication by ϕ. Define the reflection ϕ e of ϕ by n 7→ ϕ(−n), for n ∈ Z. Define P : B(Z) → L (X) by P (A) := MχeA for each subset A ⊆ Z, in which case P is a (regular) sot σ-additive spectral measure. For each ϕ ∈ ℓ∞ (Z), the spectral integral R ϕ dP = Mϕe. Since the bounded B.a. {P (A)∗∗ : A ⊆ Z} ⊆ L (ℓ∞ (Z)) is an infinite set Z and ℓ∞ (Z) is a GDP-space, it follows from Lemma 3 that Ran(P ) ⊆ L (X) is not R-bounded. What about R-boundedness of the Fourier-Stieltjes transform U of P , that is, the bounded, sot continuous homomorphism U : T → L (X) given by Z U(θ) = e−inθ dP (n) = Mϕθ , θ ∈ [0, 2π), Z

where ϕθ (n) := einθ for n ∈ Z? We proceed to show that U fails to be R-bounded. Since c0 (Z)∗∗ = ℓ∞ (Z), it follows that the family of multiplication operators {Mϕ∗∗θ : θ ∈ [0, 2π)} ⊆ L (ℓ∞ (Z)) also fails to be R-bounded. Hence, by Lemma 7 the set Λ1 (Ψ), with Ψ as given by (7), also fails to be R-bounded. This cannot be deduced directly from properties of Ran(P ) because, as noted in Example 8, it can happen that Ran(P ) is not contained in any multiple of Λ1 (Ψ). So, back to the homomorphism U. Fix N ∈ N and let {(δj1 , . . . , δjN ) : 1 ≤ j ≤ 2N } be an enumeration of all the elements in {0, 1}N . For each n = 1, . . . , N define N

θn := π

2 X j=1

δjn 2−4j .

(13)

124

B. dePagter, W. J. Ricker

Then, for 1 ≤ k ≤ 2N , we have N

24k θn = π

2 X

δjn 24(k−j)

j=1

= π

X

δjn 24(k−j) + πδkn + π

1≤j ω. We recall that, according to the first Trotter-Kato approximation theorem, (An ) converges to A in the strong resolvent sense if and only if etA = s-limn→∞ etAn uniformly for t in bounded subsets of [0, ∞); cf. [3, Theorem III.4.8], [4, Theorem 3.4.2]. In order to put formula (2.3), in Theorem 2.1, into the proper context we recall that, for a C0 -semigroup T , the modulus semigroup (if it exists) can be obtained by T ♯ (t) =

sup (γ1 ,...,γn )∈Γ

|T (γ1 t)| · · · |T (γn t)| =

s-lim (γ1 ,...,γn )∈Γ

|T (γ1 t)| · · · |T (γn t)|,

where Γ = {γ ∈ (0, 1]n ; n ∈ N, γ1 + · · · + γn = 1} (cf. [2, Theorem 2.1]).

(2.2)

134

M. Stein, J. Voigt

2.1 Theorem. Let T be a C0 -semigroup with generator A, which possesses a modulus semigroup. Then 1s (|T (s)| − I) → A♯ in the strong resolvent sense as s → 0, and T ♯ (t) = s-lim |T (t/n)|n , n→∞

(2.3)

uniformly for t in compact subsets of [0, ∞). For a generator A, one of the exponential formulas states that (n/tR(n/t, A))n tends to the semigroup generated by A in the strong operator topology, uniformly on compact intervals of [0, ∞). As a second consequence of Theorem 1.1 we obtain the following approximation for the modulus semigroup. 2.2 Theorem. Let A be the generator of a C0 -semigroup T and suppose that T possesses a modulus semigroup. Then µ2 |R(µ, A)| − µ → A♯ in the strong resolvent sense as µ → ∞, and T ♯ (t) = s-lim(n/t|R(n/t, A)|)n , n→∞

uniformly for t in compact subsets of [0, ∞).

References [1] C. D. Aliprantis and O. Burkinshaw: Positive operators. Pure and Applied Mathematics, Academic Press, 1985. [2] I. Becker and G. Greiner: On the modulus of one-parameter semigroups. Semigroup Forum 34, 185–201 (1986). [3] K.-J. Engel and R. Nagel: One-parameter semigroups for linear evolution equations. Springer, 2000. [4] A. Pazy: Semigroups of linear operators and applications to partial differential equations. Springer, 1983. [5] M. Stein and J. Voigt: Approximation of modulus semigroups and their generators. Semigroup Forum, to appear.

Martin Stein, J¨urgen Voigt Technische Universit¨at Dresden, Fachrichtung Mathematik, Institut f¨ur Analysis 01062 Dresden, Germany [email protected], [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 135-146 (2006)

Stochastic semigroups: their construction by perturbation and approximation H. R. Thieme1 (Tempe) and J. Voigt (Dresden) Abstract. The main object of the paper is to present a criterion for the minimal semigroup associated with the Kolmogorov differential equations to be stochastic in ℓ1 . Our criterion uses a weighted ℓ1 space. As an abstract preparation we present a perturbation theorem for substochastic semigroups which generalizes known results to the case of ordered Banach spaces which need not be AL-spaces We also consider extensions of Kolmogorov’s equations to spaces of measures. In an appendix we present a version of the Miyadera perturbation theorem for positive semigroups on ordered Banach spaces. Key words: semigroup, substochastic, stochastic, perturbation, approximation, Kolmogorov differential equations MSC 2000: 47D06, 47D07, 47B60, 47B65, 60J25, 60J35

1 Introduction Let X be an ordered real Banach space with a generating (reproducing) cone X+ , X = X+ − X+ , and a norm which is additive on X+ , kx + yk = kxk + kyk for all x, y ∈ X+ . A C0 -semigroup (S(t)); t > 0) (of bounded linear operators) on X is called substochastic (stochastic) if S(t) is positive and kS(t)xk 6 kxk (kS(t)xk = kxk) for all x ∈ X+ , t > 0. The norm on X+ can be uniquely extended to a positive bounded linear functional ϕ ∈ X ∗ satisfying ϕx = kxk for all x ∈ X+ . The generator A of a stochastic semigroup necessarily satisfies ϕAx = 0 for all x ∈ D(A) ∩ X+ . It is the aim of this note to indicate conditions for a C0 -semigroup defined by a limiting procedure to be stochastic. Differently from [17], [25; Sec. 2], [3] we work with an auxiliary Banach space X1 that is continuously and densely embedded into X. This space may be of its own interest: in case of the Kolmogorov differential equations (Section 4), it can be chosen as P∞ 11 ∞ := the first moment space ℓ = {x = (xj )j=0 ; kxk1 j=0 (1 + j)|xj | < ∞}. By perturbation (Section 2) we will construct substochastic C0 -semigroups on X which leave the auxiliary Banach space invariant and induce C0 -semigroups thereon. These semigroups can also obtained by an approximation procedure (Section 3). As a second application we present a measure-valued generalization of Kolmogorov’s differential equations (Section 5) whose solutions are associated with Markov transition functions. In an appendix we present a version of the Miyadera perturbation theorem for positive semigroups on ordered Banach spaces. This result is needed in Section 2. 1

partially supported by NSF grant DMS-0314529

136

H. R. Thieme, J. Voigt

2 Perturbation A generating cone X+ is always non-flat; cf. Remark A.1(a). The additivity of the norm on X+ implies that the the norm is monotone, and hence X+ is normal; cf. [10; Proposition A.2.2]. The additivity of the norm on X+ also implies that any bounded monotone sequence is convergent. 2.1 R EMARKS . (a) Let S be a positive C0 -semigroup on X, with generator A. We recall that then D(A)+ (= D(A) ∩ X+ ) is dense in X+ . Moreover, S is substochastic (stochastic) if and only if ϕAx 6 0 (ϕAx = 0) for all x ∈ D(A)+ . This follow immediately from the fact that, for x ∈ D(A)+ , one has d d kS(t)xk = ϕ(S(t)x) = ϕ(AS(t)x). dt dt Further, if D is a core for A, and ϕAx = 0 for all x ∈ D, then it follows that ϕAx = 0 for all x ∈ D(A), and that therefore S is stochastic. (b) Note that, although the norm of X is supposed to be additive on X+ , we do not suppose X to be an AL-space. In fact, the order need not be associated with a Banach lattice structure. We want to present the abstract part of the paper in this general setting, although in the applications presented in Sections 4 and 5 the space X will be an AL-space. Nevertheless, we think that it is interesting to know that certain properties are valid in the general setting we present here. 2.2 Theorem. Let S0 be a positive C0 -semigroup on X, with generator A0 . Let B : D(A0 ) → X be a positive linear operator satisfying ϕ(A0 + B)x 6 0

(x ∈ D(A0 )+ ).

(a) Then, for all x ∈ D(A0 )+ , one has Z ∞ kBS0 (s)xk ds 6 kxk. 0

For all 0 6 r < 1, the operator A0 + rB is the generator of a substochastic C0 -semigroup Sr . For 0 6 r 6 r ′ < 0 one has Sr (t) 6 Sr′ (t) (t > 0). (b) S(t) := s-limr→1 Sr (t) exists, uniformly for t in bounded subsets of [0, ∞). S thus defined is a substochastic C0 -semigroup on X. The generator of S is an extension of A0 + B. (c) S is the smallest positive semigroup (also called the minimal semigroup) whose generator is an extension of A0 + B. Proof. (a) Since B : D(A0 ) → X is positive we obtain that B(λ − A0 )−1 is positive, hence bounded (cf. [10; Proposition A.2.11]), i.e., B is A0 -bounded. Moreover, for 0 6 t < ∞, we obtain Z t Z t kBS0 (s)xk ds = ϕ(BS0 (s)x) ds 0 0 Z t Z t 6− ϕ(A0 S0 (s)x)) ds = −ϕ A0 S0 (s)x ds 0

0

= ϕ(x − S0 (t)x) = kxk − kS0 (t)xk 6 kxk

Stochastic semigroups: their construction by perturbation and approximation

and hence Z ∞

137

kBS0 (s)xk ds 6 kxk,

0

for all x ∈ D(A0 )+ (cf. [25; Lemma 1.2(a)]). Using Theorem A.2, one proves the remaining assertions in the same way as in [25; Lemma 1.2 and Proposition 1.4]. 2.3 R EMARK . The first version of a result as in Theorem 2.2 is due to Kato [17]. The application of the Miyadera perturbation theorem in this context goes back to [25]. Further developments can be found in [3], [15], [7], [6]. Next, we show a result on invariant subspaces of the dual semigroup. This result will be used in Section 5. 2.4 Proposition. Let the assumptions of Theorem 2.3 be satisfied, and let be a weakly∗ sequentially closed subspace of X ∗ which is invariant under ((λ − A0 )−1 )∗ and (B(λ − A0 )−1 )∗ . Then Y is invariant under the semigroups Sr∗ and S ∗ , with Sr and S from Theorem 2.3. Proof. It follows from the proof of Theorem A.2 that the spectral radius of F (λ) = B(λ−A0 )−1 does not exceed one. Set Ar = A0 + rB. Then (λ − Ar )−1 = (λ − A0 )−1

∞ X

(rF (λ))j .

j=0

Since Y is weakly∗ sequentially closed and invariant under ((λ − A0 )−1 )∗ and F (λ)∗ , it is invariant under ((λ − Ar )−1 )∗ . Since Sr (t) = s-limn→∞ (I − (t/n)Ar )−n , Y is invariant under Sr∗ (t), and also under S ∗ (t) because S(t) is the strong limit of Sr (t). If, in Theorem 2.2, one assumes ϕ(A0 + B)x = 0 for all x ∈ D(A0 ) ∩ X+ , one cannot conclude that S is stochastic, in general; cf. [17; § 4, Example 3]. Discussing conditions for this to hold has been a major objective in most of the pertinent work; cf. [17], [21], [25], [3], [8], [5], [4], [2], [15], [6]. Besides the development of the theory in the more general context, it was the main motivation of the present paper to indicate a new type of sufficient condition; cf. Theorem 2.7. 2.5 A SSUMPTION . Let X1 be a subspace of X such that the following hold: • There exists a norm k · k1 on X1 which makes it a Banach space. • (X1 , k · k1 ) is continuously embedded into (X, k · k) and X1 ∩ X+ is dense in X+ . • X1,+ := X1 ∩ X+ is a generating cone for X1 and k · k1 is additive on X1,+ . It follows from these assumptions that there exists a unique positive functional ϕ1 ∈ X1∗ which coincides with k · k1 on X1,+ . 2.6 Proposition. Let the assumptions of Theorem 2.2 be satisfied. Assume that X1 is a subspace X which satisfies Assumption 2.5.

138

H. R. Thieme, J. Voigt

Assume that S0 induces a (necessarily positive) C0 -semigroup Sˇ0 on (X1 , k · k1 ). Let Aˇ0 denote its generator. (Note that Aˇ0 is the restriction of A0 to D(Aˇ0 ) = {x ∈ D(A0 )∩X1 ; A0 x ∈ X1 }.) Assume that B(D(Aˇ0 )) ⊆ X1 , and assume that there is a constant c > 0 such that ϕ1 (A0 + B)x 6 ckxk1

(x ∈ D(Aˇ0 )+ ).

Then the semigroup S from Theorem 2.2 leaves X1 invariant and induces a positive C0 semigroup Sˇ on X1 . Sˇ is the smallest positive C0 -semigroup on X1 whose generator is an ˇ For x ∈ X1,+ , t > 0, we have kS(t)xk 6 ect kxk, i.e., the rescaled extension of Aˇ0 + B. ˇ t>0 is substochastic on X1 . semigroup (e−ct S(t)) Proof. Note first that the hypothesis can be reformulated as ˇ 6 0, ϕ1 ((Aˇ0 − c) + B)x

(x ∈ D(Aˇ0 )+ ),

ˇ denotes the restriction of B to D(Aˇ0 ). Therefore Theorem 2.2 can be applied to X1 where B and the restricted operators. ˇ is the generator of a Let 0 6 r < 1. Then Theorem 2.2(b) implies that (Aˇ0 − c) + r B ˇ is the generator of a positive C0 substochastic semigroup on X1 , or equivalently, that Aˇ0 + tB ct ˇ ˇ semigroup Sr on X1 satisfying kSr (t)xk 6 e kxk for all x ∈ X1,+ , t > 0. It is easy to see that Sˇr and Sr (from Theorem 2.2) coincide on X1 . Taking r → 1 one obtains the assertion. 2.7 Theorem. Let the assumptions of Proposition 2.6 be satisfied. Assume additionally that −A0 is positive and that there exists ε > 0 such that (with c > 0 from Proposition 2.6) ϕ1 (A0 + B)x 6 ckxk1 − εkA0 xk

(x ∈ D(Aˇ0 )+ ),

i.e., for all x ∈ D(A0 )+ ∩ X1 with A0 x ∈ X1 . Then the generator Aˇ of Sˇ is a restriction of A0 + B, and the generator A of S is the closure ˇ of A and of A0 + B in X. If ϕ(A0 + B)x = 0 for all x ∈ D(A0 )+ then S is stochastic. 2.8 R EMARK . The assumption that −A0 is a positive operator can be replaced by the following: If (xn ) is sequence in D(Aˇ0 )+ , xn ↑ x ∈ X1 and supn∈N kA0 xn k < ∞, then x ∈ D(A0 ). Proof of Theorem 2.7. Let (rn ) be a sequence in [0, 1), rn ↑ 1. We shall use the notation An := A0 + rn B (n ∈ N). We know that A0 + B ⊆ A and Aˇ ⊆ A. In order to show the first assertion we thus have to ˇ ⊆ D(A0 ). show D(A) Let x ∈ X1,+ , λ > max{0, c}. Then xn := (λ−An )−1 x ∈ D(Aˇ0 )+ (n ∈ N). By hypothesis, ϕ1 (−x + λxn ) = ϕ1 An xn 6 ϕ1 (A0 + B)xn 6 ckxn k1 − εkA0 xn k, εkA0 xn k 6 (λ − c)kxn k1 + εkA0 xn k 6 ϕ1 x = kxk1 . The sequence (xn ) is increasing and converges in X1 to (λ − A)−1 x. Since A0 preserves monotone sequences, (A0 xn ) is a bounded monotone sequence in X and thus has a limit in X. Since ˇ = (λ − A)−1 X1 ⊆ D(A0 ) A0 is a closed operator, (λ − A)−1 x ∈ D(A0 ). This implies D(A) since X1,+ is a generating cone for X1 . ˇ of Aˇ is a subset of D(A), invariant under S (= Sˇ on X1 ) and dense in X, The domain D(A) and therefore a core for A (cf. [20; Theorem X.49]). From Aˇ ⊆ A0 + B (⊆ A) we also obtain A = A0 + B. Now the last statement is a consequence of Remark 2.1(a).

Stochastic semigroups: their construction by perturbation and approximation

139

3 Approximation 3.1 Proposition. Let S0 , A0 , B, S be as in Theorem 2.2, in particular recall ϕ(A0 + B)x 6 0

(x ∈ D(A0 )+ ).

