Operator Theory: Proceedings of the International Conference on Operator Theory, Hammamet, Tunisia, April 30 - May 3, 2018 (De Gruyter Proceedings in Mathematics) 3110596865, 9783110596861

Table of contents :
Preface
Acknowledgment
Contents
Elliptic differential operators with periodic and oscillating decaying coefficients
Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph
Double lacunary statistically convergent sequences of weight g
Riesz probability distributions on symmetric cones
A note on traces of forms with applications to the Bessel operator
Pseudospectra of multivalued linear operators
On the frame properties of exponentials associated to analytic families of operators and application
Invariance of essential spectrum by means of generalized weak demicompactness
Some results of S-pseudospectra of bounded operators in a Hilbert space
Stabilization of a marine riser system by the use of viscoelastic material
Sequence of linear operators converging in the generalized sense and its essential pseudospectra
Left–right essential spectra of one-sided operator matrix and application
Existence of Solutions for an integral equation of Chandrasekhar type in Banach algebras with respect to the weak topology
Strong evolution problems in Musielak spaces
Numerical solution of an axisymmetric inverse heat conduction problem
Multiple solutions for nonlocal elliptic problems with concave–convex nonlinearities and weights

Citation preview

Aref Jeribi (Ed.) Operator Theory

De Gruyter Proceedings in Mathematics

|

Operator Theory |

Proceedings of the International Conference on Operator Theory, Hammamet, Tunisia, April 30 - May 3, 2018 Edited by Aref Jeribi

Editor Prof. Dr. Aref Jeribi University of Sfax Faculty of Sciences Department of Mathematics BP1171 Road Soukra Km 3.5 3000 Sfax Tunisia [email protected]

ISBN 978-3-11-059686-1 e-ISBN (PDF) 978-3-11-059819-3 e-ISBN (EPUB) 978-3-11-059690-8 Library of Congress Control Number: 2020950070 Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2021 Walter de Gruyter GmbH, Berlin/Boston Typesetting: VTeX UAB, Lithuania Printing and binding: CPI books GmbH, Leck www.degruyter.com

Preface The ICOT (International Conference on Operator Theory) provides an excellent opportunity for specialists in operator theory to exchange new ideas and applications face to face, to establish research network, and to look for future collaboration. A broad range of challenging research problems in spectral theory was covered during the conference. Themes of the ICOT 2018 focused on: operator theory, spectral theory, local spectral theory, perturbation classes, invariant subspaces, matrix of operators, semigroups, cyclic vectors, Fredholm and semi-Fredholm operators, boundary value problems, fixed point theory, functional analysis, as well as partial differential equations. This volume contains a collection of scientific papers presented at ICOT, which was held in Hammamet, Tunisia, April 30–May 03, 2018. The conference is supported by several institutions, including the following: – University of Sfax – Faculty of Sciences of Sfax (FSS) Scientific Committee Pietro Aiena, Italia Francesco Altomare, Italia Jozef Banas, Poland Aref Jeribi, Tunisia Mohammad Sal Moslehian, Iran Maher Mnif, Tunisia Ekrem Savas, Turkey Carsten Trunk, Germany Organizing Committee Fayçal Abdmouleh, Tunisia Aymen Ammar, Tunisia Salma Charfi, Tunisia Aref Jeribi, Tunisia Bilel Krichen, Tunisia Nedra Moalla, Tunisia Maher Mnif, Tunisia

https://doi.org/10.1515/9783110598193-201

Acknowledgment The ICOT is an International Conference on Operator Theory which was held in Tunisia. We would like to take this opportunity to thank once again all the participants of the conference, invited speakers, presenters, and audience alike. We would also like to extend our gratitude to all members of the Program and Organizing committees for their active help and also to reviewers of the original abstracts and the papers submitted for consideration in this book. Finally, we would particularly like to thank Pr. Salma Charfi for her valuable participation in the elaboration of this book. Sfax, Tunisia

https://doi.org/10.1515/9783110598193-202

Aref Jeribi

Contents Preface | V Acknowledgment | VII Mouez Dimassi Elliptic differential operators with periodic and oscillating decaying coefficients | 1 Hannes Gernandt and Carsten Trunk Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph | 25 E. Savaş Double lacunary statistically convergent sequences of weight g | 55 R. Zine Riesz probability distributions on symmetric cones | 63 Ali BenAmor, Rafed Moussa, and Christian Seifert A note on traces of forms with applications to the Bessel operator | 75 Aymen Ammar, Houcem Daoud, and Aref Jeribi Pseudospectra of multivalued linear operators | 85 S. Charfi and H. Ellouz On the frame properties of exponentials associated to analytic families of operators and application | 101 Imen Ferjani, Aref Jeribi, and Bilel Krichen Invariance of essential spectrum by means of generalized weak demicompactness | 115 Aymen Ammar, Ameni Bouchekoua, and Aref Jeribi Some results of S-pseudospectra of bounded operators in a Hilbert space | 121 Amirouche Berkani, Lamia Seghour, and Nasser-eddine Tatar Stabilization of a marine riser system by the use of viscoelastic material | 131

X | Contents Aymen Ammar, Aref Jeribi, and Nawrez Lazrag Sequence of linear operators converging in the generalized sense and its essential pseudospectra | 147 Sana Bouzidi and Ines Walha Left–right essential spectra of one-sided operator matrix and application | 157 Amor Fahem, Aref Jeribi, and Najib Kaddachi Existence of Solutions for an integral equation of Chandrasekhar type in Banach algebras with respect to the weak topology | 177 A. Aberqi, J. Bennouna, and M. Elmassoudi Strong evolution problems in Musielak spaces | 183 Ibtissem Djerrar, Leïla Alem, and Lahcène Chorfi Numerical solution of an axisymmetric inverse heat conduction problem | 199 Safia Benmansour, Atika Matallah, and Mustapha Meghnafi Multiple solutions for nonlocal elliptic problems with concave–convex nonlinearities and weights | 207

Mouez Dimassi

Elliptic differential operators with periodic and oscillating decaying coefficients Abstract: We study some spectral properties of an elliptic system of differential operators, H(h), with smooth and oscillating coefficients, a(hy, y), y ∈ ℝn , depending on a small positive parameter h. Under some decaying assumptions on x → a(x, y), we prove the existence of discrete eigenvalues with finite multiplicities, and for h small enough we derive optimal results on the number of eigenvalues of H(h) in an interval disjoint from its essential spectra. On the other hand, we establish strong full asymptotic expansions in powers of h of the SSF associated to H(h). Finally, if the coefficients x → a(x, y) are analytic in a complex neighborhood of ℝn , we prove the existence of shape resonances, and we give their asymptotic expansions in powers of h1/2 . Keywords: Periodic Schrödinger operator, spectral shift function, asymptotic expansions, limiting absorption theorem MSC 2010: 81Q10, 35P20, 47A55, 47N50, 81Q15