For n ∈ N let Bn : D(A0 ) → X be a positive linear operator, Bn x 6 Bx (x ∈ X+ , n ∈ N), Bn x → Bx (n → ∞) for all x ∈ D(A0 ). For n ∈ N let Sn be the smallest positive (substochastic) C0 -semigroup whose generator is an extension of A0 + Bn (see Theorem 2.2(a)). (a) Then Sn (t) 6 S(t) (n ∈ N0 ) and S(t) = s-limn→∞ Sn (t) (t > 0). (b) Assume additionally that X1 is a subspace of X satisfying Assumption 2.5. As in Proposition 2.6, assume that S0 induces a C0 -semigroup Sˇ0 on X1 , and let Aˇ0 denote its generator. Assume that Bn (D(Aˇ0 )) ⊆ X1 (n ∈ N), B(D(Aˇ0 )) ⊆ X1 , that Bn x → Bx in X1 (n → ∞) for all x ∈ D(Aˇ0 ), and that there is a constant c > 0 such that ϕ1 (A0 + B)x 6 ckxk1 (x ∈ D(Aˇ0 )+ ). Then Sn (n ∈ N) and S induce positive C0 -semigroups Sˇn (n ∈ N) and Sˇ on X1 , respectively, ˇ = s-limn→∞ Sˇn (t) in X1 (t > 0). and S(t) Proof. (a) (cf. [25; proof of Proposition 1.6]) For 0 6 r < 1 we obtain from Theorem 2.2(a) that the operators A0 + rBn (n ∈ N) and A0 + rB are generators of substochastic C0 -semigoups Sn,r (n ∈ N) and Sr , respectively. The inequalities required in the hypothesis imply Sn,r (t) 6 Sr (t) (t > 0, n ∈ N). Taking r → 1 we obtain Sn (t) 6 S(t) (t > 0, n ∈ N). A minor adaptation of [24; Theorem 1.4] to our context shows that Sn,r (t) → Sr (t) strongly (n → ∞). Let x ∈ X+ . The inequalities 0 6 S(t)x − Sn (t)x = (S(t)x − Sr (t)x) + (Sr (t)x − Sn,r (t)x) + (Sn,r (t)x − Sn (t)x) 6 (S(t)x − Sr (t)x) + (Sr (t)x − Sn,r (t)x) imply kS(t)x − Sn (t)xk 6 kS(t)x − Sr (t)xk + kSr (t)x − Sn,r (t)xk. Choosing first r close enough to 1 and then n large enough we can make the right hand side as small as we want. This implies that Sn (t) → S(t) strongly. (b) Proposition 2.6 implies that the semigroups Sn (n ∈ N) and S induce C0 -semigroups Sˇn (n ∈ N) and Sˇ on X1 , and that these semigroups are the smallest positive semigroups on X1 ˇn (n ∈ N) and Aˇ0 + B, ˇ respectively. Part (a), applied whose generators are extensions of Aˇ0 + B ˇ = s-limn→∞ Sˇn (t) (t > 0). to these semigroups, yields S(t) 3.2 R EMARK . Assuming monotonicity, i.e., Bn x 6 Bn+1 x (x ∈ D(A)+ , n ∈ N), in Proposition 3.1, one could simplify the hypothesis in part (b). In this case the convergence Bn x → Bx in X1 would follow from the remaining hypotheses.

4 Example: Kolmogorov’s differential equations The infinite system of differential equations ∞ X ′ xj = αjk xk (j = 0, 1, 2, . . .)

(4.1)

k=0

is known as Kolmogorov’s differential equations provided the coefficients αjk form a Kolmogorov matrix [16; Sec. 23.12], i.e.,

140

H. R. Thieme, J. Voigt

• for all j, k ∈ N0 , αjk > 0 if j 6= k, αjj 6 0, P∞ • j=0 αjk = 0 for all k ∈ N0 .

xj can be interpreted as the probability that the number of individuals in a population is j. As we allow the population to go extinct and be possibly resurrected by immigration, we choose the non-negative integers N0 as state space. For j 6= k, αjk is the rate at which the population size changes from k to j, while −αjj is the rate at which a population of size j changes to a size different from j. Typically Kolmogorov’s differential equations are P∞considered on the standard sequence 1 ∞ space ℓ = {x = (xj )j=0 ; kxk < ∞} with kxk = j=0 P |xj |. See [17], [16; Chap. 23], [13; XVII.9], [14; XIV.7], [25], and the literature cited there. As ∞ population j=1 jxj is the expected P∞ 11 ∞ size, the first moment space ℓ = {x = (xj )j=0 ; kxk1 < ∞} with kxk1 = j=0 (1 + j)|xj | < ∞ is also a meaningful state space. Interestingly enough, this will help us find a condition for the semigroup associated with (4.1) to be stochastic on ℓ1 . 4.1 A SSUMPTION . There exist constants c, ε > 0 such that ∞ X

jαjk 6 c(1 + k) − ε|αkk |

for all k ∈ N0 .

j=0

Notice that size k. Let

P∞

j=0 jαjk

can be interpreted as expected population growth rate at population 1

D0 = {x = (xj ) ∈ ℓ ;

∞ X

|αjj ||xj | < ∞}.

j=0

4.2 Theorem. Let (αjk ) be a Kolmogorov matrix which satisfies Assumption 4.1. Then the closure of the operator A∞ : D0 → ℓ1 , (A∞ x)j =

∞ X

αjk xk

(x = (xk ) ∈ D0 ),

k=0

is the generator of a stochastic semigroup S on ℓ1 . The semigroup S leaves ℓ11 invariant and induces a strongly continuous semigroup Sˇ on ℓ11 . The generator of Sˇ is the restriction of A∞ to ˇ = {x ∈ ℓ11 ∩ D0 ; A∞ x ∈ ℓ11 }. D(A) Moreover kS(t)xk1 6 ect kxk1 for all x ∈ ℓ11 , t > 0. This theorem is a consequence of Theorem 2.2, Proposition 2.6 and Theorem 2.7. It is useful, however, also to apply Proposition 3.1 because this provides an approximation result which allows to find conditions under which the semigroup is bounded on ℓ11 and has a strictly negative essential growth bound [19]. Let   αjj if j = k, [n] αjk if j, k < n, j 6= k, (j, k, n ∈ N0 ). αjk =  0 otherwise [n] Define An : D0 → ℓ1 analogously to A∞ with αjk replacing αjk , for n ∈ N0 . Note that in this case the operators Bn := An − A0 are bounded (positive) operators.

Stochastic semigroups: their construction by perturbation and approximation

141

4.3 Theorem. The operators An generate substochastic C0 -semigroups Sn on ℓ1 , and kSn (t)x− S(t)xk → 0 as n → ∞ for each x ∈ ℓ1 . These semigroups leave ℓ11 invariant and induce C0 semigroups on ℓ11 , and kSn (t)x − S(t)xk1 → 0 as n → ∞ for each x ∈ ℓ11 . The convergence is monotone increasing, if x is positive. In order to illustrate Assumption 4.1, we consider a birth and death process with catastrophes and immigration. We assume that (αjk ) is a Kolmogorov matrix satisfying αjk = 0

for all j, k ∈ N0 with j > k + 1.

(4.2)

This means that birth rates are such that populations can only increase by one. On the other hand the model allows drastic decreases, including catastrophes wiping out the whole population. This model was also treated in [17; § 4, Example 4]. We indicate a condition implying Assumption 4.1. 4.4 Lemma. If (4.2) is satisfied then Assumption 4.1 holds if there exists a > 1 such that k−1

X 1 (a αk+1,k − (k − j)αjk ) < ∞ s := sup k>0 k + 1 j=0

(4.3)

Proof. The matrix (αjk ) being a Kolmogorov matrix satisfying (4.2) implies ∞ X

jαjk = αk+1,k +

j=0

k−1 X

(j − k)αjk

(k ∈ N0 ).

j=0

Because of this equation Assumption 4.1 can be reformulated as (1 + ε)αk+1,k

k−1 X 6 c(1 + k) + (k − j − ε)αjk ) (k ∈ N0 ). j=0

It is not difficult to see that (4.3) implies these inequalities if (1+ε)a 6 (1−ε), i.e. 0 < ε 6 and c > 1−ε s. a

a−1 , a+1

5 A measure-valued generalization of Kolmogorov’s differential equations Let (Ω, A) be a measurable space. We consider a measure-valued generalization of Kolmogorov’s differential equations, Z Z d µ(t)(Γ) = K(Γ, x)µ(t)(dx) − K(Ω, x)µ(t)(dx) (Γ ∈ A). dt Ω Γ K is a transition measure kernel: For each x ∈ Ω, K(·, x) is a non-negative finite measure on (Ω, A) and for each Γ ∈ A, K(Γ, ·) is A-measurable on Ω. The solution µ(t) takes its values in the Banach space X of signed measures on (Ω, A) which have bounded variation. If

142

H. R. Thieme, J. Voigt

the values are probability measures, they are associated with the transition probabilities of a Markov jump process ([14; X.3], [12; 4.2]). Let h := K(Ω, ·). Then the operator A0 given by Z n o D(A0 ) := µ ∈ X ; h(x)|µ|(dx) < ∞ , Ω Z (A0 µ)(Γ) := − h(x)µ(dx), Γ

generates the C0 -semigroup S0 , Z (S0 (t)µ)(Γ) = e−th(x) µ(dx). Γ

The denseness of D(A0 ) can be seen as follows: Set Ωn = {x ∈ Ω; h(x) 6 n}. Let µ ∈ X. Define µn (Γ) = µ(Γ ∩ Ωn ). Then µn ∈ D(A0 ) and µn → µ as n → ∞ because Ω is the union of the increasing sequence of sets Ωn . We define a positive linear operator B : D(A0 ) → X, Z (Bµ)(Γ) = K(Γ, x)µ(dx). Ω

Let η : Ω → R[0, ∞) be A-measurable. If we choose X1 to be the space of signed measures µ with kµk1 = Ω (1 + η(x))|µ|(dx) < ∞, we obtain the following result from Theorem 2.2, Proposition 2.6 and Theorem 2.7. Notice that the density of X1 in X follows in the same way as the density of D(A0 ). 5.1 Proposition. (a) There exists a smallest positive C0 -semigroup S on X whose generator extends A0 + B. S is substochastic. (b) Assume that there exist positive constants c, ε such that Z (η(y) − η(x))K(dy, x) 6 c(1 + η(x)) − εK(Ω, x) for all x ∈ Ω. Ω

Then S is a stochastic C0 -semigroup which is generated by the closure of A0 + B, leaves X1 invariant and induces a C0 -semigroup on (X1 , k · k1 ). We now show that the semigroup S is associated with a Markov transition function. We first note that Y = BM(Ω), the space of bounded A-measurable functions on Ω with the supremum norm, can be identified with a subspace of X ∗ . The dominated convergence theorem implies that BM(Ω) is weakly∗ sequentially closed. BM(Ω) is invariant under ((λ−A0 )−1 )∗ and (B(λ− A0 )−1 )∗ , because f (x) , λ + h(x) Z f (y) −1 ∗ (B(λ − A0 ) ) f (x) = K(dy, x) Ω λ + h(x) ((λ − A0 )−1 )∗ f (x) =

Stochastic semigroups: their construction by perturbation and approximation

143

for all x ∈ Ω, f ∈ BM(Ω). By Proposition 2.4, BM(Ω) is invariant under S ∗ . Set Pt (Γ, x) := (S(t)δx )(Γ) (Γ ∈ A),

(5.1)

where δx is the Dirac measure concentrated at x. Then Pt (·, x) is a non-negative measure on A with values in [0, 1]. Since also Pt (Γ, ·) = S ∗ (t)χΓ and BM(Ω) is invariant under S ∗ (t), Pt (Γ, ·) ∈ BM(Ω) and Z (S(t)µ)(Γ) = Pt (Γ, x)µ(dx). Ω

The semigroup property of S implies that P satisfies the Chapman-Kolmogorov equations Z Pt+s (Γ, x) = Pt (Γ, y)Ps(dy, x), Ω

i.e., P is a Markov transition function [22; Sec. 3.2]. Since S is a C0 -semigroup, the definition (5.1) shows that for fixed x ∈ Ω, the function t 7→ Pt (Γ, x) is continuous on [0, ∞), uniformly for Γ ∈ A. If the assumption in Proposition 5.1(b) is satisfied, then Pt (·, x) is a probability measure, and therefore P is a normal Markov transition function.

Appendix: A version of the Miyadera perturbation theorem In this appendix we assume that X is an ordered Banach space with a generating (closed) cone X+ . We start with an observation that will be needed in the proof of the main result of this section. A.1 R EMARKS . (a) Under the above hypothesis the positive cone X+ is non-flat, i.e., there exists M > 1 such that for all x ∈ X, kxk 6 1, there exist x± ∈ X+ with kx± k 6 M, x = x+ − x− ; cf. [9; Proposition 19.1(d)], [10; p. 265]. This immediately implies that any linear operator C : X → X satisfying c := sup{kCxk; x ∈ X+ , kxk 6 1} < ∞ is bounded, kCk 6 2Mc. (b) Let A be the generator of a positive C0 -semigroup S on X. Then (a) can be strengthened as follows. Let C : D(A) → X be linear, A-bounded, c := sup{kCxk; x ∈ D(A)+ , kxk 6 1} < ∞. Then C uniquely extends to an operator C ∈ L(X) satisfying kCk 6 2Mc, sup{kCxk; x ∈ X+ , kxk 6 1} = c. In fact, let x ∈ D(A), kxk 6 1. There exist x± ∈ X+ , kx± k 6 M, x = x+ − x− . For λ larger than the type of S one has λ(λ − A)−1 x = λ(λ − A)−1 x+ − λ(λ − A)−1 x− , λ(λ − A)−1 x± ∈ D(A)+ . Also, λ(λ − A)−1 x± → x± (λ → ∞), and λ(λ − A)−1 x → x in the A-graph norm (λ → ∞). Taking λ → ∞ in kCλ(λ − A)−1 xk 6 c(kλ(λ − A)−1 x+ k + kλ(λ − A)−1 x− k) we obtain kCxk 6 c(kx+ k + kx− k) 6 2Mc. Now D(A) being dense implies that C extends as asserted. Since D(A)+ is dense in X+ one also obtaines the last equality.

144

H. R. Thieme, J. Voigt

A.2 Theorem. Let S0 be a positive C0 -semigroup on X, with generator A0 . Let B : D(A0 ) → X be positive. Assume that there are constants 0 < α 6 ∞, γ ∈ [0, 1) such that Z α kBS0 (t)xk dt 6 γkxk (x ∈ D(A0 )+ ). (A.1) 0

Then A0 + B is the generator of a positive C0 -semigroup S. A.3 R EMARKS . (a) If X, Y are ordered Banach spaces such that X+ is generating (i.e., X = X+ −X+ ) and Y+ is proper (i.e., Y+ ∩(−Y+ ) = {0}), then any positive linear operator T : X → Y is bounded; cf. [1; Appendix], [18; Theorem 2.1]. This implies that, in Theorem A.2, the operator B is A0 -bounded. (b) Note that (A.1) only implies Z α kBS0 (t)xk dt 6 2Mγkxk (x ∈ D(A0 )) 0

(with a proof as in Remark A.1(b)), so the assertion of Theorem A.2 is not a direct consequence of the Miyadera perturbation theorem; cf. [23; Theorem 1], [11; chap. 3, Theorem 3.14]. (For the application of the Miyadera perturbation theorem the constant 2Mγ would have to be < 1.) (c) The proof will show that  sup kS(t)xk; 0 6 t < α, x ∈ X+ , kxk 6 1  1 sup kS0 (t)xk; 0 6 t < α, x ∈ X+ , kxk 6 1 . 6 1−γ Proof of Theorem A.2. As mentioned above in Remark A.3(a), the operator B is A0 -bounded. By induction we define strongly continuous mappings Sn : [0, α) → L(X) satisfying Z t Sn (t)x := Sn−1 (t − s)BS0 (s)x ds (x ∈ D(A0 ), 0 6 t < α), (A.2) 0

kSn (t)xk 6 γ n kxk

(x ∈ X+ , 0 6 t < α),

for all n ∈ N. Indeed, the linear mappings Sn (t) defined by (A.2) belong to L(DA0 , X), where DA0 denotes D(A0 ) provided with the graph norm. The induction hypothesis implies Z t n−1 kSn (t)xk 6 γ kBS0 (s)xk ds 6 γ n kxk (x ∈ D(A0 )+ ). 0

Then use Remark A.1(b). P The series S(t) := ∞ n=0 Sn (t) is norm convergent, uniformly for 0 6 t < α, and as a consequence, S is strongly continuous. The proof that S satisfies the semigroup property on [0, α) and that S can be extended to a C0 -semigroup is the same as in [23; proof of Theorem 1]. It is easy to see that the generator A of S is an extension of A0 + B (cf. [23; Lemma 3]). Therefore (λ − A0 )−1 = (λ − A)−1 (I − B(λ − A0 )−1 ) for large λ. In order to show D(A) = D(A0 ) (then A = A0 + B) it therefore is sufficient to show spr(B(λ − A0 )−1 ) < 1 (spr denoting the spectral radius). Let γ ′ ∈ (γ, 1). The estimate

Stochastic semigroups: their construction by perturbation and approximation

145

as on [23; p. 168] shows that there exists λ > 0 such that kB(λ − A0 )−1 xk 6 γ ′ kxk for all x ∈ D(A0 )+ . Since B(λ − A0 )−1 is bounded we obtain kB(λ − A0 )−1 xk 6 γ ′ kxk for all x ∈ X+ . Therefore k(B(λ − A0 )−1 )n xk 6 γ ′n kxk for all x ∈ X+ , n ∈ N (it is at this point where the positivity of B is used), and finally k(B(λ − A0 )−1 )n k 6 2Mγ ′n (n ∈ N). These inequalities show spr(B(λ − A0 )−1 ) 6 γ ′ . Acknowledgement. The authors thank Wolfgang Arendt for useful comments.