1 Introduction In this paper we review some results on the spectral and scattering theory of elliptic differential operator with oscillatory and decaying coefficients. Let H = ∑|α|≤2m aα (y)𝜕yα be an elliptic differential operator with periodic coefficients (i. e., y 󳨃→ aα (y) is periodic with respect to a lattice Γ in ℝn ). We assume that H is self-adjoint on H 2m (ℝn ) := {u ∈ L2 (ℝn ); 𝜕yα u ∈ L2 (ℝn ), ∀|α| ≤ 2m}. Then by standard results (see, e. g., [27]) the spectrum of H as an operator in L2 (ℝn ), is equal to ∞

σ(H) = σess (H) = ⋃ Λj , j=1

n

where Λj := [aj , bj ] = μj (ℝ /Γ ), and (μj (ξ ))j≥1 are the eigenvalues of operator H on L2loc (ℝn ) with domain 2m

α

∗

2

n

ℋξ = {u|E ; 𝜕x u ∈ Lloc (ℝ ); u(x + γ) = e

iξ ⋅γ

u(x), ∀γ ∈ Γ, ∀|α| ≤ 2m}.

Here Γ∗ is the dual reciprocal lattice and E a fundamental domain of Γ in ℝn . The closed interval Λj is known as the “jth Bloch band” in the spectrum of H. Consider the Acknowledgement: The author wishes to thank Aref Jeribi for the invitation to the International Conference on Operator Theory, held in Hammamet, April 30 – May 03, 2018, Tunisia. Mouez Dimassi, Université Bordeaux, I. M. B. 33405 Talence, France, e-mail: [email protected] https://doi.org/10.1515/9783110598193-001

2 | M. Dimassi perturbed operator acting on L2 (ℝn ), namely H(h) = H + ∑ bα (hy, y)𝜕yα |α|≤2m

(h ↘ 0),

(1.1)

where bα (x, y) is a smooth function, tending to zero as |x| tends to infinity and Γ-periodic with respect to y. When b0 (x, y) = b0 (x) is independent of y and h = 1, the spectral properties of the perturbed periodic Schrödinger operator H(h) = −Δ + a0 (y) + b(hy) have been intensively studied in the last 20 years (see [2, 1] and the references given there). In particular, the asymptotic behavior of the number of eigenvalues near the edges of gaps ai and bj of H = −Δ + a0 (y) was studied in [25]. On the other hand, the distribution of the discrete spectrum of the operator Hμ = −Δ+a0 (y)+μb0 (y) in a gap was well studied when μ tends to infinity, see [2, 1]. The semiclassical case, H(h) with h ↘ 0, was considered in [7, 8, 10]. The case where b(x, y) depends both on x and y, and a0 (y) ≡ 0 independent of the semiclassical parameter h, has been studied recently in [26] (see also [3, 22]). In particular, the asymptotic behavior of the number of eigenvalues of −Δ + b(y, y) in ]−∞, −τ[ has been established when τ ↘ 0. The existence of eigenvalues in a gap of the 1D-periodic Schrödinger operator with fast oscillating potential (i. e., Q(h) = −𝜕x2 + a0 (x) + b0 (x, hx ), h ↘ 0) was studied in [17] (see also [3, 6, 16, 18, 15]). Fix a band [am , bm ] of the periodic Hamiltonian −𝜕x2 + a0 (x), and assume that am is a border of a spectral gap. For h small enough, V. Duchêne, I. Vukicevic, and M. I. Weinstein (see [17]) proved that the operator Q(h) has a simple discrete eigenvalue near am . Notice that, by the Lieb–Thirring trace inequalities, this result is false for higher dimension (n ≥ 3). On the other hand, our problem (1.1) is completely different since the number of eigenvalues in each gap tends to infinity when h tends to zero (see Corollary 2.3). In fact, for h small enough the scaling (1.1) is that of semiclassical analysis. This paper is concerned with the scattering theory of the operator H(h) when h tends to zero. First, for all f ∈ C0∞ (ℝ), we show that tr(f (H(h)) − f (H)) is of trace class and we study its asymptotic behavior as h tends to zero. As a consequence, we prove that in each gap there exist 𝒪(h−n ) eigenvalues counted with their multiplicities. In particular, for an interval [a, b] disjoint from the essential spectra of H(h), we give a sharp estimate on the number of eigenvalues of H(h) in [a, b] for h small enough. On the other hand, if the coefficients x → b(x, y) are analytic in a complex neighborhood of ℝn , we prove the existence of shape resonances, and we give their asymptotic expansions in powers of h1/2 . In this work, we are strongly inspired by the recent progress in the analysis of the resonances, the spectral shift function, and trace formulae for Schrödinger operators (see [32, 4] and the references given there). Our approach is quite similar to those in [7, 8], and based on the Hamiltonian effective method. Some of the results in this paper are announced in [12]. Notations Throughout this paper we adopt the following notations. Let fh be a scalar-valued h-depending function, we write fh = 𝒪(hN ) (respectively fh = 𝒪(h∞ )) if and only if

Elliptic differential operators with periodic and oscillating decaying coefficients | 3

there exists CN , hN > 0 (respectively for all N ∈ ℕ there exists CN > 0) such that |fh | ≤ CN hN for all h ∈ ]0, hN [. The notation A ≡ B means A − B = 𝒪(h∞ ). In particular, N j j N+1 we write fh ≡ ∑∞ ). The scalar j=0 aj h if, for each N ∈ ℕ, we have fh − ∑j=0 aj h = 𝒪 (h product in a Hilbert space ℋ will be denoted by ⟨⋅, ⋅⟩. The set of linear bounded operators from ℋ1 to ℋ2 is denoted by ℒ(ℋ1 , ℋ2 ), and ℒ(ℋ1 ) when ℋ1 = ℋ2 . For z ∈ ℂ, we 1 ̂ use the standard notation ⟨z⟩ = (1 + |z|2 ) 2 . When A is a square matrix, tr(A) denotes its trace.