References [1] W. A RENDT, Resolvent positive operators and integrated semigroups, Semesterbericht Funktionalanalysis, Sommersemester 1984, T¨ubingen, pp. 73–101. [2] L. A RLOTTI and J. BANASIAK, Strictly substochastic semigroups with application to conservative and shattering solutions to fragmentation eqations with mass loss, J. Math. Anal. Appl. 293, No. 2 (2004), 693–720. [3] J. BANASIAK, On an extension of the Kato-Voigt perturbation theorem for substochastic semigroups and its application, Taiwanese J. Math. 5, No. 1 (2001), 169–191. [4] J. BANASIAK, A complete description of dynamics generated by birth-and-death problem: a semigroup approach, Mathematical Modelling of Population Dynamics, Banach Center Publications, vol. 63, Inst. of Math., Polish Academy of Sciences, Warszawa, 2004. [5] J. BANASIAK, Conservative and shattering solutions for some classes of fragmentation models, Math. Models Methods Appl. Sci. 14, No. 4 (2004), 1–19. [6] J. BANASIAK and L. A RLOTTI, Perturbations of Positive Semigroups with Applications, Springer, London 2006 [7] J. BANASIAK and M. L ACHOWICZ, Around the Kato generation theorem for semigroups, preprint. [8] J. BANASIAK and W. L AMB, On the application of substochastic semigroup theory to fragmentation models with mass loss, J. Math. Anal. Appl. 284 (2003), 9–30. [9] K. D EIMLING, Nonlinear Functional Analysis, Springer, Berlin 1985. [10] B. DE PAGTER, Ordered Banach spaces, One-Parameter Semigroups (Ph. Cl´ement, H.J.A.M. Heijmans, S. Angenent, C.J. van Duijn, B. de Pagter, eds.), 265-279, NorthHolland, Amsterdam 1987. [11] K.-J. E NGEL and R. NAGEL, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York 2000. [12] S. N. E THIER , T. G. K URTZ, Markov Processes. Characterization and Convergence, Wiley, New York 1986 [13] W. F ELLER, An Introduction to Probability Theory and Its Applications, Vol. I, 3rd edition, John Wiley & Sons, New York 1968.

146

H. R. Thieme, J. Voigt

[14] W. F ELLER, An Introduction to Probability Theory and Its Applications, Vol. II, John Wiley & Sons, New York 1965. [15] G. F ROSALI, C. V. M. VAN DER M EE, and F. M UGELLI, A characterization theorem for the evolution semigroup generated by the sum of two unbounded operators, Math. Methods Appl. Sci. 27, No. 6 (2004), 669–685. [16] E. H ILLE and R. S. P HILLIPS, Functional Analysis and Semi-Groups, AMS, Providence 1957. [17] T. K ATO, On the semi-groups generated by Kolmogoroff’s differential equations, J. Math. Soc. Japan 6 (1954), 1-15. [18] M. A. K RASNOSEL’ SKIJ, J E . A. L IFSHITS, and A. V. S OBOLEV, Positive Linear Systems – The Method of Positive Operators –, Heldermann, Berlin 1989. [19] M. M ARTCHEVA, H. R. T HIEME, and T. D HIRASAKDANON, Positive semigroups on sequence spaces and continuous-time Markov chains, preprint. [20] M. R EED and B. S IMON, Methods of modern mathematical physics II: Fourier analysis, self-adjointness, Academic Press, New York 1975. [21] G. E. H. R EUTER, Denumerable Markov processes and the associated contraction semigroups on l. Acta Math. 97 (1957), 1–46. [22] K. TAIRA, Semigroups, Boundary Value Problems and Markov Processes, Springer, Berlin 2004 [23] J. VOIGT, On the perturbation theory for strongly continuous semigroups, Math. Ann. 229 (1977), 163-171. [24] J. VOIGT, Absorption semigroups, their generators, and Schr¨odinger semigroups. J. Funct. Anal. 67 (1986), 167–205. [25] J. VOIGT, On substochastic C0 -semigroups and their generators, Transp. Theory Stat. Phys. 16 (1987), 453-466.

H. R. Thieme Department of Mathematics and Statistics Arizona State University Tempe, AZ 85287-1804, USA [email protected] J. Voigt Technische Universit¨at Dresden Fachrichtung Mathematik, Institut f¨ur Analysis, 01062 Dresden, Germany [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 147-154 (2006)

Generalized Ideal Spaces and Applications to the Superposition Operator Martin V¨ath (Berlin)1 Abstract. Motivated by a typical application of the superposition (Nemytskij) operator in partial differential equations, we discuss why it is necessary to generalize the notion of ideal spaces in the case of measurable vector functions. Generalized ideal spaces are introduced by axioms which are necessary and sufficient for the required results. Some open problems concerning the relation of these axioms are formulated. Key words: Generalized ideal space, Vitali’s convergence theorem, Nemytskij operator, superposition operator, automatic continuity of operators, vector function MSC 2000: primary 47H30, 46E30; secondary 46E40, 46G10

Let S be a measure space. For simplicity, we will assume that mes S < ∞. (Note that this can always be arranged for σ -finite measure spaces by passing to an equivalent measure). By M , we denote the set of all (classes of) measurable functions x : S → R, equipped with the usual topology. Definition 1. A preideal space is a normed space X ⊆ M with the property that the relations x ∈ X, y ∈ M and |y(s)| ≤ |x(s)| (almost everywhere) imply y ∈ X and kyk ≤ kxk. If X is a Banach space, it is called ideal space. Ideal spaces are among the most important examples of Banach lattices. They have been intensively studied in a famous series of papers by W. A. J. Luxemburg and A. C. Zaanen (see [13] for some summary) and somewhat independently by P. P. Zabrejko [14]. However, as soon as vector functions x : S → U with a Banach space (U, | · |) are concerned, there are at least two canonical extensions. 1. One approach is to use the above definition, only replacing | · | by the norm. We will still call these spaces preideal/ideal spaces. 2. Another approach is to require instead that the relations x ∈ X and λ ∈ L∞ (S, R) imply λ x ∈ X and kλ xk ≤ kλ k∞ kxk. For distinction with the above definition, we will call these spaces preideal∗ /ideal∗ spaces or L∞ -modules. Note that the first definition could be formulated analogous to the second by replacing only L∞ (S, R) with L∞ (S, L (U)) (where L (U) denotes the Banach space of bounded operators of U), see [4]. Of course, each ideal space is an ideal∗ space and the converse does not hold. 1 The

paper was written in the framework of a Heisenberg fellowship (Az. VA 206/1-1). Financial support by the DFG is gratefully acknowledged.

148

M. Väth

We remark that a third approach could consist in assuming that U is a Banach lattice and to replace | · | in the first definition by the corresponding lattice operation. This was done in [5, 6], but we will not proceed this line here any further. Ideal spaces in the above sense were studied by the author [11], and ideal∗ spaces of functions (with values in Rn ) by H. T. Nguyˆen˜ and P. P. Zabrejko, see e.g. [1, 2, 3, 7, 8, 9]. Even in case U = R2 , an ideal∗ space X lacks any natural order. A replacement for the order are so-called m-units: A measurable (in the Kuratowski-Ryll-Nardzewski sense) multivalued function Φ : S ( U is called an m-unit for X if Φ(s) contains 0 and is even, closed, bounded, and convex, and if all measurable selections of Φ belong to X. The family of m-units is partially ordered by inclusion. Measurable selections of a fixed m-unit might be considered as a substitute of order-bounded sets. In the special case of ideal spaces, it suffices to consider only those m-units whose values are balls around 0 and thus one has an easier description of the “order”. For this reason, it turns out that most results of the scalar case carry over to ideal spaces of vector functions. On the other hand, they usually carry over to ideal∗ spaces of vector functions only if dimU < ∞. This is true in particular for the following two results which are in a sense crucial for the whole theory. Theorem 1. Each ideal space is continuously embedded into the space M of measurable functions, i.e. kxn − xk → 0 implies xn → x in measure. Theorem 2 (Vitali’s convergence theorem). If xn converges to x in measure and the set M := {x1 , x2 , . . . } has absolutely continuous norm in an ideal space X, i.e. sup kχD xk → 0

as

mes D → 0,

(1)

x∈M

then kxn − xk → 0. Vitali’s convergence theorem is especially useful if X is regular, i.e. if {x} has absolutely continuous norm for each x ∈ X. For example, the spaces L p ([0, 1],U) are regular if and only if p < ∞. More general, Orlicz spaces LΦ ([0, 1],U) are regular if and only if Φ satisfies a so-called ∆2 -condition. In regular spaces, Vitali’s convergence theorem implies Lebesgue’s dominated convergence theorem: If xn ∈ X converges to x in measure and if |xn (s)| ≤ |y(s)| where y ∈ Y (and thus {y} has absolutely continuous norm and so has {x1 , x2 , . . . }) then kxn − xk → 0. Moreover, in regular spaces each norm-convergent sequence has equicontinuous norm so that in view of Theorem 1 the conditions of Vitali’s convergence theorem are necessary and sufficient in regular spaces. However, Theorem 1 holds also in non-regular spaces. In the case of L p -spaces, Theorem 1 is rather trivial (and a popular exercise), but in the general setting it is far from being obvious. As remarked above, Theorems 1 and 2 both carry over for ideal spaces of functions with values in a Banach space U, but for ideal∗ spaces, they hold only if dimU < ∞. In fact, if dimU = ∞, the following regular ideal∗ space satisfies neither of the above theorems. Example 1. Let B be a Hamel base for U, and pick a sequence of pairwise different elements e1 , e2 , . . . ∈ B, without loss of generality |en | = 1. Define a map J : B → B by putting J(en ) := nen and J(e) := e for each e ∈ B \ {e1 , e2 , . . . }. Then J extends to a linear map J : U → U. Clearly,

Generalized Ideal Spaces and Applications to the Superposition Operator

149

J is a bijection, and even an isometry if we equip the image space with the different norm |u|∗ := |Ju|. Hence, (U, | · |) and (U, | · |∗ ) are both Banach spaces but with nonequivalent norms. We conclude that the identity map i : (U, | · |) → (U, | · |∗ ) is unbounded, since otherwise the open mapping theorem would imply that i−1 is bounded. Hence, we also find a sequence un ∈ U with |un | = 1 and |un |∗ → 0. Consider now S := {0} and the regular ideal∗ space X with kxk := |x(0)|∗ . The sequence xn (0) := un converges to 0 in X but not in measure. Moreover, the sequence yn (0) := en /n converges to 0 in measure and has equicontinuous norm, but kyn k = 1 6→ 0. Hence, Theorems 1 and 2 both fail for X. Another result concerns the superposition operator Fx(s) := f (s, x(s)) generated by a Carath´eodory function f . By the latter we mean as usual that f ( · , u) is measurable for each u ∈ U and that f (s, · ) is continuous for almost all s ∈ S. The following result is known in this case. Theorem 3. Let X and Y be ideal spaces. If Y is regular and F : X → Y is a superposition operator generated by a Carath´eodory function, then F is continuous. We remark that the proof of Theorem 3 as given in [11] follows essentially from Theorem 1 and 2. In fact, using the completeness of X and regularity of Y one can show that {Fx1 , Fx2 , . . . } has equicontinuous norm if kxn − xk → 0; hence, in view of Theorem 1 and 2, it suffices to observe that F is continuous in measure (since F is a Carath´eodory function). Similarly as Theorems 1 and 2, also Theorem 3 holds for ideal spaces of vector functions, but for ideal∗ spaces only for functions with values in a finite-dimensional space. Example 2. Let X be as in Example 1, and let Y be the ideal space with the norm kxk := |x(0)| . Then for the Carath´eodory function f (s, u) := u the (linear!) superposition operator Fx(s) = x(s) acts from the ideal∗ space X into the regular ideal space Y and from the ideal space Y into the regular ideal∗ space X, but none of these mappings is continuous, as is shown by the sequences xn and yn of Example 1. It is a somewhat unsatisfactory situation that the theory so far splits into two parts: the theory of ideal spaces (where infinite-dimensional U is allowed) and the theory of ideal∗ spaces restricted to finite-dimensional U. This is not only a theoretical drawback but has much practical relevance. Consider e.g. a smooth bounded domain S ⊆ Rn and the operator F0 u(s) := f (s, u(s), ∇u(s))

(2)

150

M. Väth

which arises naturally in the context of partial differential equations. Since F0 can be written as a composition of the local operator Gu(s) := (u(s), ∇u(s)) and of the superposition operator F(x, y)(s) := f (s, x(s), y(s)), one would like to have a continuity result for F on a “natural” space which contains the range of G. In the case of Sobolev spaces, such a space typically has the form X = L p (S,U) × Lq (S,V ) with V = U n and different powers p, q ≥ 1, because by the Sobolev embedding theorems, the function u is “smoother” (i.e. integrable to a larger power) than ∇u. Of course, we understand here X as a space of functions (x, y) : S → U ×V . For X = L p (S,U) × Lq (S,V ), one would expect continuity of F : X → Lr (S,W ) if f is a Carath´eodory function satisfying an asymmetric growth estimate | f (s, u, v)| ≤ a(s) + b |u| p/r + c |v|q/r

(3)

with some a ∈ Lr (S,W ) so that indeed F acts from X into Lr (S,W ). Unfortunately, the space X is not an ideal space even in case U = V = R, because (in case p > q, say) there is a function y ∈ Lq (S, R) which does not belong to L p (S, R). Hence, although |(y, 0)(s)| = |(0, y)(s)| almost everywhere, we have (0, y) ∈ X but (y, 0) ∈ / X. Thus Theorem 3 cannot be used to prove the continuity of F under the growth hypothesis (3). Also the “ideal∗ -version” of this result cannot be used if e.g. dimU = ∞. One could of course try to prove this result directly, but it would be better to have a more general theorem. Our aim thus is to find a definition of a class of “generalized ideal spaces” which is large enough to contain all ideal spaces (of arbitrary vector functions), ideal∗ spaces of vector functions with values in finite-dimensional spaces, and spaces like L p (S,U) × Lq (S,V ), but on the other hand small enough that Theorems 3 holds for this class of spaces. Of course, we would prefer that Theorem 1 and 2 hold as well for this class. For a start, it appears a good idea to forget completely about ideal and ideal∗ spaces and to consider a class of spaces as general as possible to formulate the problem. In order to define the notion “regular space” or, more general, “absolutely continuous norm” by (1) in a reasonable way, it seems that we should restrict attention to the following class of spaces. Definition 2. A normed space of (classes of) measurable functions x : S → U is called a projectable space if for each x ∈ X and each measurable set D ⊆ S we have χD x ∈ X and kχD xk ≤ kxk. Of course, each ideal∗ space is a projectable space. An example of a projectable space which is not an ideal∗ space is the subspace X ⊆ L p of all integrable simple functions. The following result gives a necessary criterion for a space Y to satisfy Theorem 3 whenever there exists at least one reasonable superposition operator from a preideal space into Y . Proposition 1. Let Y be a projectable space. Assume that there is a preideal space X of full support and a continuous superposition operator F : X → Y which is generated by a function of the form f (s, u) := ψ (s, |u|)u where r 7→ ψ (s, r)r is nondecreasing, continuous from the left, and strictly positive on (0, ∞) for almost all s, and where ψ ( · , r) is measurable for all r > 0. Then for each sequence yn ∈ Y which converges almost everywhere to 0 and each set E ⊆ S with mes E > 0 there is a set D ⊆ E with mes D > 0 such that kχD yn k → 0.

Generalized Ideal Spaces and Applications to the Superposition Operator

151

Proof. Put ϕs (r) := ψ (s, r)r and ϕs (∞) := supr>0 ϕs (r). Then

ϕs−1 (ρ ) := max {r ∈ [0, ∞] : ϕs (r) ≤ ρ } defines a function on [0, ∞] which is monotonically nondecreasing and continuous from the right, see e.g. [12, Proposition 8.2]. The functions

λn (s) := ϕs−1 (|yn (s)|) and zn (s) := λn (s)

yn (s) |yn (s)|

(put zn (s) := 0 for those s with yn (s) = 0) are measurable, because we have by [12, (8.7)] for each r > 0 {s : λn (s) < r} = {s : ϕs (r) > |yn (s)|} , and the latter sets are measurable. Since ϕs−1 is continuous from the right at 0 and ϕs−1 (0) = 0, we have for almost all s that zn (s) → 0. By Egorov’s theorem, the convergence is uniform on a set E0 ⊆ E with mes E0 > 0. Since X has full support, there is a function x ∈ X with mes(E0 ∩ supp x) > 0. There is some j and a subset D ⊆ E0 of positive measure such that |x(s)| ≥ 1/ j on D. We put xn := χD zn . Then xn ∈ X and kxn k → 0. Moreover, Fxn (s) = χD (s)ψ (s, λn (s))zn (s) = χD (s)ϕs (λn (s))

yn (s) = χD (s)yn (s). |yn (s)|

Since F is continuous by hypothesis, we conclude kχD yn k → 0. To define generalized ideal spaces, we use a somewhat weaker formulation of the above necessary condition as an axiom. (By Egorov’s theorem, this axiom is easily seen to be equivalent to the above condition under our hypothesis mes S < ∞). Definition 3. A generalized preideal space is a projectable space X with the property that for each sequence xn ∈ X which converges uniformly to 0 and each set E ⊆ S with mes E > 0 there is a set D ⊆ E with mes D > 0 such that kχD xn k → 0. If in addition X is complete, we call X a generalized ideal space. For the case that the set D ⊆ E can even be chosen independent of the sequence xn , we say that X locally contains L∞ ∩ X. Indeed, this condition means that for each set E ⊆ S with mes E > 0 there is a set D ⊆ E with mes D > 0 such that the space L∞ (D,U) (considered as a subspace of L∞ (S,U) by trivial extension) has the property that the embedding i : (L∞ (D,U) ∩ X, k · k∞ ) → (X, k · k) is continuous. Clearly, if X locally contains L∞ ∩ X then X is a generalized preideal space. One should expect that the converse is false, but the author does not know an example. Problem 1. Does each generalized (pre)ideal space X locally contain L∞ ∩ X? For Theorem 1 another hypothesis is required. Proposition 2. Let X be a projectable space which is continuously embedded into M . Whenever xn ∈ X ∩ L∞ (S,U) is such that kxn − xk → 0 for some x ∈ X and if xn → y uniformly, then y ∈ X.