2 Preliminaries and results Let {v1 , v2 , . . . , vn } be a basis in ℝn , and let {v1∗ , . . . , vn∗ } ⊂ ℝn be the dual basis satisfying n

vj ⋅ vi∗ = 2πδij , i, j = 1, . . . , n. Let Γ = ⊕ ℤvi (respectively Γ∗ ) be the lattice generated by i=1

(v1 , . . . , vn ) (respectively (v1∗ , . . . , vn∗ )). In the following, E and E ∗ denote the fundamental domains for Γ and Γ∗ , respectively. Without any loss of generality, we may identify E and E ∗ with 𝕋 = ℝn /Γ and 𝕋∗ = ℝn /Γ∗ . Let p(y, η) ∈ C ∞ (ℝ2n ) and q(x, y, η) ∈ C ∞ (ℝ3n ) be two real-valued functions satisfying: q(x, y, η) = ∑ bα (x, y)ηα ,

p(y, η) = ∑ aα (y)ηα ,

|α|≤2m

|α|≤2m

aα (y + γ) = aα (y), ∀α, β,

bα (x, y + γ) = bα (x, y),

∀γ ∈ Γ,

󵄨 󵄨 lim supy∈ℝn 󵄨󵄨󵄨𝜕xα 𝜕yβ bα (x, y)󵄨󵄨󵄨 = 0

|x|→∞

p2m (y, η) = ∑ aα (y)ηα ≥ C0 |η|2m ,

(2.1) (2.2) (2.3) (2.4)

|α|=2m

p2m (y, η) + q2m (x, y, η) ≥ C1 |η|2m . According to the Weyl quantization, we associate to p(y, η) and Q(x, y, η) := p(y, η) + q(x, y, η) the differential operators Hu(y) = pw (y, Dy )u(y) = ∫ ∫ ei(y−x)η p( 𝕋∗ ℝn

H(h)u(y) = ∫ ∫ ei(y−x)η Q(h 𝕋∗

ℝn

dx dη x+y , η)u(x) , 2 (2π)n

dx dη x+y x+y , , η)u(x) , 2 2 (2π)n

(2.5) (2.6)

Assumption (2.4) yields that H and H(h) are self-adjoint on H 2m (ℝn ). Fix (x, ξ ) in ℝ2n , and define P1 (x, ξ ) := Qw (x, y, Dy + ξ ),

P0 (ξ ) = P w (y, Dy + ξ ),

(2.7)

4 | M. Dimassi from L2 (𝕋) into L2 (𝕋) as unbounded operators with domain H 2m (𝕋). The operator Pj is semibounded and self-adjoint. By Rellich theorem, (P1 (x, ξ ) + i)−1 and (P0 (ξ ) + i)−1 are compact operators. Let (Ψ)j≥1 and (φ)j≥1 be an orthonormal basis of eigenfunctions of P1 and P0 , respectively. We enumerate the associated eigenvalues according to their multiplicities: λ1 (x, ξ ) ≤ λ2 (x, ξ ) ≤ ⋅ ⋅ ⋅

and

μ1 (ξ ) ≤ μ2 (ξ ) ≤ ⋅ ⋅ ⋅ .

(2.8)

Using that e−iy⋅γ Pj eiy⋅γ = Pj (⋅, ξ + γ ∗ ), we deduce that the band functions ξ 󳨃→ λm (x, ξ ) and ξ → μm (ξ ) are Γ∗ -periodic. Analytic perturbation theory yields that (x, ξ ) → λj (x, ξ ) and ξ → μj (ξ ) are continuous. According to Floquet theory (see [27, 31]), the spectrum of H consists of the bands Λm := μm (𝕋∗ ), m = 1, 2, . . . , given by ∗

∗

σ(H) = ⋃ Λm . m≥1

Assumption (2.3) and the Weyl criterion show that the essential spectra of H and H(h) coincide: σess (H) = σess (H(h)) = ⋃ Λm . m≥1

For fixed x, we let ρ(⋅, x) and ρ0 (⋅) be the densities of states corresponding to P1 and P0 (see [31]): ρ(t, x) := ∑

∫

l≥1 {k∈E ∗ ; λ (x,k)≤t} l

ρ0 (t) := ∑

∫

l≥1 {k∈E ∗ ; μ (k)≤t} l

̄ dk,

̄ dk,

(2.9) (2.10)

̄ = (2π)−n dk. where dk Remark 2.1. 1. Notice that in the case where Q(x, y, η) = φ0 (x)p(y, η) + φ1 (x), we have λj (x, ξ ) = φ0 (x)μj (ξ ) + φ1 (x). In particular, ρ1 (t, x) = ρ0 (φ0 (x)t + φ1 (x)). 2.

If p(y, ξ ) = p(ξ ) = p2m (ξ ) = ∑|α|=2m aα ηα (aα is constant) then ρ0 (t) = cn (2π)−n max(t, 0)m/n , where cn = vol({η ∈ ℝn ; ∑|α|=2m aα ηα ≤ 1}).

Elliptic differential operators with periodic and oscillating decaying coefficients | 5

2.1 Spectrum in the gaps Fix an interval [a, b] in the gaps of H, σ(H) ∩ [a, b] = 0.

(2.11)

Here we discuss the nature and the behavior of the spectrum of H(h) when h is small enough. First we show that, for h small enough, the perturbation creates discrete eigenvalues with finite multiplicities, and we give an estimate of the number of these eigenvalues inside [a, b]. Proposition 2.2. Under the assumptions (2.1), (2.2), (2.3), and (2.4), the operator f (H(h)) is of trace class for all f ∈ C0∞ (]a, b[; ℝ), ∃a1 , a2 , . . . ∈ ℝ such that ∞

tr[f (H(h))] ∼ ∑ aj hj−n , j=0

h ↘ 0,

(2.12)

with a0 = ∑ ∬ f (λk (x, ξ )) dx dξ̄ = − ∫ ∫ f 󸀠 (t)ρ(t, x)dt dx. k≥1 ℝn ×E ∗

(2.13)

ℝnx ℝ

x

Let us enumerate the eigenvalues of H(h) in [a, b] according to their multiplicities ϵ1 (h) ≤ ϵ2 (h) ⋅ ⋅ ⋅ ≤ ϵj (h) ≤ ⋅ ⋅ ⋅. Denote N[a,b] (h) = #{j; ϵj (h) ∈ [a, b]}. Theorem 2.3. The following estimate holds: N[a,b] (h) = h−n ∫ ρ(b, y) − ρ(a, y) dy + o(h−n ),

as h ↘ 0.

(2.14)

ℝn

For t = a, b, we denote ∞

Σt = ⋃{(y, k) ∈ ℝn × E ∗ ; λl (y, k) = t}. l=1

Under the following condition, we shall improve Theorem 2.3: (H) For (x0 , ξ0 ) ∈ Στ , λj (x0 , ξ0 ) = τ is simple and noncritical, i. e., λj−1 (x0 , ξ0 ) < λj (x0 , ξ0 ) < λj+1 (x0 , ξ0 ),

and

∇x,ξ λj (x0 , ξ0 ) ≠ 0.

Proposition 2.4. Under assumption (H), for h small enough, we have N[a,b] (h) = h−n ∫ [ρ(b, y) − ρ(a, y)] dy + o(h−n+1 ), ℝn

as h ↘ 0.

(2.15)

6 | M. Dimassi

2.2 Spectral shift function Fix now an energy λ ∈ σ(H). To study the spectrum of H(h) near λ, we investigate the scattering function corresponding to the pair (H(h), H). For that, we suppose that for all |γ| ≤ 2m there exists δ > n such that ∀α, β ∈ ℕn , ∃Cα,β > 0, ∀x, y ∈ ℝn ,

󵄨 󵄨󵄨 α β −δ 󵄨󵄨𝜕x 𝜕y bγ (x, y)󵄨󵄨󵄨 ≤ Cα,β ⟨x⟩ .

(2.16)

Inequality (2.16) enables us to define the spectral shift function (SSF for short), ξ (μ, h) ∈ 𝒟󸀠 (ℝ), related to operators H(h) and H, following the general theory (see [33, 23] and the references given there) by the equality ∫ ξ (t; h)f 󸀠 (t) dt = tr[f (H(h)) − f (H)],

∀f ∈ C0∞ (ℝ).

(2.17)

ℝ

Without any loss of generality, we may assume that ξ (μ; h) = 0 for μ < inf(σ(H(h))). The SSF may be considered as an extension of the eigenvalues counting function. It is one of the main objects in the scattering theory. For a recent account of the theory, we refer the reader to [32, 4, 33, 24, 30]. Proposition 2.5. Let f ∈ C0∞ (ℝ). Under the assumptions (2.16), (2.1), (2.2), and (2.4), the operator [f (H(h)) − f (H)] is of trace class. Moreover, ∃a1 , a2 , . . . ∈ ℝ such that for h small enough we have ∞

tr(f (H(h)) − f (H)) ∼ ∑ al h−n+l ,

(2.18)

a0 = ∫ ∫ f 󸀠 (t)[ρ0 (t) − ρ(t, y)] dt dy.

(2.19)

l=0

where

ℝnx

ℝ

Theorem 2.6. Suppose that p(y, ξ ) = ∑|α|=2m aα ηα , and q(x, y, ξ ) = V(x, y) is independent of η. Under the hypotheses of the above theorem, the following estimate holds for any λ > 0: ξ (λ; h) = (2πh)−n ∫ [ρ(λ, x) − cn (2π)−n λn/2 ] dx + o(h−n ).

(2.20)

ℝn

2.3 Resonances First, let us recall the definition of resonance (for more details, we refer to [20]). From now on, we fix λ0 in the interior of σ(H), and we let 𝒜 be a dense subset in L2 (ℝn ). For

Elliptic differential operators with periodic and oscillating decaying coefficients | 7

Φ, Ψ ∈ 𝒜, we define the functions 𝒢Φ,Ψ and KΦ,Ψ from ℂ+ := {z ∈ ℂ; Im z > 0} into ℂ by −1

𝒢Φ,Ψ (z) = ⟨(z − H) Φ, Ψ⟩,

−1

KΦ,Ψ (z) = ⟨(z − H(h)) Φ, Ψ⟩.

Suppose that 𝒢Φ,Ψ extends analytically in a complex neighborhood Ωλ0 of λ0 , and suppose that KΦ,Ψ is meromorphic on Ωλ0 . We call resonances of H(h) near λ0 the poles of KΦ,Ψ in Ωλ0 . From now on we make use of the following conditions: (H1) For all |γ| ≤ 2m, there exist δ0,γ , δ1,γ > 0 such that t 󳨃→ bγ (t, y) has a holomorphic extension on D(δ0,γ ) = {z ∈ ℂn ; | Im z| ≤ δ0,γ ⟨Rez⟩}, and the following estimate holds uniformly for z ∈ D(δ0,γ ): 󵄨󵄨 󵄨 −δ 󵄨󵄨V(z, y)󵄨󵄨󵄨 ≤ C⟨z⟩ 1,γ .

(2.21)

(H2) There exists a unique m ∈ ℕ \ {0} such that

Σλ0 := {(x, ξ ) ∈ ℝn × E ∗ ; λm (x, ξ ) = λ0 } = (x0 , ξ0 ) ∪ Ω, Πλ0 := {ξ ∈ E ∗ ; μm (ξ ) = λ0 },

where (x0 , ξ0 ) ∈ ̸ Ω is a connected. (H3) If μm (ξ ) = λ0 , then ∇ξ μm (ξ ) ≠ 0. 󸀠󸀠 (H4) ∇λm (x0 , ξ0 ) = 0, λx,ξ (x0 , ξ0 ) is positive or negative in the sense of square matrices. In addition, we assume that the set Ω is nontrapping, i. e., there exists an escape function G such that 𝜕λ 𝜕G 𝜕G 𝜕λm {λm , G} := m − > c0 > 0, ∀(x, ξ ) ∈ Ω. (2.22) 𝜕ξ 𝜕x 𝜕ξ 𝜕x Assumption (H2) implies that, if λm (x󸀠 , ξ 󸀠 ) = λ0 (resp. μm (ξ 󸀠 ) = λ0 ), then the eigenvalue λm (x󸀠 , ξ 󸀠 ) (resp. μm (ξ 󸀠 )) is simple. Thus, it follows from (H1)–(H2) and the analytic perturbation theory (see [20]) that (x, ξ ) 󳨃→ λm (x, ξ ) (resp. ξ 󳨃→ μm (ξ )) is holomorphic in a neighborhood of (x󸀠 , ξ 󸀠 ) (resp. ξ 󸀠 ). There is no loss of generality in assuming 󸀠󸀠 󸀠󸀠 (x0 , ξ0 ) = (0, 0) and λm (0, 0) > 0 (the case λm (0, 0) < 0 is quite similar). Denote by 󸀠󸀠 (0, 0), and set κ1 , . . . , κn , ζ1 , . . . , ζn the eigenvalues of the symmetric square matrix 21 λm n