152

M. Väth

Proof. Since X is continuously embedded into M , we have xn → x in M , and so x = y, i.e. y = x ∈ X. We will actually require slightly less. Definition 4. A projectable space is embeddable if for each sequence xn ∈ X ∩ L∞ (S,U) which converges uniformly to some y and which converges in X (to a possibly different function) the following holds. Each set E ∈ Σ with µ (E) > 0 contains some set D ∈ Σ with µ (D) > 0 such that χD y ∈ X. Note that this definition is rather analogous to the definition of a generalized preideal space. It seems therefore natural to conjecture that the two definitions are not independent of each other. However, the connection between the two definitions is not obvious. Problem 2. Is each generalized ideal space (or maybe even each generalized preideal space) embeddable? The author expects that the answer is negative, but it seems not easy to construct a counterexample. The class of spaces introduced so far is large enough for our purpose: Theorem 4. 1. Each preideal space X is an embeddable generalized preideal space which locally contains L∞ ∩ X. 2. Each preideal∗ space X of functions with values in Rn is an embeddable generalized preideal space which locally contains L∞ ∩ X. 3. If Xk are generalized preideal spaces, contain locally L∞ ∩ Xk , or are embeddable, then X1 × X2 × · · · × Xn has the same property. Theorem 4 is proved in [10]; except for the second claim the proofs are straightforward. The proof of the second claim is rather technical: roughly speaking, it is proved by showing some auxiliary claims by induction on max{esssup dim span {x(s) : x ∈ X0 } : X0 ⊆ X finite}. It turns out that the above “necessary” properties are already sufficient in order to prove the required analogue of Theorems 1–3. This is the main result of [10] (in the single-valued case). Theorem 5.

1. Theorem 1 holds for each embeddable generalized ideal space.

2. Theorem 2 holds for each generalized ideal space, in case x = 0 even for each generalized preideal space. 3. Theorem 3 holds whenever X is an embeddable generalized ideal space and Y is a regular generalized preideal space. Idea of the proof. The first claim follows from the closed graph theorem for the embedding X ,→ M . The second claim is proved very similarly to the case of ideal spaces. The last claim follows esssentially from the other two claims (as explained earlier for ideal spaces). Example 3. The operator (2) is continuous from W 1,q (S,U) into Lr (S,V ) if the Carath´eodory qn function f : S ×U ×U n → V satisfies (3) with p = n−q (if q < n) or some p ∈ (1, ∞) (if q ≥ n).

Generalized Ideal Spaces and Applications to the Superposition Operator

153

It is known that Theorem 1 holds also for preideal spaces, although the proof is much harder in this case, because the closed graph theorem cannot be used. Therefore, it is natural to ask: Problem 3. Does Theorem 1 hold for (embeddable) generalized preideal spaces? We point out that even if the answer to this question is positive, one may not drop the completeness hypothesis for X in the last part of Theorem 5. This is not surprising, because a special case of this result means that the linear superposition operator Fx(s) := c(s)x(s) is automatically bounded if it acts from an embeddable generalized ideal space X into a regular generalized preideal space. Of course, such a multiplication operator need not be bounded in incomplete spaces. For example, for the subspace X ⊆ L p ([0, 1]) of functions vanishing in a neighborhood of 0, the operator Fx(s) = x(s)/s maps X into itself but is unbounded. We close with the remark that many results of this paper have an extension when the measure space is not σ -finite and, moreover, even if the “norms” are not homogeneous or only quasi-pseudonorms. In particular, M itself becomes a generalized ideal space in this extended sense. Moreover, the results about the superposition operator hold also for multivalued functions f : S × U ( V . Note, however, in contrast to what is claimed in [2], the corresponding superposition operator is practically never upper semicontinuous, not even in measure. However, it is lower semicontinuous, and it is upper and lower semicontinuous in the ε -sense if f (s, · ) has the corresponding property. For further details, we refer to [10].

References [1] Appell, J., Nguyˆen˜ , H. T., and Zabre˘ıko, P. P., Multivalued superposition operators in ideal spaces of vector functions I, Indag. Math. 2 (1991), no. 4, 385–395. [2]

, Multivalued superposition operators in ideal spaces of vector functions II, Indag. Math. 2 (1991), no. 4, 397–409.

[3]

, Multivalued superposition operators in ideal spaces of vector functions III, Indag. Math. 3 (1992), no. 2, 1–9.

[4] Dirr, G. and V¨ath, M., Continuity of near-duality maps and characterizations of ideal spaces of measurable functions, Recent Trends in Nonlinear Analysis (Appell, J., ed.), Festschrift Dedicated to Alfonso Vignoli on the Occassion of His Sixtieth Birthday, Birkh¨auser, 2000, 139–148. [5] Mullins, C. W., Linear functionals on vector valued K¨othe spaces, Proc. Madras Conference Functional Analysis 1973 (Berlin) (Garnir, H. G., Unni, K. R., and Williamson, J. H., eds.), Lect. Notes Math., no. 399, Springer, 1974, 380–381. [6]

, Order vector valued K¨othe spaces, J. London Math. Soc. (2) 13 (1976), no. 1, 34–40.

[7] Nguyˆen˜ , H. T., The superposition operator in Orlicz spaces of vector valued functions (in Russian), Dokl. Akad. Nauk BSSR 31 (1987), 197–200.

154

[8]

M. Väth

, The theory of semimodules of infra-semiunits in ideal spaces of vector-valued functions, and its applications to integral operators (in Russian), Dokl. Akad. Nauk SSSR 317 (1991), 1303–1307, Engl. transl.: Soviet Math. Dokl. 43 (1991), 615–619.

[9] Nguyˆen˜ , H. T. and Zabre˘ıko, P. P., Cones of vector functions in Orlicz spaces of vector functions (in Russian), Vesc¯ı Akad. Navuk BSSR Ser. F¯ız.-Mat. Navuk (1990), no. 3, 30– 34. [10] V¨ath, M., Continuity of single- and multivalued superposition operators in generalized ideal spaces of measurable vector functions, (submitted). [11] [12]

, Ideal spaces, Lect. Notes Math., no. 1664, Springer, Berlin, Heidelberg, 1997. , Integration theory. A second course, World Scientific Publ., Singapore, New Jersey, London, Hong Kong, 2002.

[13] Zaanen, A. C., Integration, North-Holland Publ. Company, Amsterdam, 1967. [14] Zabre˘ıko, P. P., Ideal spaces of functions I (in Russian), Vestnik Jaroslav. Univ. 8 (1974), 12–52.

Martin V¨ath Univ. of W¨urzburg Dept. of Mathematics Am Hubland D-97074 W¨urzburg Germany

Current address: Free Univ. of Berlin Mathematics and Computer Science (WE1) Secr. B. Fiedler Arnimallee 2-6 D-14195 Berlin Germany

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 155-172 (2006)

Finite Elements in Vector Lattices Martin R. Weber (Dresden) Abstract. In vector lattices of continuous functions on a locally compact Hausdorff space a natural and important role play the finite functions, i.e. continuous functions with a compact support. Finite and totally finite elements are the abstract notion of such functions in arbitrary Archimedean vector lattices. This survey describes the main properties of finite and totally finite elements in arbitrary and normed vector lattices and in their sublattices. For most of the classical vector lattices the collection of all finite elements is indicated. Finite and totally finite elements and their relations to finite functions are studied in vector lattices of continuous functions. In special vector lattices of regular operators the first results on their finite elements will be presented. Then there is investigated the interesting question on the behaviour of finite elements, when the vector lattice is represented (isomorphically) as a vector lattice of continuous functions on a locally compact space, where one expects that finite elements are mapped into finite functions. Special situations are discussed where this is the case. By making use of the space of maximal ideals, in particular, a topological characterization of finite and totally finite elements by means of compact subsets is given. Key words: Vector lattice, Dedekind completeness, Finite element, Maximal ideals, Hull-kernel topology, Representation of vector lattices, Banach lattice, Regular operator, Modulus of an operator, Finite rank operator MSC 2000: 46B42, 47B60, 47B65, 54A05

1 Introduction Let S be a locally compact (noncompact) Hausdorff space and X(S) a vector lattice of continuous functions on S, i.e. X(S) ⊂ C(S). Functions with a compact support are of special interest and one might ask for some vector lattice characterization of such a function ϕ. This is easy to do for a positive function: the family of the infima of all multiples of ϕ with any positive function x ∈ X(S) should be majorized by one and the same function, of course with a constant depending on x (see figure 1). In general, the moduli of the functions have to be used. In the sequel the abstract version of this description leads to the Definition 1 of a finite element, already in an arbitrary Archimedean vector lattice. If the majorizing element itself is a finite element then ϕ is called totally finite, see Definition 2. The notion of a finite and a totally finite element in (abstract) Archimedean vector lattices E was introduced by B. M. Makarov and the author in 1973 (see [8]) and in 1975 (see [9]). Finite and totally finite elements in vector lattices are thoroughly studied in a series of papers, see [8], [9], [10], [11], [3], [4], [5], [16]. In connection with finite and totally finite elements several questions are quite natural and this survey will answer some of them, of course, under appropriate additional conditions and in

156

M. R. Weber

most cases in a condensed manner. To remove or weaken the latter seems to be desirable. Although as the rule, for the proofs we refer to the references it will be clear that special techniques have to be applied in order to establish the formula (3) contained in the main Definition 1, when the finiteness of some class of elements has to be proved. Moreover, some analysis of the formula (3) will help to derive more information about the structure of the finite elements in many special cases. Only some typical proofs (Theorems 4, 5, 10 and 14) are provided in order to demonstrate these two ideas. First of all, the relations of finite and totally finite elements in vector subspaces of a given vector lattice are of interest (§2). As the rule, an additional structure of the vector lattice will give some, and sometimes even exhaustive, information on the finite elements. Another cycle of problems is the study of both finite functions and finite elements in vector lattices of continuous functions on a locally compact Hausdorff space (§2.2), where one expects very close relations between. Although, in general, both notions are different a very mild condition ensures at least that any finite function is a finite element. However it is hard for a finite element to be a finite function. If E is a Banach (or vector) lattice and H is a vector sublattice of E then it is a natural question to establish the relations between Φ1 (E) and Φ1 (H), i.e. we ask, whether (or under which conditions) do the following relations hold ? a) Φ1 (H) ⊂ Φ1 (E),

b) Φ1 (E) ∩ H ⊂ Φ1 (H),

c) Φ1 (E) ∩ H = Φ1 (H).

(1)

In §3 we present some first results on finite elements in vector lattices of regular operators. It turns out that the finiteness of a finite rank operator is closely related to the finiteness of all elements constituting such an operator. These investigations have to be understood as a starting point of a systematic study of finite elements in particular vector lattices of operators. §4 is devoted to the representation theory of vector lattices containing finite elements as vector lattices of continuous functions. It is very natural, at least under some conditions, to expect that finite elements are isomorphically represented as finite functions. An Archimedean vector lattice possessing a sufficient number of finite elements allows (under some additional conditions) a representation as a vector lattice of (everywhere finite-valued) continuous functions on a locally compact σ-compact Hausdorff space such that all finite elements are represented as finite functions. Finally, for an Archimedean vector lattice we study the space of maximal ideals equipped with the hull-kernel topology. Since representations of vector lattices are actually constructed on topological spaces that are homeomorphic to subsets of the space of maximal ideals of the vector lattice (§5), one might expect to obtain some further information on finite elements by more detailed investigation of that topological space. This gives the possibility for an abstract characterization of finite and totally finite elements by means of the compactness of a special subsets in this space which can be assigned to each element. Recall some definitions, notations and elementary facts in an Archimedean vector lattice (E, E+ ) which will be used further on, where in the most cases we refer to [1] and [12]. • A set B ⊂ E is called a band if it is an order closed ideal, that is the limit (in E) of any order convergent net of the ideal B belongs to B. • Two elements x, y ∈ E are called disjoint written as x ⊥ y, if |x| ∧ |y| = 0.

Finite Elements in Vector Lattices

157

• For any nonempty subset A ⊂ E denote by Ad the set {x ∈ E : x ⊥ y for any y ∈ A}. The set Add is known as the band generated by A, the smallest band that contains A. If A consists of one single element x, the band generated by {x} is called a principle band. • A band B in E is a projection band if E = B ⊕ B d . In this case any element x ∈ E has a unique representation x = x1 + x2 , where x1 ∈ B and x2 ∈ B d . The map PB : E → E defined by PB (x) = x1 for any x ∈ E = B ⊕ B d is a positive projection. In a Dedekind complete vector lattice any band is a projection band. • If {u}dd is a projection band then P{u} is denoted by Pu . In this case for each element x ≥ 0 there exists the element sup{x ∧ n|u|} and, Pu (x) (for x ≥ 0) is calculated by the formula Pu (x) = sup{x ∧ n|u|} .

(2)

• A vector lattice E is said to have the principal projection property (PPP), if {x}dd is a projection band for each x ∈ E. Any σ-Dedekind complete vector lattice has the (PPP). • An element u ∈ E+ is a order unit, if for each x ∈ E there is a λ ∈ R with −λu ≤ x ≤ λu (or equivalently, |x| ≤ λu). • An element e ∈ E+ is a weak order unit, if x ∈ E and x ⊥ e imply x = 0, i.e. {e}dd = E. • An element 0 < a ∈ E is called an atom of E, if whenever 0 ≤ b ≤ a one has b = λa. A Banach lattice is said to be atomic if for each x > 0 there is an atom a such that 0 < a ≤ x. • A vector lattice E not possessing any order unit is called of type (Σ) if E contains a sequence of elements (en )∞ n=1 with the following property  e1 ≤ e2 ≤ · · · ≤ en ≤ · · · , ′ (Σ ) for any x ∈ E there exist n ∈ N and C > 0 such that |x| ≤ Cen .

2 Finite and totally finite elements in arbitrary Archimedean vector lattices 2.1 Definition of finite and totally finite elements Let E be an Archimedean vector lattice. Definition 1. An element ϕ ∈ E is called finite, if there is an element z ∈ E satisfying the following condition: for any element x ∈ E there exists a number Cx > 0 such that the inequality |x| ∧ n|ϕ| ≤ Cx z holds for all n ∈ N.

(3)

158

M. R. Weber

The element z is called an E-majorant or briefly a majorant of the finite element ϕ, see figure 1. The set of all finite elements of a vector lattice E is denoted by Φ1 (E). Obviously, Φ1 (E) is an ideal, i.e. a solid (sometimes also called normal) linear subspace of E. The trivial cases for Φ1 (E) to be even a projection band in E are Φ1 (E) = E and Φ1 (E) = {0}. The general case is considered in [4], Thm.2.8: Theorem 1. The ideal Φ1 (E) is a projection band of the vector lattice E if and only if E = E1 ⊕ E0 , where Φ1 (E1 ) = E1 and Φ1 (E0 ) = {0}. In this case E1 = Φ1 (E). nϕ

ϕ nϕ

x

ϕ

x

cx z z x ∧ nϕ

x ∧ nϕ

ϕ

Figure 1: Finite element ϕ with majorant z The special class of finite elements characterized by possessing at least one E-majorant, which itself is a finite element, in general, turns out to be different from Φ1 (E). Definition 2. A finite element ϕ ∈ E is called totally finite if it has an E-majorant z belonging to Φ1 (E). The set of all totally finite elements of a vector lattice E is also an ideal which will be denoted by Φ2 (E). Obviously, there hold the inclusions {0} ⊂ Φ2 (E) ⊂ Φ1 (E) ⊂ E, which might be proper (see [16]): {0} ⊆ Φ2 (E) ⊆ Φ1 (E) ⊆ E possible case 6= 6= 6= yes =

=

6=

yes

6=

=

6=

yes

6=

=

=

yes

6=

6=

=

no

=

6=

=

no

=

6=

6=

no

=

=

=

E = {0}.

In [9] it was shown that any element ϕ ∈ Φ2 (E) possesses an E-majorant which itself is a totally finite element, see also § 5. It is clear that Φ1 (E) = E implies Φ2 (E) = Φ1 (E). Each atom in a vector lattice is a totally finite element with itself as a majorant ([3], Prop.2.2).