n

ℛ := {∑(2pj + 1)ζj κj ; (p1 , . . . , pn ) ∈ ℕ }. j=1

Fix C0 > 0, and let e1 < e2 ≤ e3 ≤ ⋅ ⋅ ⋅ ≤ eN0 be all the elements of ℛ in [0, C0 ] counted with their multiplicity. We can now formulate our main result of this subsection. Proposition 2.7. Assume (H1)–(H4). For h small enough, the operator H(h) has exactly N0 resonances (ej (h))1≤j≤N0 in the disk D(λ0 , C0 h) = {z ∈ ℂ; |z − λ0 | < C0 h} (counted with their algebraic multiplicity), and for every fixed j we have the following full asymptotic expansion: ∞

l

ej (h)∼λ0 + hej + ∑ ej,l h1+ 2 , l=1

ej,l ∈ ℝ

(as h ↘ 0).

(2.23)

8 | M. Dimassi

2.4 Outline of the proofs and comments Assume that p and q (see (2.1)) are independent of the variable y (see (2.7)), then H(h) = p(Dy ) + Qw (hy, Dy ). By the change of variable x = hy, H(h) is unitarily equivalent to ̃ H(h) = p(hDx ) + Qw (x, hDx ).

(2.24)

̃ In this case, H(h) is the standard Schrödinger operator with small parameter, and then Theorems 2–7 are proved by many authors (see [13, 28] and the references therein). If p and q depend on the periodic variable y, the situation is more complicated. In fact, in the coefficients bγ (hy, y), we have two scale of variables y and x = hy. When h is small enough, x = hy and y are completely different. Thus, we cannot consider H(h) as a semiclassical operator in the standard sense. Here we use the V. S. Buslaev approach (see [5]) based on a two-scale expansion method (i. e., the coordinates y and x = hy are regarded as independent parameters). Then, as in [7, 8, 11], by constructing a Grushin problem (see Section 3.1), we show that the spectral study of equation (H(h) − z)u = 0, near any fixed energy level z, can be reduced to the study of a finite system of h-pseudodifferential operators 1,w 1,w 1,w Eeff (x, hDx , z; h) = E0,eff (x, hDx , z) + hE1,eff (x, hDx , z) + ⋅ ⋅ ⋅ ,

acting on L2 (𝕋∗ ; ℂN ) (see Proposition 3.2). Hence we provide a trace formula based on 1,w the effective Hamiltonian Eeff (x, hDx , z; h) (see (4.14) and (4.22)), which allows as to prove Proposition 2.2, Theorem 2.3, Proposition 2.4, Proposition 2.5, and Theorem 2.6, using the standard approach of semiclassical analysis. Notice that the characteristic 1 set Σz := {(y, η); det E0,eff (y, η, z) = 0} plays a fundamental role in the study of the 1,w 1,w 1 operator Eeff (y, hDy , z; h). Here E0, (y, η, z) is the principal symbol of Eeff (y, hDy , z, h). + From (3.17) we deduce ∞

Σz = ⋃{(y, η); λl (y, η) = z}. l=1

For this reason, all our results will be related to λj (y, η). Next, let us sketch the proof of Proposition 2.7. From the nontrapping condition (2.22) on Ω, we deduce that a study 1 of Eeff (y, η, z; h) near (y0 , η0 ) suffices to prove our main result. Assuming that (y0 , η0 ) = (0, 0), assumptions (H2) and (H4) allow us to prove that n

󵄨 󵄨3 1 E0,eff (y, η, λ0 ) = λ0 − ∑(κl yl2 + ζl η2l ) + 𝒪(󵄨󵄨󵄨(y, η)󵄨󵄨󵄨 ), l=1

when (y, η) → (0, 0). This shows that the resonances of H(h) near λ0 coincide, modulo 𝒪(h3/2 ), with the eigenvalues of the operator n

∑(κl yl2 + ζl h2 D2yl ). l=1

Elliptic differential operators with periodic and oscillating decaying coefficients | 9

3 Effective Hamiltonian theory 3.1 Grushin problem Here, we introduce the notion of Grushin problem and effective Hamiltonian method. Let Ki , i = 1, 2, 3 be three Hilbert spaces, and let Ω be an open bounded set in ℂ. For G ∈ ℒ(K1 , K3 ), assume that there exist R+ ∈ ℒ(K1 , K2 ) and R− ∈ ℒ(K2 , K3 ) such that the linear operator G−t T+

𝒢 (t) = (

T− ) : K1 × K2 → K3 × K2 0

is invertible for t ∈ Ω. We denote by K(t) K− (t)

ℰ (t) = (

K+ (t) ) Eeff (t)

its inverse. We call Eeff (t) the effective Hamiltonian corresponding to G, and 𝒢 (t) is known as a Grushin problem. Using that ℰ ∘ 𝒫 = I and 𝒢 ∘ ℰ = I, an easy computation shows that (G − t)

is bijective if and only if Eeff (t) is bijective,

dim ker(G − t) = dim ker(Eeff (t)), −1

(G − t) −1 Eeff (t)

=

−1 = K(t) − K+ (t)Eeff (t)K− (t), −1 −T+ (G − t) T− .

(3.1) (3.2) (3.3) (3.4)

Since z 󳨃→ (G − t) is analytic, it follows that K(t), K± (t) and Eeff (t) are also analytic in t ∈ Ω. Since T± are t-independent, it follows that 𝜕t Eeff (t) = K− (t)K+ (t).

(3.5)

3.2 Classes of symbols and notations Let ℋN be the space of Hermitian N × N matrices endowed with the norm ‖ ⋅ ‖N×N , where for A ∈ ℋN , ‖A‖N×N := sup{w∈ℝN ; |w| 0 such that 󵄩󵄩 γ β 󵄩 −m−|γ| , 󵄩󵄩𝜕η 𝜕y H(y, η)󵄩󵄩󵄩ℋN ≤ Cγ,β ⟨η⟩

⟨η⟩2 = 1 + |η|2 .

(3.6)

10 | M. Dimassi When N = 1 or m = 0, we write Sm (𝕋∗ ×ℝn ) or S(𝕋∗ ×ℝn ; ℋN ) instead of Sm (𝕋∗ ×ℝn ; ℋ1 ) and S0 (𝕋∗ × ℝn ; ℋN ). If H depends on h ∈ ]0, h0 ] and possibly on other parameters as well, we require (3.6) to hold uniformly with respect to these parameters. Let H(y, η; h) ∈ Sm (𝕋∗ × ℝn ; ℋN ), we write ∞

H(y, η; h) ∼ ∑ Hl (y, η)hl in Sm (𝕋∗ × ℝn ; ℋN ) l=0

if and only if for every p ∈ ℕ, h−(p+1) (H − ∑pl=0 Hl hl ) ∈ Sm (𝕋∗ × ℝn ; ℋN ). We use the standard Weyl quantization of symbols. More precisely, if H ∈ Sm (𝕋∗ × ℝn ; ℋN ) then H w (x, hDx ) is the operator defined by H w (y, hDy )u(y) =

1 t+y , η)u(t) dt dη, ∫ ∫ ei(y−t).η/h H( (2πh)n 2

(3.7)

𝕋∗ ×ℝn

for all u ∈ C ∞ (𝕋∗ ; ℂN ). We will occasionally use the shorthand notations Opw h (H) = H w = H w (y, hDy ) when there is no ambiguity. If m ≥ 0, then H w (y, hDy ) ∈ ℒ(L2 (𝕋∗ ); L2 (𝕋∗ )) (see Theorem 5.2).

3.3 Effective Hamiltonian As indicated in the introduction, we will reduce in this subsection the spectral study of H(h) near an energy z to the study of a system of h-pseudodifferential operators. The proofs of the results in this subsection are quite similar to those in [19] (see also [13, 14]). For the reader’s convenience, let us fix the notations and explain the main change in this paper. Let 𝒯Γ be the distribution in 𝒮 󸀠 (ℝ2n ) given by 𝒯Γ (x, y) =

α∗ 1 ∑ ei(x−hy) h , n vol(E)h α∗ ∈Γ∗

where E is the fundamental domain of Γ introduced in Section 2. For m ∈ ℕ, and ξ ∈ ℝn , we denote by 𝕃m and K0 the Hilbert spaces with their standard norms: 𝕃m := {u(t)𝒯Γ (t, y󸀠 ); 𝜕tα u ∈ L2 (ℝn ), ∀α, |α| ≤ m}, K0 = L2 (𝕋),

Km,η = {u ∈ K0 ; (Dy + η)α u ∈ K0 , ∀|α| ≤ m}.

According to Proposition 13.5 in [13], the operator H(h) acting on L2 (ℝn ) with domain H 2 (ℝn ) is unitarily equivalent to ℍ1 (h) := Qw (x, y, hDx + Dy ),

(3.8)

acting on 𝕃0 with domain 𝕃2 . Moreover, the following results are proved in [19]. First, we construct a Grushin problem on the symbolic level.

Elliptic differential operators with periodic and oscillating decaying coefficients | 11

Theorem 3.1 ([19]). There exist an integer N, an operator r+ ∈ ℒ(L2 (𝕋); ℂN ), and a neighborhood Ω of I such that P1 (y, η) − z r+

r+∗ ) : H 2m (𝕋) × ℂN → L2 (𝕋) × ℂN 0

𝒫 (y, η, z) := (

(3.9)

is invertible for z ∈ Ω with bounded inverse ℰ (y, η, z) := (

e(y, η, z) e− (y, η, z)

e+ (y, η, z) ). eeff (y, η, z)

(3.10)

∗ Here eeff ∈ S(ℝ2d y,η ; ℋN ) is Γ -periodic with respect to η.

Let Ω and N be the open set and the integer given by Proposition 3.1. Next, we construct a Grushin problem for the corresponding Weyl operator 𝒫 1 (z, h) := 𝒫 w (y, hDy , z) where ℍ1 (h) − z R+

1

𝒫 (z, h) = (

R∗+ ) : 𝕃2m × L2 (𝕋∗ ; ℂN ) → 𝕃0 × L2 (𝕋∗ ; ℂN ). 0

(3.11)

We have Proposition 3.2 ([19]). For h small enough, the operator 𝒫 1 (z, h) is invertible with bounded two-sided inverse 1

ℰ (z, h) := (

E 1 (z, h)

E−1 (z, h)

E+1 (z, h)

(3.12)

). 1 Eeff (z, h)

1,w 1 (y, hDy ; z, h) with Here Eeff (z, h) = Eeff 1 1 Eeff (y, η; z, h) ∼ ∑ E0,eff (y, η; z) hl in S0 (𝕋∗ × ℝn ; ℋN ) l≥0

(3.13)

1 where E0,eff (y, η, z) = eeff (η, −y, z) is given by Theorem 3.1.

Remark 3.3. (a) According to Proposition 2.1 in [19], the operator R+ is only related to the nonperturbed operator H. Thus, we may take the same R+ for both the perturbed and nonperturbed operator. (b) Let 𝒫 0 (z, h) and ℰ 0 (z, h) be the operators given by Proposition 3.1 corresponding to the nonperturbed Hamiltonian pw (y, Dy + hDx ): ℍ0 (h) − z R+

0

R∗+ ), 0

𝒫 (z, h) = (

0

E 0 (z, h) E−0 (z, h)

ℰ (z, h) := (

E+0 (z, h) ). 0 Eeff (z, h)

0 0 For abbreviation, we denote Eeff (z, h), E±0 (z, h), and E 0 (z, h) simply by Eeff , E±0 , and 0 E . It follows from (3.1), (3.2), (3.3), (3.4), and Propositions 3.1–3.2 that

(ℍj (h) − z)

−1

j

= E j − E+j (Eeff ) E−j , −1

(3.14)

12 | M. Dimassi j

−1

(Eeff )

j 𝜕z Eeff

=

= −R+ (ℍj (h) − z) R∗+ , −1

E−j E+j ,

(3.15) (3.16)

det(eeff (y, η, z)) = 0

iff z ∈ σ(P1 (y, η)) iff ∃l ∈ ℕ s. t. z = λl (y; η),

C −1 󵄩 󵄩󵄩 , 󵄩󵄩(eeff (y, η, z)) 󵄩󵄩󵄩N×N ≤ | Im z|

(3.17) (3.18)

dim ker(P(y, η) − z) = dim ker(eeff (y, η, z)).