Finite Elements in Vector Lattices

159

It is easy to show that Φ1 (E) = Φ2 (E) = E for the vector lattice E = c00 of all real sequences with finite support and for the vector lattice E = K(S) of all finite (i.e. with a compact support) continuous functions on a locally compact topological Hausdorff space S, see also the vector lattice in Corollary 6. Below (§2.4) we will detect vector lattices E with Φ1 (E) = {0}. From the definitions one has Theorem 2. If a vector lattice E has an order unit, then Φ1 (E) = Φ2 (E) = E. As a consequence, for the following classical vector lattices one immediately obtains that any element is finite: If E is one of the vector lattices c, l∞ , L∞ (µ) or C(K) (where K is a compact Hausdorff space), then Φ1 (E) = Φ2 (E) = E. Example for a finite element not being totally finite. Let be T = [−2, 2] \ {1, 21 , 13 , 41 , . . .}. The required vector lattice will be constructed by the help of the following functions eν (t) =

ν X k=1

1 , (t ∈ T ) |kt − 1|

ν = 1, 2, . . . .

The vector lattice E consists of all functions f on [−2, 2] restricted to T such that – f is continuous on [−2, 2] except a finite number of points

1 n

– for any n there exists the finite limit lim1 f (t)|nt − 1|. t→ n

Then E is vector lattice (even a uniformly complete and of type (Σ)) with the property Φ1 (E) 6= Φ2 (E), since one has (see [10]) (i) If ϕ ∈ E and ∃δ > 0 such that ϕ(t) = 0 for all t ∈ T ∩ [0, δ), then ϕ ∈ Φ1 (E). (ii) E ∋ ψ is totally finite (i. e. ψ ∈ Φ2 (E)) if and only if ∃δ > 0 such that ψ(t) = 0 for all t ∈ T ∩ (−δ, δ).

2.2 Finite elements and finite functions in vector lattices of continuous functions Let E(S) be vector lattice of continuous functions on some locally compact Hausdorff space S, i.e. E(S) ⊂ C(S). The finite functions in E(S) are K(S) ∩ E(S). As usual the finite elements in E(S) are denoted by Φ1 (E(S)). If E(S) = C(S) then Φ1 (C(S)) = K(S). In general, a)

K(S) ∩ E(S) * Φ1 (E(S)) and

b)

Φ1 (E(S)) * K(S) ∩ E(S).

An example for case a) is the vector lattice E of all continuous functions on [0, ∞) vanishing at 0. The finite function  if t ∈ [0, 1]  t, 2 − t, if t ∈ (1, 2] ϕ(t) =  0, if t ∈ (2, ∞)

160

M. R. Weber

belongs to E but fails to be a finite element. Indeed, if z would be an p E-majorant of ϕ then |x| ∧ nϕ ≤ cx z for some cx > 0 and any n ∈ N. The function x (t) = z(t) belongs also to 0 √ E and z ∧ nϕ ≤ c0 z for all n ∈ N. Since ϕ(t) > 0 on (0, 1) one p0 one has p √with some c0 > has ( z ∧ nϕ) (t) = z(t) and so z(t) ≤ c0 z(t) on (0, 1). This implies 0 < c12 ≤ z(t) what 0 contradicts to z(t) −→ 0. t→0 An example for case b) provides the vector lattice E of all continuous functions x on [1, ∞) × [1, ∞) such that there exist n ∈ N and λ > 0 with the property |x(t, s)| ≤ λtn for ∀(t, s) ∈ [1, ∞) × [1, ∞). In this vector lattice an element x belongs to Φ1 (E) if and only if x(t, s) = 0 on a set [a, ∞) × [1, ∞), where 0 < a = a(x). Clearly, not all such functions have a compact support. The following conditions on a vector lattices E(S) ⊂ C(S) turn out to be very important not only for a detailed investigation of them but also for the representation theory of general vector lattices containing finite elements: Condition (⋆): For ∀s ∈ S there ∃ x ∈ E(S) such that x(s) 6= 0. Condition (Φ): Any finite element of E(S) is a finite function. Condition (α): For any ordered pair of points s0 , s1 ∈ S (s0 6= s1 ) there ∃ a finite function x ∈ E(S) with x(sk ) = k, k = 0, 1. The condition (⋆) avoids the case a), i.e. there holds: If a vector lattice E(S) of continuous functions on a locally compact Hausdorff space S satisfies the condition (⋆), then any finite function of E(S) is a finite element of the vector lattice E(S). Obviously, the condition (Φ) avoids the case b). In [8] there are proved the following properties for a vector lattice E(S) which satisfies the condition (α): 1) If K ⊂ S is compact and s0 ∈ / K then ∃ a finite function x0 ∈ E(S) such that x0 (s0 ) = 0 and x0 (s) ≥ 1 on K 2) If F ⊂ S is closed and s1 ∈ / F then ∃ a finite function x ∈ E(S) such that x(s1 ) = 1 and x(s) = 0 on F 3) Let E(S) be uniformly complete. Then together with any finite function x0 ∈ E(S) the vector lattice E(S) contains all finite functions x ∈ C(S) such that supp(x) ⊂ supp(x0 ) 4) Let f be any discrete linear functional on E(S), i. e. f (x ∨ y) = max{f (x), f (y)}. If f does not vanish on K(S) ∩ E(S) then f = c δs for some point s ∈ S and c ∈ R+ 5)

Let S be locally compact, σ-compact. Let the vector lattice E(S) be uniformly complete, of type (Σ) and additionally satisfy the conditions (Φ) and (α)

=⇒

For any discrete linear functional f there ∃ a finite element x0 ∈ E(S) such that f (x0 ) 6= 0. (Then f = c δs also holds for some s ∈ S and c ∈ R+ )

2.3 Finite elements in arbitrary vector lattices We continue the study of finite and totally finite elements in vector lattices. For a given vector sublattice X of a vector lattice E an element z ∈ E+ is called a generalized order unit for X if

Finite Elements in Vector Lattices

161

for each x ∈ X there is a real number Cx > 0 with |x| ≤ Cx z. Note that E belongs then to the ideal generated in E by z and that z is not required to belong to X+ = X ∩ E+ . Theorem 3. Let E be a vector lattice. If ϕ ∈ E is a finite element then {ϕ}dd has a generalized order unit and {ϕ}dd ⊂ Φ1 (E). The converse statement of Theorem 2 is also true if E contains a weak order unit. Corollary 1. Let E be a vector lattice with a weak order unit. Then Φ1 (E) = Φ2 (E) = E if and only if E has an order unit. A weak order unit of a vector lattice E fails to be an order unit in general, even if Φ1 (E) = Φ2 (E) = E and E has an order unit. For example, u = (1, 21 , . . . , n1 , . . .) is a weak order unit but not an order unit in the vector lattice E = c. In a vector lattices with (PPP) the finite elements can be characterized as follows. Theorem 4. Let E be a vector lattice with (PPP). Then for an element ϕ ∈ E the following statements are equivalent: 1) ϕ is a finite element of E. 2) {ϕ}dd has a generalized order unit z ∈ E+ . 3) {ϕ}dd has an order unit z0 ∈ {ϕ}dd . Proof. 2) ⇒ 3). If z ∈ E+ is a generalized order unit of {ϕ}dd then for each x ∈ {ϕ}dd , there is a real positive number Cx such that |x| ≤ Cx z. Let Pϕ be the band projection from E onto {ϕ}dd . Then |x| = Pϕ |x| ≤ Pϕ (Cx z) = Cx Pϕ z = Cx z0 , where z0 = Pϕ z ∈ {ϕ}dd . This implies that z0 is an order unit of {ϕ}dd . 3) ⇒ 1). Since Pϕ |x| ∈ {ϕ}dd for arbitrary x ∈ E there is a positive number Cx such that Pϕ |x| ≤ Cx z0 . Then by using the formula (2) one has |x| ∧ n|ϕ| ≤ sup{|x| ∧ n|ϕ|} = Pϕ |x| ≤ Cx z0

for all

n ∈ N.

This implies that ϕ is a finite element of E. 1) ⇒ 2) is precisely Theorem 3. As a corollary one obtains Corollary 2. Let E be a vector lattice E with (PPP). Then Φ1 (E) = Φ2 (E) and Φ1 (E) has the (PPP). Theorem 4 shows that the finiteness of an element in an Archimedean vector lattice can be detected by the properties of the principle band it generates. Namely, let ϕ ∈ E be such that {ϕ}dd is a projection band. Then the element ϕ is finite if and only if {ϕ}dd has an order unit. In particular, if E is a σ-Dedekind complete vector lattice then Φ1 (E) = E if and only if each principal band possesses an order unit. So, if for an σ-Dedekind complete vector lattice E one has Φ1 (E) 6= E then there exists at least one principal band without order unit.

162

M. R. Weber

2.4 Finite elements in Banach lattices Without (PPP), the structure of a Banach lattice helps to describe the finite elements. Theorem 5. Let E be a Banach lattice and ϕ ∈ E. Then the following statements are equivalent: 1) ϕ is a finite element. 2) The closed unit ball B({ϕ}dd ) of {ϕ}dd is order bounded in E. 3) {ϕ}dd has a generalized order unit. Proof. We show only the equivalence of 1) and 3). For details see [3] and [4]. So, it is to show that the element ϕ is finite, if {ϕ}dd has a generalized order unit. In fact, let z ∈ E+ be a generalized order unit of {ϕ}dd . Define a norm on {ϕ}dd by kxkz = inf{λ > 0 : |x| ≤ λz},

x ∈ {ϕ}dd .


Then by Theorem 12.20, [1], the space {ϕ}dd , k · kz is an AM-space, where |x| ≤ kxkz z holds. Since the band {ϕ}dd is closed in E ([12], Prop.1.2.3) ({ϕ}dd , k · k) also is a Banach space. The open mapping theorem implies that the norms k·k and k·kz are equivalent on {ϕ}dd . In particular, there is a C > 0 such that kxkz ≤ Ckxk for all x ∈ {ϕ}dd . Now |x| ≤ kxkz z for each x ∈ {ϕ}dd , implies kxkz ≤ C, i.e. |x| ≤ Cz, for each x ∈ {ϕ}dd with kxk ≤ 1. If x ∈ E is now an arbitrary element then 0 ≤ |x| ∧ n|ϕ| ≤ |x| (and hence k|x| ∧ n|ϕ|k ≤ kxk) implies |x| ∧ n|ϕ| ≤ kxkCz for all n ∈ N, which means that ϕ is finite. The principal band generated by a finite element may fail to possess an order unit as the following example shows. Let E = C[0, 1] and H = {x ∈ E : x(t) = 0, ∀t ∈ [0, 21 ]}. Then Φ1 (E) = E, the ideal H is a principal band, moreover, H = {ϕ}dd for any ϕ ∈ H satisfying ϕ(t) 6= 0 for t ∈ ( 21 , 1] and H does not possess any order unit. However, each function z ∈ E with z(t) > 0 for t ∈ ( 21 − δ, 1] is a generalized order unit, where δ is some positive sufficiently small number. For details see [3]. For a Banach lattice E denote by ΓE the set of all atoms of E with norm 1. As was mentioned after Definition 2 the inclusion ΓE ⊂ Φ1 (E) holds. Theorem 6. Let the norm of the Banach lattice E be order continuous. Then 1) Φ1 (E) = Φ2 (E) = span(ΓE )1 , 2) Φ1 (E) is closed in E if and only if ΓE is a finite set, particularly, Φ1 (E) = E if and only if E is finite dimensional. For the following classical vector lattices one immediately obtains the following information on the finite elements: a) If E is one of the vector lattices c0 or lp with 1 ≤ p < ∞ then Φ1 (E) = Φ2 (E) = span(ΓE ) = span{ek : k = 1, 2, . . .}, where ek ∈ E is the sequence which k ′ s term equals 1 and all others are 0, b) If E = Lp (a, b) with 1 ≤ p < ∞ then Φ1 (E) = {0}. The vector lattice of all finite continuous functions on R is an example of a vector lattice possessing the property Φ1 (E) = Φ2 (E) = E. In the next theorem the class of Banach lattices with this property is characterized. 1

If ΓE = ∅ then we define span(ΓE ) = {0}.

Finite Elements in Vector Lattices

163

Theorem 7 (Characterization of Banach lattices with Φ1 (E) = Φ2 (E) = E). For a Banach lattice E the following statements are equivalent: 1) Φ1 (E) = Φ2 (E) = E. 2) E is lattice isomorphic to an AM-space and each pricipal band has a generalized order unit. Another result for a vector lattice E to satisfy Φ1 (E) = E uses the structure of a strict inductive limit (see [8], Thm.5.3). A vector lattice is called of type (LF ), if it is the strict inductive limit of an increasing sequence of F -spaces, where the topology is defined by a sequence of monotone seminorms. Theorem 8. Let E be a vector lattice of type (Σ) and of type (LF ). Then E = Φ1 (E).

2.5 Finite elements in sublattices We start with three natural situations, where a given vector lattice is embedded into another vector lattice. b denotes the Dedekind completion of an Archimedean vector lattice E then Φ1 (E) = If E b ∩ E, i.e. the relation c) of (1) is true. This follows from a general result which can be Φ1 (E) proved for any majorizing vector sublattice H of a vector lattice E (see [4], Thm.2.3). If E denotes the norm completion of a normed vetor lattice E then E is norm dense in E, however, in general, Φ1 (E) ⊂ Φ1 (E) does not hold. Whether the inclusion Φ1 (E) ∩ E ⊂ Φ1 (E) is true or not, is not known. The norm completion of the vector lattice c00 equipped with the supremum norm is the Banach lattice c0 . In this case Φ1 (c00 ) = c00 = Φ1 (c0 ). On the other hand the Banach lattice L1 (0, 1) is the norm completion of the vector lattice C[0, 1] equipped with the integral norm induced from L1 (0, 1). In this case Φ1 (C[0, 1]) = C[0, 1] and Φ1 (L1 (0, 1)) = {0}. If E is a Banach lattice, E ′′ its bidual and j : E → E ′′ the canonical embedding, then j(Φ1 (E)) ⊂ Φ1 (E ′′ ). After identification of j(E) with E this result can be written as Φ1 (E) ⊂ Φ1 (E ′′ ) ∩ E. The equation however is not true, in general, as the vector lattice c0 shows. For the details see [4], Thm.2.10. Observe that the inclusion a) of (1) may not hold if H is an arbitrary ideal of a Banach lattice E or if H is a norm closed sublattice which is the range of a positive projection on E. It holds if H is a closed ideal. The inclusion b) of (1) may not hold if H is a closed ideal or a band in E. It holds if H is a vector sublattice which is the range of a positive projection. The relation c) (and therefore also a) and b)) holds if H is a projection band: Theorem 9. If H is a projection band in a vector lattice E and PH the band projection from E
onto H, then PH Φ1 (E) = Φ1 (E) ∩ H = Φ1 (H). For an example and also for a description of the finite elements in the direct sums c0 (I, Ei), ℓp (I, Ei) for p ∈ [1, ∞) and ℓ∞ (I, Ei ) of Banach lattices Ei , where I is an arbitrary index set, we refer to [2], [4].

164

M. R. Weber

The indicated spaces are defined as follows. Let be x = (xi )i∈I , xi ∈ Ei . Then  c0 (I, Ei ), if ∀ε > 0 ∃ finite set Iε ⊂ I with kxi k < ε, ∀i ∈ / Iε     P ℓp (I, Ei ), if kxi kp < ∞ x = (xi )i∈I ∈ i∈I     ℓ∞ (I, Ei ), if sup kxi k < ∞ . i∈I

The linear operations and the order are understood to be the point- or coordinatewise ones, the norms are defined by  sup kxi k, if x ∈ c0 (I, Ei ), ℓ∞ (I, Ei )   i∈I   p1 kxk = k(xi )i∈I k = P  p  , if x ∈ ℓp (I, Ei ), 1 ≤ p < ∞ kxi k i∈I

An element (ϕi )i∈I belongs to Φ1 (c0 (I, Ei )) and Φ1 (ℓp (I, Ei )) if and only if each ϕi ∈ Φ1 (Ei ) for all i ∈ I and ϕi = 0 for all but finite many i ∈ I. An element (ϕi )i∈I is in Φ1 (ℓ∞ (I, Ei )) if and only if ϕi ∈ Φ1 (Ei ), ∀i ∈ I and there exist 0 ≤ zi ∈ Ei such that each closed unit ball B({ϕi }dd ) is a subset of [−zi , zi ] and sup kzi k < ∞. i∈I

3 Finite elements in vector lattices of regular operators 3.1 Regular operators on Banach lattices For two vector lattices E, F , where F is Dedekind complete the vector lattice Lr (E, F ) is Dedekind complete. So, Corollary 2 implies Φ1 (Lr (E, F )) = Φ2 (Lr (E, F )). A consequence of Theorem 4 is that Orth(E) = {I}dd = {T ∈ Lr (E) : − λI ≤ T ≤ λI} ⊂ Φ1 (Lr (E)) = Φ2 (Lr (E)), in particular, the identity operator IE is a finite element in Lr (E) with itself as an Lr (E)majorant, provided the Banach lattice E is Dedekind complete. That is for any U ∈ Lr (E) there exists a positive number cU such that |U| ∧ nIE ≤ cU IE

for all

n∈N.

(4)

Theorem 10. Let E and F be Banach lattices such that F is Dedekind complete and let T : E → F be a lattice isomorphism (onto F ). Then T is a finite element in Lr (E, F ). Moreover, T is a majorant of itself. Proof. Since T is a lattice isomorphism, E is Dedekind complete as well. The order continuity of T (as any lattice isomorphism) guarantees the equalities |T U| = T |U| and T (U1 ∧ U2 ) = (T U1 ) ∧ (T U2 ) for any U, U1 , U2 ∈ Lr (E).

(5)

If S ∈ Lr (E, F ) then, obviously, T −1 S ∈ Lr (E) and therefore (4) implies |T −1 S| ∧ nIE ≤ cT −1 S IE ,

for

n ∈ N.

(6)

Finite Elements in Vector Lattices

165

If now the (positive) operator T is applied to the inequality (6), then by means of (5)
T |T −1 S| ∧ nIE = |S| ∧ nT ≤ cT −1 S T for n ∈ N

follows, which shows that the operator T is a finite element in Lr (E, F ) with itself as a majorant. An application of techniques and results from §2 is now

Theorem 11. If E and F be Banach lattices such that F is Dedekind complete. Let be H a band of F and P : F → H the band projection. Then 1) Lr (E, H) is a projection band of Lr (E, F ). 2) Φ1 (Lr (E, H)) = Φ1 (Lr (E, F )) ∩ Lr (E, H) = {P T : T ∈ Φ1 (Lr (E, F ))}. For Banach lattices E and F define the mapping P : Lr (E, F ′′ ) → Lr (F ′ , E ′ ) by P(T ) = T ′ J for T ∈ Lr (E, F ′′ ), where T ′ : F ′′′ → E ′ is the adjoint operator to T and J : F ′ ֒→ F ′′′ the canonical embedding2. The following fact is established in [6], Thm.5.6 and will be used to
prove the next theorem:
P is an isometric lattice isomorphism from Lr (E, F ′′ ), k · kr onto Lr (F ′ , E ′ ), k · kr , where k · kr denotes the regular operator norm, respectively. Theorem 12. Let E and F be Banach lattices, and P as above. Denote A = Lr (E, F ′′ ) and B = Lr (F ′ , E ′ ). Then 1) T ∈ A is finite if and only if P(T ) is finite in B, i.e. P (Φ1 (A)) = Φ1 (B). 2) U ∈ L+ (E, F ′′ ) is an A-majorant of T if and only if P(U) is a B-majorant of P(T ). Note that if T ∈ Lr (E, F ) then jT ∈ A, where again j : F ֒→ F ′′ denotes the canonical inclusion mapping. Since T ′ = P(jT ) we have Corollary 3. Let E and F be Banach lattices and T ∈ Lr (E, F ). Then T ′ is finite in L (F ′ , E ′ ) if and only if jT is finite in Lr (E, F ′′ ). r

As a special case, when F is a reflexive Banach lattice, we obtain that the finiteness of an operator T : E → F can be characterized by the finiteness of its adjoint T ′ . Corollary 4. If F is a reflexive Banach lattice then for each Banach lattice E, an operator T ∈ Lr (E, F ) is a finite element if and only if T ′ is finite in Lr (F ′ , E ′ ), i. e.

Φ1 Lr (F ′ , E ′ ) = T ′ : T ∈ Φ1 Lr (E, F ) . Moreover, U ∈ L+ (E, F ) is an Lr (E, F )-majorant of T if and only if U ′ is an Lr (F ′ , E ′ )majorant of T ′ .

For finite elements in the vector lattice of regular operators defined on an AL-space there are some results gathered in the next theorem ([5], Thm.2.7), where the vector lattice F is always assumed to be Dedekind complete. 2

where the notation J is chosen only for stressing the slightly different situation.

166

M. R. Weber

Theorem 13 (Regular operators on AL-spaces). 1) E is an AL-space and F is an AM-space with order unit

=⇒

Φ1 (Lr (E, F )) = Lr (E, F ).

2) E is an AL-space and F is an AM-space

⇐=

Φ1 (Lr (E, F )) = Lr (E, F ).

3) Let E be a Dedekind complete Banach lattice.
r Then Φ1 L (E) = Lr (E) if and only if dim E < ∞.


The Banach lattice F may fail to have an order unit even if Φ1 Lr (E, F ) = Lr (E, F ), the Banach lattice E is an AL-space and F is an AM-space. For an example see [5] and [3].

3.2 Finite rank operators in Lr (E, F ) Now we consider the finite rank operators in Lr (E, F ), where E, F are vector lattices and F is Dedekind complete. n P Let be ψ1′ , . . . , ψn′ ∈ E ′ and ϕ1 , . . . , ϕn ∈ F and T = ψi′ ⊗ ϕi . Each finite rank operator i=1

possesses a compact modulus (see [1], Thm.5.7), which is dominated by the operator n X

|ψi′ | ⊗ |ϕi |

i=1

however, need not coincide with it. In [5] §4 there is proved the following result. Theorem 14 (Finite rank operators). n P Let T = ψi′ ⊗ ϕi belong to Lr (E, F ). Let at least either ψ1′ , . . . , ψn′ be pairwise disjoint in i=1

E ′ or ϕ1 , . . . , ϕn be pairwise disjoint in F . Then T ∈ Φ1 (Lr (E, F ))

⇐⇒

ψi′ ∈ Φ1 (E ′ ) and ϕi ∈ Φ1 (F ), i = 1, . . . , n.

For rank one operators we get that ψ ′ ⊗ ϕ is a finite element in Lr (E, F ) if and only if ψ ′ ∈ Φ1 (E ′ ) and ϕ ∈ Φ1 (F ). In order to prove at least one special case of the formulated theorem and also to demonstrate how from the defining inequality (3) of a finite element one can derive additional information on the structure of the finite elements in some given vector lattice we show that ψ ′ ∈ Φ1 (E ′ ) and ϕ ∈ Φ1 (F ) provided the rank one operator T = ψ ′ ⊗ ϕ is a finite element in Lr (E, F ) with a majorant U ∈ L+ (E, F ). Namely, without loss of generality we may assume ψ ′ ≥ 0 and ϕ ≥ 0. Then for each S ∈ Lr (E, F ) there is a positive constant cS such that |S| ∧ nT ≤ cS U

for all n ∈ N.

(7)

Finite Elements in Vector Lattices

167

In particular, if S = ψ ′ ⊗ h ∈ Lr (E, F ), where h ∈ F then it follows from (7) and Theorem 1.16 of [1] that
cS Ux ≥ |S| ∧ nT (x) k k nX o X = inf (|S|xi) ∧ (nT xi ) : xi ∈ E+ , xi = x, n ∈ N = inf

= inf

i=1 k nX

i=1 k nX

i=1

′

′

(ψ (xi )|h|) ∧ (ψ (xi )nϕ) : xi ∈ E+ ,

k X

xi = x, n ∈ N

i=1

ψ ′ (xi )(|h| ∧ (nϕ)) : xi ∈ E+ ,

i=1

k X

xi = x, n ∈ N

i=1

o

o

= ψ ′ (x)(|h| ∧ (nϕ)) for all x ∈ E+ and n ∈ N. One finds some x0 ∈ E+ such that ψ ′ (x0 ) = 1 and has |h| ∧ (nϕ) ≤ cS Ux0

for

n ∈ N,

which implies that ϕ is a finite element of F . On the other hand, if S = h′ ⊗ ϕ, where h′ ∈ E ′ then again by means of (7) for arbitary k P x, xi ∈ E+ with xi = x one has i=1




|h′ | ∧ nψ ′ (xi ) ϕ ≤ |h′ |(xi ) ϕ ∧ nψ ′ (xi ) ϕ = (|S|xi ) ∧ (nT xi )

for each 1 ≤ i ≤ k, and thus (|h′ | ∧ nψ ′ )(x) ϕ ≤

k P

(|S|xi) ∧ (nT xi ). It follows from Theorem

i=1

1.16 of [1] that (|h′ | ∧ nψ ′ ) (x) ϕ ≤ (|S| ∧ (nT )) (x) ≤ cS Ux for all x ∈ E+ and n ∈ N. With some u′ ∈ F+′ such that u′ (ϕ) = 1 one has (|h′ | ∧ nψ ′ ) (x) ≤ u′ (cS Ux) = cS (U ′ u′ )(x)

x ∈ E+ , n ∈ N.

Therefore |h′ | ∧ nψ ′ ≤ cS (U ′ u′ ) = cS v ′

n ∈ N,

where v ′ = U ′ u′ ∈ E+′ . This shows that ψ ′ is finite in E ′ .

4 Representation of vector lattices containing finite elements Let E be an Archimedean vector lattice and S a topological Hausdorff space. A vector lattice E(S) ⊂ C(S) is called a representation3 of the vector lattice E, if there is a vector lattice isomorphism i : E → E(S). In many situations representations of Archimedean vector lattices (Dedekind complete or not) are considered preferably on compact (extremally disconnected) 3

More exactly, the representation of E should be understood as the pair (E(S), i).

168

M. R. Weber

spaces S, where sometimes continuous functions are allowed, that might take on even infinite values on nowhere dense subsets of S, see e.g. [13]. For Archimedean vector lattices containing nontrivial finite elements it seems to be natural if one looks for representations consisting of continuous functions on a locally compact space. This would give the possibility to represent finite elements of the vector lattice as finite functions. A representation E(S) is called a (⋆)-representation, if the vector lattice E(S) satisfies the condition (⋆), see § 2.2. Analogously, (Φ)- and (α)- and also (Φα)-representations are defined. Based on the S.Kakutani-H.F.Bohnenblust-M.G.Krein-S.G.Krein-Theorem ([1], Thm.12.28) a vector lattice E of type (Σ) possesses a (⋆)-representation consisting of bounded functions if there exists a monotone norm on E, ([8], Thm.1.3). A vector lattice of type (Σ) can not have a (⋆)-representation on a compact space S ([15], Lemma 7). The next results show that for vector lattices of type (Σ) there exist more qualified representations such that any member of a given countable collection of finite elements will be mapped by means of an isomorphism on a finite function. For the details (and other kinds of representations) see [14], [8], [15], [11]. Theorem 15. Let E be a vector lattice of type (Σ). Let E admit a monotone norm and let {ϕn }n∈N be a sequence of finite elements of E. Then there exists a (⋆)-representation on some σ-compact (locally compact) space S such that each element ϕn , n ∈ N is represented as a finite functions on S. Theorem 16. Let E be a vector lattice of type (Σ). Then for E to possess a (Φα)-representation on some σ-compact space S it is sufficient and, in case of the uniform completeness of E, also necessary, that there exists a sequence of finite elements {ϕn }n∈N in E satisfying the condition: for any discrete (see §2.2) functional f there exists a number n such that f (ϕn ) 6= 0. Corollary 5. Let E be a vector lattice of type (Σ) such that E = Φ1 (E). Then E possesses a (Φα)-representation E(S) on some σ-compact space S. If moreover, E is uniformly complete, then E(S) = K(S). Together with Theorem 8 from Corollary 5 follows now Corollary 6. Let E be a vector lattice of type (Σ) and of type (LF ). Then E has a representation as K(S) on some locally compact σ-compact space S (see [7]). Indeed, the vector lattice E is uniformly complete since it is the strict inductive limit of its subspaces of type (F ). The equality E = Φ1 (E) is established in Theorem 8. In the next section among others we study the relation between the space S of a representation E(S) and the space of all maximal ideals of E topologized in an appropriate manner, where the latter space or some of its subspaces turn out to be homeomorphic to S.

Finite Elements in Vector Lattices

169

5 The space of maximal ideals of a vector lattice For an Archimedean vector lattice E denote by M(E) the set of all maximal ideals of E. For any discrete functional f 6= 0 on E one has f −1 (0) ∈ M(E) and, vice versa, any M ∈ M(E) defines a discrete functional4 (up to a constant coefficient). The collection M(E) will be equipped with the hull-kernel topology τhk by defining the closure of any subset A ⊂TM(E): a maximal ideal M0 ∈ M(E) belongs to the closure of A (i.e. M0 ∈ A), iff M0 ⊃ M. M ∈A

If (E(S), i) is a (⋆)-representation of E then for each point s ∈ S the sets Hs = {x ∈ E(S) : x(s) = 0}

and i−1 (Hs ) are maximal ideals in E(S) and E, respectively. The canonical map κ : S → M(E) defined by κ(s) = i−1 (Hs ) is always continuous. Under additional conditions the spaces S and M(E) might be even canonically homeomorphic. Important subsets of M(E) are obtained as follows. For any x ∈ E put Gx = {M ∈ M(E) : x ∈ / M}. Then the setS suppM(x) = Gx is called the abstract support of the element
x. For A ⊂ E put G(A) = Gx and use the special notation MΦ (E) = G Φ1 (E) for x∈A

A = Φ1 (E). The main properties of the topology τhk : a) Gx is a τhk -open set for each x ∈ E

b) The system {Gx }x∈E is a basis of the toplogy τhk
c) M(E), τhk is a Hausdorff space

d) MΦ (E) is the largest locally compact subspace contained in M(E) and the system {Gϕ }ϕ∈Φ1 (E) is a basis of τhk in MΦ (E).

will be used both for a deeper study of finite elements in vector lattices (see [9], [10]) as well as for further developing of the representation theory in [8] and [11]. Theorem 17 (Topological characterization of finite and totally finite elements). For a radical-free Archimedean vector lattice E the following assertions hold: 1) ϕ ∈ Φ1 (E) ⇐⇒ suppM(ϕ) is a compact set with respect to τhk 2) z ∈ E is an E-majorant of the finite element ϕ ⇐⇒ suppM(ϕ) ⊂ Gz 3) ψ ∈ Φ2 (E) ⇐⇒ suppM(ψ) is compact and suppM(ψ) ⊂ MΦ (E) 4) Φ2 (E) = Φ1 (Φ2 (E)), i.e. each totally finite element has a totally finite majorant. 4

Concerning discrete functionals or maximal ideals in the vector lattice E we assume not only the existence of them but in most cases also that there are sufficient many to separate the vectors of E T (i.e. f (x) = 0 for all discrete functionals f implies x = 0), or equivalently, that E is radical-free, i.e. R = R(E) = {M : M ∈ M(E)} = {0}. For example, vector lattices of type (Σ) and vector lattices such that Φ1 (E) = E are radical-free. In case of R(E) 6= {0} the already radical-free vector lattice E/R will be considered instead of E, because the spaces M(E) and M(E/R) are then homeomorphic.

170

M. R. Weber

Many information on the finite elements of E are related to topological properties of M(E) and of the subspace MΦ (E). The compactness of M(E) e.g. implies the existence of an oder unit in E and therefore Φ1 (E) = E. The σ-compactness (and non-compactness) of MΦ (E) is necessary and sufficient for Φ2 (E) to be a vector lattice of type (Σ). Especially for vector lattices of type (Σ) there is known quite a lot about MΦ (E). In that case its closedness implies its σ-compactness and by the properties 2) and 3) of Theorem 17 the equality Φ2 (E) = Φ1 (E). For MΦ (E) = MΦ (E) in M(E) we mention the following result Theorem 18. Let E be of type (Σ). For the closedness of MΦ (E) in M(E) it is necessary and in case of the uniform completeness of E also sufficient that there holds both conditions: (i) Φ2 (E) is a vector lattice of type (Σ) and (ii) Φ1 (E) = Φ2 (E). All the conditions in that theorem, i.e. type (Σ), uniform completeness, (i) and (ii) are essential, as exhaustive examples in [10] demonstrate. Now coming back to the question of representation we are able to add some further results, where the underlying space is homeomorphic to a subspace of M(E). Based on the next theorem more qualified representations of vector lattices containing sufficiently many finite elements can be constructed, see [11]. Theorem 19. Let (E(S), i) be an (α)-representation of E and E0 = {x ∈ E : ix ∈ K(S)}. Then 1) E0 is an ideal of E and E0 ⊂ Φ2 (E), 2) E0 is order dense5 in E, i.e. x ⊥ y for all x ∈ E0 implies y = 0, 3) E0 is embeddable into a vector lattice with order unit, 4) S is homeomorphic to G(E0 ) (and also to M(E0 )). An ideal E0 ⊂ E is called an R-base (representation base), if E0 satisfies the first three conditions of Theorem 19. Definition 3. A representation (E(S), i) of E is called an representation by means of the Rbase E0 if iE0 ⊂ K(S) (i.e. iE0 is a vector lattice of finite functions) and satisfies the condition (α). Theorem 19 implies that a representation by means of any R-base E0 is always a representation on G(E0 ) = M(E0 ). Theorem 20 (Existence of base representations). Let E be a vector lattice and E0 an R-base in E. Then there exists a representation by means of E0 . If (E(S), i) is this representation then iE0 = K(S) if and only if E0 is uniformly complete. 5

see [17], Thm.23.3.

Finite Elements in Vector Lattices

171

Analogously to Theorem 16 one has now Theorem 21. For the existence of a (Φα)-representation (E(S), i) for a vector lattice E on a σcompact space S it is sufficient and, if E is of type (Σ) and uniformly complete, also necessary that the space M(E) is locally compact and σ-compact.