(3.19)

Remark 3.4. For fixed (y, η, t0 ) ∈ ℝ2n × ℝ, we denote d = dim ker(eeff (y, η, t0 )). Let λ1 (t0 ) = λ2 (t0 ) = ⋅ ⋅ ⋅ = λd (t0 ) be the eigenvalues of eeff (y, η, t0 ). Ordinary analytic perturbation theory (see [21]) shows that for z near t0 , eeff (y, η, z) has exactly d eigenvalues, (λl (z))1≤l≤d , which can be chosen analytic with respect to z near t0 . From (3.18) we deduce that 󵄨󵄨 󵄨 󵄨󵄨λl (z)󵄨󵄨󵄨 ≥ Cl | Im z|, which yields λl󸀠 (t0 ) ≠ 0,

for all 1 ≤ l ≤ d.

Thus, z 󳨃→ det eeff (y, η, z) has a zero t0 of multiplicity d.

4 Proofs 4.1 Proof Proposition 2.2 Under assumption (2.11), there exists a constant C such that −1 󵄩 󵄩 suξ 󸀠 ∈ℝn ,z∈[a,b] 󵄩󵄩󵄩(z − pw (y, Dy + ξ 󸀠 )) 󵄩󵄩󵄩ℒ(L2 (𝕋);H 2m (𝕋)) < C.

On the other hand, assumption (2.3) and (2.4) yield −1 󵄩 󵄩 supξ 󸀠 ∈ℝn ,z∈[a,b] 󵄩󵄩󵄩qw (x󸀠 , y, Dy + ξ 󸀠 )(z − pw (y, Dy + ξ 󸀠 )) 󵄩󵄩󵄩ℒ(L2 (𝕋);L2 (𝕋)) → 0,

(4.1)

as |x󸀠 | tends to infinity. Now let W(x󸀠 ) ∈ C ∞ (ℝnx ; [0, 1]) with W = 1 if |x 󸀠 | > 2R and W(x󸀠 ) = 0 if |x󸀠 | < R. Put ̂ 󸀠 , ξ 󸀠 ) = pw (y, Dy + ξ 󸀠 ) + W(x󸀠 )qw (x󸀠 , y, Dy + ξ 󸀠 ). P(x ̂ 󸀠 , ξ 󸀠 )) is uniformly invertible from L2 (𝕋) into L2 (𝕋) It follows from (4.1) that (z − P(x ̂ 󸀠 , ξ 󸀠 ), and let for z ∈ [a, b] provided that R is large enough. Apply Theorem 3.1 to P(x

Elliptic differential operators with periodic and oscillating decaying coefficients | 13

êeff (x󸀠 , ξ 󸀠 , z) be given by its inverse (see (3.10)). Making use of (3.17) and the above remark, we deduce that there exits C > 0 such that for all (x 󸀠 , ξ 󸀠 , z) ∈ ℝn × 𝕋∗ × [a, b], we have 󵄨 1 󵄨󵄨 󸀠 󸀠 󵄨󵄨det êeff (x , ξ , z)󵄨󵄨󵄨 ≥ . C

(4.2)

1 1 Êeff (x󸀠 , ξ 󸀠 , z; h) = êeff (ξ 󸀠 , −x 󸀠 , z) + Eeff (x󸀠 , ξ 󸀠 , z; h) − E0,eff (x 󸀠 , ξ 󸀠 , z)

(4.3)

Next, let 1 = êeff (ξ 󸀠 , −x 󸀠 , z) + ∑ hj Ej,eff (x󸀠 , ξ 󸀠 , z). j≥1

It follows from (4.2) that there exists C, h0 > 0 such that for 0 < h < h0 and for all (x󸀠 , ξ 󸀠 , z) ∈ 𝕋∗ × ℝn × [a, b], we have 1 󵄨󵄨 󵄨 󸀠 󸀠 . 󵄨󵄨det Êeff (x , ξ , z; h)󵄨󵄨󵄨 ≥ 2C

(4.4)

Combining this with Theorem 5.4, we deduce that Êeff is invertible and its inverse is analytic with respect to z near [a, b]. In particular, 󵄩󵄩 ̂ −1 󵄩󵄩 󵄩󵄩(Eeff ) 󵄩󵄩ℒ(L2 (𝕋∗ ;ℂN )) = 𝒪(1).

(4.5)

By construction of W, we have 1 Eeff (x󸀠 , ξ 󸀠 , z; h) = Êeff (x󸀠 , ξ 󸀠 , z; h)

󵄨 󵄨 for large 󵄨󵄨󵄨ξ 󸀠 󵄨󵄨󵄨.

(4.6)

Now let f ∈ C0∞ (ℝ; ℝ), and let ̃f ∈ C0∞ (ℂ) satisfy the following properties: suppf ̃ ⊂ (a, b) + i[−1, 1],

(4.7)

̃f = f

(4.8)

on ℝ,

N

𝜕z ̃f (z) = 𝒪N (| Im z| ),

for all N ∈ ℕ.

(4.9)

We call f ̃ an almost analytic extension of f . By Helffer–Sjöstrand formula (see, e. g., [13]), we have f (ℍ1 (h)) = −

1 −1 ∫ 𝜕z ̃f (z)(z − ℍ1 (h)) L(dz), π

(4.10)

where L(dz) = dxdy with ℂ ∼ ℝ2x,y . Combining (3.14) with 1 (Eeff )

−1

−1 1 −1 1 −1 = Êeff − (Eeff ) (Eeff − Êeff )Êeff ,

−1 1 1 and using the fact that Êeff , E , E+ , E−1 are analytic in a complex neighborhood of [a, b], we get

f (ℍ1 (h)) = −

1 1 −1 ̂ 1 ̂ −1 1 ) (Eeff − Eeff )Eeff E− )L(dz). ∫ 𝜕z ̃f (z)(E+1 (Eeff π

(4.11)

14 | M. Dimassi We recall that if K(z) is analytic near suppf ̃ then ∫ 𝜕z ̃f (z)K(z)L(dz) = 0.