References [1] C.D. Aliprantis and O. Burkinshaw. Positive Operators. Academic Press, Inc., London, 1985. [2] Z.L. Chen. On week sequential precompactness in Banach lattices. Chinese J. Contemporary Math., 20 (4):477–486, 1999. [3] Z.L. Chen and M.R. Weber. On finite elements in vector lattices and Banach lattices. Math. Nachrichten, 279, No.5-6, 495–501 (2006). [4] Z.L. Chen and M.R. Weber. On finite elements in sublattices of Banach lattices. Math. Nachrichten, to appear [5] Z.L. Chen and M.R. Weber. On finite elements in lattices of regular operators. Prepr., Techn. Univ. Dresden, No.MATH-AN-06-03, 2003. . [6] Z.L. Chen and A.W. Wickstead. Some applications of Rademacher sequences in Banach lattices. Positivity, 2:171-191, 1998. [7] I. Kawai. Locally convex lattices. J. Math. Soc. Japan, 9, no.3-4, 1957. ¨ [8] B.M. Makarow and M. Weber. Uber die Realisierung von Vektorverb¨anden I. (Russian). Math. Nachrichten, 60:281–296, 1974. [9] B.M. Makarow and M. Weber. Einige Untersuchungen des Raumes der maximalen Ideale eines Vektorverbandes mit Hilfe finiter Elemente I. Math. Nachrichten, 79:115–130, 1977. [10] B.M. Makarow and M. Weber. Einige Untersuchungen des Raumes der maximalen Ideale eines Vektorverbandes mit Hilfe finiter Elemente II. Math. Nachrichten, 80:115–125, 1977. ¨ [11] B.M. Makarow and M. Weber. Uber die Realisierung von Vektorverb¨anden III. Math. Nachrichten, 68:7–14, 1978. [12] P. Meyer-Nieberg. Banach Lattices. Springer-Verlag, Berlin, Heidelberg, New York, 1991. [13] B.Z. Vulikh. Introduction to the Theory of Partially Ordered Spaces. Wolters-Nordhoff, Groningen, 1967. ¨ [14] M. Weber. Uber eine Klasse von K-Linealen und ihre Realisierung. Wiss. Zeitschrift TH Karl-Marx-Stadt, XIII, Heft 1: 159–171, 1971.

172

M. R. Weber

¨ [15] M. Weber. Uber die Realisierung von Vektorverb¨anden II. Math. Nachrichten, 65:165– 177, 1975. [16] M.R. Weber. On finite and totally finite elements in vector lattices. Analysis Mathematica, 21:237–244, 1995. [17] A.C. Zaanen. Introduction to Operator Theory in Riesz Spaces. Springer-Verlag, Berlin, Heidelberg, New York, 1997.

Martin R. Weber Fachrichtung Mathematik Technische Universit¨at Dresden D - 01062 Dresden e-mail: [email protected]

PROCEEDINGS Positivity IV - Theory and Applications Dresden (Germany), 173-182 (2006)

When are ultrapowers of normed lattices discrete or continuous? Witold Wnuk1 (Pozna´n) and Bła˙zej Wiatrowski (Pozna´n) Abstract. The paper is devoted to a study of some aspects of the order structure of ultrapowers. The topics considered are the following. First, a characterization of normed lattices whose ultrapowers are continuous is presented. Second, a description of discrete elements in ultrapowers is shown. Third, we answer the question when the kernel of the limit over an ultrafilter is a σ-ideal. Key words: normed lattice, continuous Riesz space, discrete Riesz space, ultraproduct of normed spaces. MSC 2000: Primary 46A40, 46B42, 46B08.

1 Introduction Investigations of ultraproducts of Banach spaces brought a lot of useful methods and results which were applicable, with a success, to problems in the local theory of Banach spaces and in operator ideals on them. Systematic studies of this special sort of quotients started in early seventies after the paper [2] of D. Dacunha-Castelle and J.L. Krivine. This direction of research was continued later by many authors such as C.W. Henson ([5], [6]), B. Sims ([14]), S. Heinrich ([4]). The papers mentioned above contain fundamental results concerning ultraproducts of classical Banach spaces: Lp (µ), ℓ∞ , C(S), and Orlicz spaces as well, which are also important examples of Banach lattices. As far as we know ultraproducts of general normed lattices, from the point of view of their order structure, were considered very seldom. Let us mention a short note [15] where it was shown that every ultraproduct of a family of normed lattices with respect to a countably incomplete ultrafilter satisfies the σ-Lebesgue property. It follows that a normed lattice embeds, as a vector sublattice, into a normed lattice satisfying the σ-Lebesgue property. An analysis of conditions implying continuity of ultrapowers and a description of discrete elements there are the main aim of our paper. The reader interested in basic facts and terminology concerning ultraproducts and Banach lattices is referred to B. Sims’ and S. Heinrich’s papers [4], [14], and to the books [1], [11], [16]. We recall some concepts needed for this paper. Let P(I) be the power set of an infinite set I. A filter A ⊂ P(I) is said to be an ultrafilter if each filter containing A coincides with A. Ultrafilters can be characterized as follows (see [3] Theorem 9.10). Theorem 1.1 Let A be a filter in P(I). Then the following are equivalent: (a) A is an ultrafilter. 1

The research of the first author was supported in part by Komitet Bada´n Naukowych (State Committee for Scientific Research), Poland, grant no P 03A 022 25.

174

W. Wnuk, B. Wiatrowski

(b) ∀A,B⊂I A ∪ B ∈ A =⇒ A ∈ A or B ∈ A. (c) ∀A⊂I A ∈ A or I r A ∈ A. (d) A set B ⊂ I belongs to A if A ∩ B 6= ∅ for all A ∈ A. (e) Given pairwise disjoint sets J1 , J2 , . . . , Jn ⊂ I such that k ∈ {1, 2, . . . , n} such that Jk ∈ A.

Sn 1

Jk = I there exists a unique

Countably incomplete ultrafilters are of special importance for the theory of ultraproducts. The countable incompleteness means exactly that the ultrafilter A contains a decreasing sequence of sets with empty intersection, i.e., A is not closed under the formation of countable intersections. Every ultrafilter A ⊂ P(N) containing none one-point set belongs to the class of countably incomplete ultrafilters. If {i0 } ∈ A for some i0 ∈ I, then A is said to be trivial. If a family of real numbers (xi )i∈I is bounded and A ⊂ P(I) is an ultrafilter then there exists a unique number x such that {i : |xi − x| < ε} ∈ A for every ε > 0. The number x, denoted by limA xi , is called the limit of (xi ) over the ultrafilter A. There holds {limA xi } = T J∈A {xi : i ∈ J}. Properties of limA and the limit of sequences are similar. Below we mention a few of them which are useful in further considerations: • limA is additive and homogeneous, • limA is monotone, i.e., if families (xi )i∈I , (yi )i∈I ⊂ R are bounded and {i : xi 6 yi } ∈ A, then limA xi 6 limA yi , • limA xi > a implies {i : xi > a} ∈ A, • if f : R2 → R is continuous and families (xi )i∈I , (yi)i∈I ⊂ R are bounded then f (limA xi , limA yi ) = limA f (xi , yi ). For every countably incomplete ultrafilter A there exists a family (xi )i∈I of positive real numbers with limA xi = 0. Indeed, fix numbers 0 < an → 0 and put ( 1 for i ∈ I r I1 xi = ak for i ∈ Ik r Ik+1 , T where I1 ! I2 ! · · · ∈ A, n In = ∅. Let Ei = (Ei , k · ki ), i ∈ I, be a family of normed lattices. Define Y ℓ∞ (Ei , I) = {(xi ) ∈ Ei : sup kxi ki 6 ∞}. i∈I

i

It is clear that ℓ∞ (Ei , I) is a Riesz space under the natural order induced from the product Q ∞ i∈I Ei and the equality k(xi )k = supi kxi ki defines a monotone norm on ℓ (Ei , I). For an ∞ ultrafilter A ⊂ P(I) we will denote by NA the following subspace of ℓ (Ei , I) NA = {(xi ) ∈ ℓ∞ (Ei , I) : lim kxi ki = 0}. A

The subspace NA is a closed ideal and NA is order dense when none one-point set belongs to A.

When are ultrapowers of normed lattices discrete or continuous?

175

The ultraproduct (Ei )A of the family of normed lattices Ei , i ∈ I, with respect to the ultrafilter A is the quotient Riesz space ℓ∞ (Ei , I)/NA equipped with the quotient norm which can be expressed by the formula kQ((xi ))k = lim kxi ki , A

where Q : ℓ∞ (Ei , I) → (Ei )A is the canonical quotient map. If Ei = E for every i ∈ I and a certain E, then we speak about an ultrapower (E)A of E. There is an isometric embedding of E onto a Riesz subspace of (E)A which is defined by x −→ Q((xi )), where xi = x for all i ∈ I. A notion of a discrete element will be very important for our investigations. Let us recall that a strictly positive element e ∈ E is discrete if |y| 6 e implies y = te for some real number t. In the class of Archimedean Riesz spaces discrete elements have a useful characterization: e is discrete if and only if the interval [0, e] does not contain two nonzero disjoint elements (see [10] Theorem 26.4). A Riesz space E is said to be discrete if every positive element in E majorizes a discrete element, or equivalently: E possesses a complete disjoint system consisting of discrete elements. Classical Banach sequence spaces ℓp , c0 are standard examples of discrete Riesz spaces. On the other hand E is called continuous if E does not contain discrete elements. Spaces Lp [0, 1], C[0, 1] belong to the class of continuous Riesz spaces. Finally, E is heterogeneous if it is neither discrete nor continuous. Products of a discrete space and a continuous one are the simplest examples of heterogeneous spaces. Our notation is standard. Let us Q only explain that 1A stands for the characteristic function of a set A. Moreover, if x = (xi ) ∈ i∈I Ei , where Ei are linear spaces, then for A ⊂ I a notation x1A means x1A (i) = xi for i ∈ A and x1A (i) = 0 whenever i ∈ / A.

2 A characterization of continuous ultrapowers We start considerations with a lemma introducing a special type of a norm. Lemma 2.1 For a normed lattice (E, k · k) the following conditions are equivalent. (∗) ∃d>0 ∀x>0 ∃x1 ,x2 ∈[0,x]

x1 ∧ x2 = 0 and min(kx1 k, kx2 k) > dkxk.

(∗∗) ∃c>0 ∀ε>0 ∀x>0 ∃x1 ,x2 ∈[0,x]

kx1 ∧ x2 k < ε and min(kx1 k, kx2 k) > ckxk.

Proof. (∗) =⇒ (∗∗) obvious. x (∗∗) =⇒ (∗) Let 0 < x, ε = 41 c. There exist y1 , y2 ∈ [0, kxk ] such that ky1 ∧y2 k < 14 c and min(ky1k, ky2 k) > c. Put xi = kxk(yi − y1 ∧ y2 ) for i = 1, 2. Hence xi ∈ [0, x], x1 ∧ x2 = 0 and moreover kxi k > kxk(kyik − ky1 ∧ y2 k) > 43 ckxk. It is sufficient to take d = 43 c. Let us note that condition (∗) implies continuity of E. Moreover, conditions (∗) and (∗∗) are preserved whenever the original norm is replaced by an equivalent monotone norm. A characterization of normed lattices having all ultrapowers continuous goes as follows. Theorem 2.2 If A is a countably incomplete ultrafilter, then the ultrapower (E)A is continuous if and only if (E, k · k) satisfies condition (∗).

176

W. Wnuk, B. Wiatrowski

Proof. =⇒ Suppose that (∗) does not hold. Therefore, for every n there exists 0 6 an ∈ E such that min(kyk, kzk) < n1 kan k for disjoint y, z ∈ [0, an ]. Replacing, if necessary, an by kan k−1 an we can assume kan k = 1. Let (In ) be a decreasing sequence of sets from A with empty intersection. Define xi = a1 for i ∈ I r I1 , and xi = an for i ∈ In r In+1 . We obtain limA kxi k = 1, i.e., (xi ) ∈ / NA. Let Q : ℓ∞ (E, I) → (E)A be the quotient map. We claim that Q((xi )) is discrete. Indeed, suppose Q((yi )), Q((zi )) ∈ [0, Q((xi ))], Q((yi )) ∧ Q((zi )) = 0. Without loss of generality we can assume yi , zi > 0. Elements yi = (yi − yi ∧ zi ) ∧ xi , zi = (zi − zi ∧ yi) ∧ xi are disjoint and they belong to [0, xi ]. Additionally, Q((yi )) = Q((yi )) and Q((zi )) = Q((zi )). Putting ci = 1 for i ∈ I r I1 and ci = n1 for i ∈ In r In+1 we get min(kyi k, kzi k) 6 ci . Therefore min(limA kyi k, limA kzi k) = limA min(kyi k, kzi k) 6 limA ci = 0. In other words (yi ) ∈ NA or (zi ) ∈ NA, i.e., Q((yi )) = 0 or Q((zi )) = 0. ⇐= Fix 0 < Q((xi )). We can assume xi > 0. There holds limA kxi k > 0. According to (∗) intervals [0, xi ] contain disjoint yi , zi satisfying kyi k, kzik > dkxi k. Hence Q((yi )) ∧ Q((zi )) = 0, Q((yi )), Q((zi )) ∈ [0, Q((xi ))] and Q((yi )) 6= 0, Q((zi )) 6= 0, because min(limA kyi k, limA kzi k) > d limA kxi k > 0. Finally, none element Q((xi )) is discrete. Many classical Banach lattices satisfy condition (∗) as examples below show. Examples of Banach lattices satisfying (∗). 1. Let E be a σ-Dedekind complete Banach lattice whose dual space E ∗ is continuous. Condition (∗) is satisfied for d = 41 . Fix 0 < x ∈ E. By Lozanovskii’s theorem (see [12] Corollary 4) we are able to find Pn disjoint y1 , y2 , . . . , yn ∈ E+ such that 1 yi = x and kyi k < 12 kxk. It has to be n > 2. P Let j be the smallest natural number for which k j1 yi k > 41 kxk. There holds j < n, P because the case j = n implies kxk 6 k n−1 yi k + kyn k < 41 kxk + 12 kxk < kxk. 1 Pn 1 If j = 1, then putting x1 = y1 , x2 = 2 yi we obtain kx1 k > 4 kxk and kx2 k > 1 1 kxk − ky1 k > kxk − 2 kxk > 4 kxk. P P If j > 1, then because of j 6 n − 1 we can define x1 = j1 yi , x2 = nj+1 yi , and we get P kx1 k > 41 kxk, kx2 k > kxk − k j−1 yi k − kyj k > kxk − 41 kxk − 21 kxk = 14 kxk. 1 Remark. The class of Banach lattices considered in the previous example contains continuous Banach lattices with order continuous norms.

2. Let ϕ and ϕ∗ be a convex Orlicz function and the complementary function to ϕ, respectively. Consider the Orlicz space Lϕ [0, 1] equipped with the Orlicz norm Z 1 kf k = sup{ |f (t)g(t)|dt : ̺ϕ∗ (g) 6 1}, 0

where ̺ϕ∗ (g) = d ∈ (0, 21 ).

R1 0

ϕ∗ (|g(t)|)dt. The space (Lϕ [0, 1], k · k) satisfies (∗) with an arbitrary

Indeed, fix 0 < f ∈ Lϕ [0, 1] and ε ∈ (0, 1). Choose gf such that ̺ϕ∗ (gf ) 6 1 and R1 R |f (t)gf (t)|dt > (1 − ε)kf k. A measure mf (A) = A |f (t)gf (t)|dt is atomless, and 0 so there exists a set C for which mf (C) = mf ([0, 1] r C) = 21 mf ([0, 1]). Taking

When are ultrapowers of normed lattices discrete or continuous?

177

f1 = f 1C , f2 = f 1[0,1]rC we obtain f1 , f2 ∈ [0, f ], f1 ∧ f2 = 0, kf1 k > mf (C) = 1 m ([0, 1]) > 12 (1 − ε)kf k. Similarly, kf2 k > 21 (1 − ε)kf k. 2 f Remark. If ϕ does not satisfy the ∆2 -condition for large t, then the norm k · k is not order continuous and the dual (Lϕ [0, 1])∗ is not continuous (see [12] Corollary 7 or [16] Theorem 133.6). Therefore Example 2 is not included in Example 1. 3. Let S be a compact space without isolated points. The space C(S) equipped with the sup norm k · k∞ satisfies (∗) with an arbitrary d ∈ (0, 1). Fix ε ∈ (0, 1) and 0 < f ∈ C(S). Let f attends its norm on t ∈ S. Since t ∈ S r {t}, we can find s 6= t such that f (s) > (1 − ε)f (t). By Urysohn’s Lemma there exists a continuous function h : S → [0, 1] taking value 1 at the point t and value 0 at s. Define f1 = f h−f h∧f (1−h), f2 = f (1−h)−f h∧f (1−h). Then f1 , f2 ∈ [0, f ], f1 ∧f2 = 0, f1 (t) = f (t), f2 (s) = f (s), and so kf1 k∞ = kf k∞ , kf2 k∞ > (1 − ε)kf k∞ . It is obvious that negation of condition (∗) is of the following form ∼ (∗) :

∀d>0 ∃x>0 ∀x1 ,x2 ∈[0,x]

x1 ∧ x2 = 0 =⇒ min(kx1 k, kx2 k) < dkxk.

Clearly, discrete normed lattices satisfy ∼ (∗). Below we show not so obvious examples of normed lattices which do not satisfy (∗). 4. Suppose a normed lattice E = (E, k · k) admits a norm one real-valued (order) homomorphism h. Define a sequence (k · kn ) of norms, equivalent to k · k, by kxkn = 2nh(|x|) + kxk. A space F = (⊕(E, k · kn ))ℓ1 with a standard norm k(xn )kF = satisfy (∗).

P∞ 1

kxn kn does not

Suppose, contrary to our claim, that F satisfies (∗) with a constant d. Choose x0 ∈ E+ such that kx0 k = 1, h(x0 ) > 21 . Hence kx0 kn > n + 1. Putting x0 1{n} = (0, . . . , 0, x0 , 0. . . . ) ∈ F (x0 is the n-th term), we can find disjoint xn1 , xn2 ∈ [0, x0 ], for which there holds min(kxn1 1{n} kF , kxn2 1{n} kF ) > dkx0 1{n} kF . Since min(h(xn1 ), h(xn2 )) = 0, we can assume h(xn2 ) = 0. For this reason kxn2 kn = kxn2 k 6 kx0 k = 1 and 1 > min(kxn1 kn , kxn2 kn ) = min(kxn1 1{n} kF , kxn2 1{n} kF ) > dkx0 1{n} kF = dkx0 kn > d(n + 1), a contradiction. Moreover the space F is continuous if and only if E is continuous. 5. Let ω : (0, 1] → (0, ∞) be a bounded function vanishing near zero, i.e., limt→0+ ω(t) = 0. For every f ∈ C[0, 1] define kf kω = |f (0)| + sup |f (t)| ω(t) 0 0. For every n there exists δn > 0 such that ω(t) < n1 whenever t ∈ (0, δn ). Choose a function f ∈ C[0, 1]+ satisfying the following conditions: f (0) = 1 = kf k∞ = sup06t61 |f (t)|, f (t) = 0 for t > δn . Let f1 , f2 ∈ [0, f ] be disjoint and such that min(kf1 kω , kf2 kω ) > d. We can assume f1 (0) = 1. Therefore f2 (0) = 0 and f2 (t) = 0 if t > δn . Moreover 0 6 f2 (t) 6 1. Finally d 6 kf2 kω 6 sup0 ε} ∈ A. Consider a set Mε =

\

1 {i : |xi (j)| < ε}. 8 j∈J

S It has to be Mε ∈ A. If Mε were not in A there would be j∈J {i : |xi (j)| > 81 ε} = I rMε ∈ A. Therefore for every i ∈ / Mε we are able to choose j(i) ∈ J with |xi (j(i))| > 81 ε. Fixing j0 ∈ J we define a function f0 : I → J putting f0 (i) = j(i) for i ∈ / Mε and f0 (i) = j0 for i ∈ Mε . But 1 {i : |xi (f0 (i))| > 16 ε} ⊃ I r Mε implies limA |xi (f0 (i))| > 0 contrary to (‡). By the Fatou property for every i ∈ Nε ∩ Mε ∈ A there exists a finite set ∆i ⊂ J and ji ∈ ∆i such that k

X

j∈∆i

1 xi (j)ej k > ε and 4

k

X

j∈∆i r{ji }

1 xi (j)ej k 6 ε. 4

When are ultrapowers of normed lattices discrete or continuous?

179

Moreover X 1 ε ε. For elements x1i = j∈∆i xi (j)ej , x2i = supj ∈∆ / i xi (j)ej 1 3 1 2 1 2 1 2 1 there holds xi = xi + xi , xi ∧ xi = 0, kxi k > 4 ε, kxi k > kxi k − kxi k > ε − 8 ε > 14 ε. Let y k = (yik )i∈I , k = 1, 2, where yik = xki for i ∈ Nε ∩ Mε and yik = 0 whenever i ∈ / Nε ∩ Mε . 1 k k ∞ k k Since 0 6 y 6 x then y ∈ ℓ (E, I) and y ∈ / NA because limA kyi k > 4 ε. Disjointness of 1 2 1 2 y and y gives Q(y ) ∧ Q(y ) = 0. Finally Q(y 1 ) + Q(y 2 ) = Q(x1Nε ∩Mε ) = Q(x), i.e., Q(x) is not discrete. Suppose that Q(x) is discrete in (E)A. We can assume, as usually, x > 0. From previous considerations we deduce that 0 6= Q(x) ∧ Q((ef (i) )) for some f ∈ J I . We have already known that Q((ef (i) )) is discrete. Hence, for some numbers r, s > 0 there holds sQ(x) = Q(x) ∧ Q((ef (i) )) = rQ((ef (i) )). It suffices to take t = rs . Remark. Let us note that for f, g ∈ J I the equality Q((ef (i) )) = Q((eg(i) )) is equivalent to {i : f (i) = g(i)} ∈ A. A corollary below is an immediate consequence of Theorem 3.2. Corollary 3.3 The band {Q((ef (i) )) : f ∈ J I }dd is the discrete part of (E)A. Next result shows that ultrapowers of discrete AM-spaces are discrete too. Corollary 3.4 If H is a regular discrete Riesz subspace in an AM-space E with a strong unit, then (H)A is discrete. Therefore the ultrapowers (c0 (Γ))A, (c(Γ))A, (ℓ∞ (Γ))A are discrete. Proof. Let (ej )j∈J be a complete disjoint system in H consisting of norm one discrete elements and let F (J) denote a family of nonempty finite subsets in J. If x ∈ H+ , then x = supj x(j)ej where the supremum can be taken in H as well as in E because H is regular in E. According to Proposition P 10.12 from [13] the AM-norm in E has the Fatou property, and so kxk = sup∆∈F (I) k j∈∆ x(j)ej k = sup∆∈F (I) maxj∈∆ x(j) = supj x(j). Fix 0 < x = (xi )i∈I ∈ ℓ∞ (H, I) r NA. For every i ∈ I choose j(i) ∈ J such that xi (j(i)) > 21 kxi k. Since xi (j(i)) 6 supi kxi k < ∞, then limA xi (j(i)) exists. Consider e = (xi (j(i))ej(i) ) ∈ ℓ∞ (H, I) and let Q : ℓ∞ (H, I) → (H)A be the quotient map. There holds Q(e) 6= 0 because limA xi (j(i)) > 21 limA kxi k > 0. By Lemma 3.1 and Theorem 3.2 Q(e) = (limA xi (j(i)))Q((ej(i) )) is discrete in (H)A and, clearly, Q(e) 6 Q(x). We have just shown that every element 0 < Q(x) ∈ (H)A majorizes a discrete element, i.e., (H)A is discrete. Theorem 3.2 and Corollary 3.3 can be applied to investigations of the order structure of ultrapowers of Orlicz spaces. Corollary 3.5 If ϕ is a convex Orlicz function and A is a countably incomplete ultrafilter, then (ℓϕ )A is heterogeneous. T Proof. Let (In ) ⊂ A, I1 ) I2 ) . . . , n In = ∅. The continuity, unboundedness of ϕ, and the condition ϕ(0) = 0 imply the existence of numbers an for which ϕ(an ) = n1 . The

180

W. Wnuk, B. Wiatrowski

monotonicity of ϕ and the equivalence ϕ(t) = 0 ⇐⇒ t = 0 ensure 0 < an → 0. Let an ∈ (0, 1) for n > n0 . Define yn = (an , an , . . . , an , 0, 0, . . . ) ∈ ℓϕ . Since ϕ increases, then {z } | n−times

1 2

nϕ(2an ) > 1, and so < kyn k 6 1, where k · k is the Luxemburg norm. Put xi = 0 for i ∈ I r In0 and xi = yn for i ∈ In r In+1 , n > n0 . We obtain kxi k 6 1, {i : kxi k > 1 } = In0 ∈ A. Hence x = (xi ) ∈ ℓ∞ (ℓϕ , I) r NA. Let f : I → N be a function. For 2 every ε > 0 there is n1 > n0 such that an < εϕ−1 (1) whenever n > n1 . In consequence {i : kxi (f (i))ef (i) k < ε} = {i : xi (f (i)) < εϕ−1 (1)} ⊃ In1 +1 , i.e., limA kxi (f (i))ef (i) k = 0. Finally Q(x) ∧ Q((ef (i) )) = Q((min(xi (f (i)), 1)ef (i) )) = Q((xi (f (i))ef (i) )) = 0. According to Corollary 3.3 Q(x) does not belong to the discrete part of (ℓϕ )A, and so the ultrapower is heterogeneous. Remark. In a special case ϕ(t) = tp , p > 1, C.W. Henson (see [6]) obtained stronger and deeper result: if A does not contain finite sets, then (ℓϕ )A is order isometric to [ℓp (Γ) ⊕ (Lp [0, 1])A]p , where the ultrapower (Lp [0, 1])A is order isometric to the ℓp -sum of c-copies of Lp ([0, 1]c) (c = 2ℵ0 ).

4 When is NA a σ-ideal? If Ei = (Ei , k · ki ) are non-trivial normed lattices, i.e., Ei 6= {0} Q and A is non-trivial, then c0 (Ei , I) NA ℓ∞ (Ei , I), where c0 (Ei , I) = {(xi ) ∈ i∈I Ei : ∀ε>0 {i : kxi ki > ε} is finite}. Therefore under the above assumptions NA is never a band in ℓ∞ (Ei , I). On the other hand, if A is trivial, i.e., {i0 } ∈ A for some i0 ∈ I, then NA is a band because in this case NA = {(xi ) ∈ ℓ∞ (Ei , I) : xi0 = 0}. Below we discuss a problem when NA is a σ-ideal. Theorem 4.1 Let card A be a non-measurable cardinal. The ideal NA is a σ-ideal if and only if A is trivial. Proof. Suppose that NA is a σ-ideal and A is non-trivial. Let ℓ∞ (I) = {(ai ) ∈ RI : supi |ai | < ∞} and NA∼ = {(ai ) ∈ ℓ∞ (I) : limA |ai | = 0}. We claim that NA∼ is a σ-ideal. Indeed, fix x = (xi ) ∈ ℓ∞ (Ei , I)+ r NA and note the following equivalence ∀(ai )∈ℓ∞ (I)

lim |ai | = 0 ⇐⇒ lim kai xi ki = 0. A

A

Let an = (an (i)) ∈ NA∼ , an ↑ a = (ai ) ∈ ℓ∞ (I). Elements yn = (an (i)xi )i∈I ∈ NA increase, in ℓ∞ (Ei , I), to y = (a(i)xi )i∈I . It follows that y ∈ NA, and so a ∈ NA∼ . Now we check that A is closed under the formation ofTcountable intersections. TnOn the ∞ contrary, suppose that A contains a sequence (In ) with I0 = 1 In ∈ / A. Sets Jn = k=1 Ik ∩ (I r I0 ) belong to A and I r Jn ↑ I. Moreover I r Jn ∈ / A. Hence 1I\Jn ∈ NA∼ which ∼ contradicts the fact that NA is a σ-ideal because 1IrJn ↑ 1I ∈ / NA∼ . T T Since A is non-trivial, then I r F ∈ A for every finite set F ⊂ I. Consequently A∈A A ⊂ A ∈ A} is a maximal ideal of sets closed under F ⊂I,F finite I \ F = ∅. Therefore J = {I \ A :S the formation of countable unions. Moreover, B∈J B = I, and card J = card A. By Ulam’s Theorem (see [7] Theorem 3, p. 289) the ideal J is closed under the formation of Tcard J-unions. Thus A is closed under the formation of card A-intersections. Finally, ∅ = A∈A A ∈ A, a contradiction.

When are ultrapowers of normed lattices discrete or continuous?

181

Acknowledgment. We are extremely grateful to Professor Lech Drewnowski for suggestions concerning Example 4.

References [1] C. Aliprantis, O. Burkinshaw, Locally solid Riesz spaces with applications to economics, Mathematical Surveys and Monographs vol 105, American Mathematical Society, 2003. [2] D. Dacunha-Castelle and J.L. Krivine, Applications des ultraproduits à l’étude des espaces et des algèbras de Banach, Studia Math. 41 (1972), 315-334. [3] B.A. Davey and H.A. Priestley, Introduction to lattices and order, Cambridge University Press, Cambridge, 1992. [4] S. Heinrich, Ultraproducts in Banach space theory, J. Reine Angew. Math. 313 (1980), 72–104. [5] C.W. Henson, Ultraproducts of Banach spaces, The Altgeld Book 1975–1976, University of Illinois, Functional Analysis Seminar. [6] C.W. Henson, Nonstandard hulls of Banach spaces, Israel J. Math. 25 (1976), 108–144. [7] K. Kuratowski and A. Mostowski, Set theory, Monografie Matematyczne vol. 27, PWN Polish Scientific Publishers, Warszawa, 1966. (Polish) [8] J. Lindenstaruss and L. Tzafriri, Classical Banach spaces I. Sequence spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 92, Springer-Verlag, Berlin Heidelberg New York, 1977. [9] J. Lindenstaruss and L. Tzafriri, Classical Banach spaces II. Function spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 97, Springer-Verlag, Berlin New York, 1979. [10] W.A.J. Luxemburg and A.C. Zaanen, Riesz spaces I, North Holland, Amsterdam London, 1971. [11] P. Meyer-Nieberg, Banach lattices, Springer-Verlag, Berlin Heidelberg New York, 1991. [12] B. de Pagter and W. Wnuk, Some reamrks on Banach lattices with non-atomic duals, Indag. Math. NS 1 (1990), 391–396. [13] H.U. Schwarz, Banach lattices and operators, Teubner-Texte zur Mathematik, band 71, B.G. Teubner Verlagsgesellschaft, Leipzig, (1984). [14] B. Sims, "Ultra"-techniques in Banach Space Theory, Queen’s Papers in Pure and Applied Mathematics No. 60, Queen’s University, Kingston Ontario Canada, 1982. [15] W. Wnuk, Ultraproducts of F -lattices satisfy the σ-Lebesgue property, Atti Sem. Mat. Fis. Univ. Modena 51 (2003), 173-177.

182

W. Wnuk, B. Wiatrowski

[16] A.C. Zaanen, Riesz spaces II, North Holland, Amsterdam London, 1983.

Witold Wnuk Faculty of Mathematics and Computer Science, A. Mickiewicz University, Umultowska 87, 61-614 Pozna´n, Poland E-mail: [email protected] Bła˙zej Wiatrowski The Co-operative Banks Group, Gospodarczy Bank Wielkopolski SA, Information Technology Department, Miel˙zy´nskiego 22, 61-725 Pozna´n, Poland E-mail: [email protected]

List of Participants Name

City

Country

e-mail

Alekhno, Egor A.

Minsk

Belarus

[email protected]

Alpay, Şafak

Ankara

Turkey

[email protected]

Amor, Fethi Ben

La Marsa

Tunisia

[email protected]

Ando, Tsuyoshi

Hokaido

Japan

[email protected]

Arendt, W.

Ulm

Germany

[email protected]

Bartoszek, Wojciech

Gdansk

Poland

[email protected]

Becker, Richard

Paris

France

[email protected]

Boulabiar, Karim

La Marsa

Tunisia

karim.boulabiar.ipest.rnu.tn

Buskes, Gerard

USA

[email protected]

Byrne, J.

Oxford, Mississippi Belfast

Irland

[email protected]

Chen, Zi Li

Chengdu Sichuan China

[email protected]

Chil, E.

Monfery

Tunisia

[email protected]

Drnovsek, Roman

Ljubljana

Slovenia

[email protected]

Eisner, Tatjana

Tübingen

Germany

[email protected]

Emelyanov, E. Yu.

Ankara

Turkey

[email protected]

Ercan, Zafer

Ankara

Turkey

[email protected]

van Gaans, Onno

Jena

Germany

[email protected]

Grobler, J. J.

Potchefstroom

South Africa [email protected]

Herzog, Gerd

Karlsruhe

Germany

[email protected]

Kalauch, Anke

Dresden

Germany

[email protected]

Keicher, Vera

Tübingen

Germany

[email protected]

König, H.

Saarbrücken

Germany

[email protected]

Kusraev, Anatoly G.

Vladikavkaz

Russia

[email protected]

Kutateladze, S. S.

Novosibirsk

Russia

Lemmert, Roland

Karlsruhe

Germany

[email protected] [email protected] [email protected]

Lenski, Wlodzimierz

Zielona Góra

Poland

[email protected]

184

List of Participants

Name

City

Country

e-mail

Luxemburg, Willem

Caltech

USA

[email protected]

Mısırlıoğlu, Tunç

Istanbul

Turkey

[email protected]

Nowak, M.

Zielona Góra

Poland

[email protected]

dePagter, Ben

Delft

Polat, Faruk

Ankara

The [email protected] Netherlands Turkey [email protected]

Polyrakis, Yannis

Athen

Greece

[email protected]

Rhandi, A.

Marrakesch

Morocco

[email protected]

Ricker, Werner

Eichstätt

Germany

[email protected]

Scheffold, Egon

Darmstadt

Germany

[email protected]

Schep, Anton R.

Columbia, SC

USA

[email protected]

Stein, Martin

Dresden

Germany

[email protected]

Thieme, H.

Tempe

USA

[email protected]

Troitsky, V. G.

Alberta

Canada

[email protected]

Tschichholz, Ingo

Dresden

Germany

[email protected]

Väth, Martin

Würzburg

Germany

[email protected]

Voigt, Jürgen

Dresden

Germany

[email protected]

Weber, M. R.

Dresden

Germany

[email protected]

Wickstead, A. W.

Belfast

Irland

[email protected]

Wnuk, Witold

Poznan

Poland

[email protected]