(4.12)

1 Using Theorem 5.3, we deduce that (Eeff − Êeff ) is of trace class. Combining this with 1 1 1 the identity 𝜕z Eeff = E− E+ , we get for Im z ≠ 0, 1 1 ̂ −1 1 tr(E+1 (Eeff ) (Êeff − Eeff )Eeff E− ) −1

(4.13)

1 1 ̂ −1 1 = tr((Eeff ) (Êeff − Eeff )Eeff 𝜕z Eeff ). −1

According to (4.6), the set 1 K := Π󸀠ξ (supp(E0,eff (x 󸀠 , ξ 󸀠 , z) − êeff (ξ 󸀠 , −x 󸀠 , z)))

is compact. Let χ ∈ C0∞ (ℝnξ ), with χ = 1 near K, and put χ̂ = χ w (hDx󸀠 ). Since K ∩ supp(1 − χ) = 0, Theorems 5.1 and 5.3 yield 󵄩 󵄩󵄩 ̂ 1 1 −1 󵄩󵄩(Eeff − Eeff )Êeff 𝜕z Eeff (1 − χ̂)󵄩󵄩󵄩tr ≡ 0. From (3.15) we have 󵄩󵄩 1 −1 󵄩󵄩 −1 󵄩󵄩(Eeff ) 󵄩󵄩 = 𝒪(| Im z| ). Consequently, 󵄩 󵄩󵄩 1 −1 ̂ −1 1 ∞ −1 1 󵄩󵄩(Eeff ) (Eeff − Eeff )Êeff 𝜕z Eeff (1 − χ̂)󵄩󵄩󵄩tr = 𝒪(h | Im z| ), which, together with (4.11) and (4.13), yields tr[f (ℍ1 (h))] ≡ −

1 1 −1 ̂ 1 ̂ −1 1 tr[∫ 𝜕z ̃f (z)(Eeff ) (Eeff − Eeff )Eeff 𝜕z Eeff χ̂L(dz)]. π

Dividing the right-hand side of the above equality into two terms and using the −1 ̂ fact (4.12), as well as that Êeff 𝜕z Eeff is analytic in z, we obtain tr[f (ℍ1 (h))] ≡ −

1 1 −1 1 ) 𝜕z Eeff χ̂L(dz)]. tr[∫ 𝜕z ̃f (z)(Eeff π

(4.14)

The following result may be proved in much the same way as in [8]. Proposition 4.1. There is r 1 (x󸀠 , ξ ; h) ∼ ∑j≥0 hj rj (x 󸀠 , ξ 󸀠 ) in S0 (𝕋∗ × ℝn , ℋN ) such that 1 󸀠 󸀠 Opw h (r (x , ξ ; h)) = −

1 π

∫

1 1 𝜕z ̃f (z)(Eeff ) 𝜕z Eeff L(dz). −1

| Im z|≥hδ

In particular, r0 (x󸀠 , ξ 󸀠 ) = −

1 −1 1 1 (x 󸀠 , ξ 󸀠 , z)) 𝜕z E0,eff (x󸀠 , ξ 󸀠 , z)L(dz). ∫ 𝜕z ̃f (z)(E0,eff π

(4.15)

Elliptic differential operators with periodic and oscillating decaying coefficients | 15

Let us return to the proof of Proposition 2.2. We decompose (4.14) into two terms: tr[f (ℍ1 (h))] ≡ −

1 1 −1 1 ) 𝜕z Eeff χ̂L(dz)] tr[∫ 𝜕z ̃f (z)(Eeff π

(4.16)

ℐ1

−

1 1 −1 1 tr[∫ 𝜕z ̃f (z)(Eeff ) 𝜕z Eeff χ̂L(dz)] = (1) + (2), π ℐ2

where ℐ1 = {z; |ℑz| ≥ hδ } and ℐ2 = {z ∈ ℂ; |ℑz| ≤ hδ }. Since 𝜕z ̃f (z) = 𝒪N (| Im z|∞ ), it follows that (2) ≡ 0. Therefore, tr[f (ℍ1 (h))] ≡ −

1 1 −1 1 ) 𝜕z Eeff χ̂L(dz)], tr[∫ 𝜕z ̃f (z)(Eeff π ℐ1

which, together with Proposition 4.1, gives (2.18). It remains to compute a0 . We have ̂ 0 (x󸀠 , ξ 󸀠 )] dx󸀠 dξ 󸀠 = ∫ ∫ tr[r ̂ 0 (x 󸀠 , ξ 󸀠 )] dx 󸀠 dξ 󸀠 a0 = ∫ ∫ tr[r E ∗ ×ℝn

= ∫ ∫ (− E ∗ ×ℝn

E ∗ ×ℝn

1 −1 1 1 ̂ 0,eff (X)) 𝜕z E0,eff (X)]L(dz)) dx󸀠 dξ 󸀠 , ∫ 𝜕z ̃f (z)tr[(E π

where X = (x󸀠 , ξ 󸀠 , z). Therefore, a0 = ∬ (− E ∗ ×ℝn

1 𝜕z det E0,eff (x󸀠 , ξ 󸀠 , z) 1 L(dz)) dx 󸀠 dξ 󸀠 . ∫ 𝜕z ̃f (z) 1 π (x󸀠 , ξ 󸀠 , z) det E0,eff

Now to end the proof we need the following result. For the proof, we refer to [12]. Proposition 4.2. Let (zl )l≥1 be all the zeros of a holomorphic function g defined in a neighborhood of supp(̃f ). Then g 󸀠 (z) −1 L(dz) = ∑ f (zk ). ∫ 𝜕z ̃f (z) π g(z) k≥1 1 Applying Proposition 4.2 to g(z) = det(E0,eff (x󸀠 , ξ 󸀠 , z)), and using Remark 3.4 and (3.17), we get

a0 = ∑ ∬ f (λl (y, η)) dy dη, l≥1 ℝn ×E ∗ x

which, together with (2.9), completes the proof of Proposition 2.2.

(4.17)

16 | M. Dimassi

4.2 Proof of Theorem 2.3 For ϵ > 0, let fϵ , fϵ ∈ C0∞ (ℝ; [0, 1]) satisfy 1[a+ϵ,b−ϵ] ≤ fϵ ≤ 1[a,b] ≤ fϵ ≤ 1[a−ϵ,b+ϵ] .

(4.18)

Using that tr[fϵ (H(h))] ≤ N[a,b] (h) ≤ tr[fϵ (H(h))], we deduce lim lim((2πh)n tr[fϵ (H(h))]) ≤ lim(2πh)n N[a,b] (h) ϵ↘0 h↘0

h↘0

≤ lim lim((2πh)n tr[fϵ (H(h))]). ϵ↘0 h↘0

By Proposition 2.2 and (4.18), we have lim lim((2πh)n tr[fϵ (H(h))]) = lim lim((2πh)n tr[fϵ (H(h))]) ϵ↘0 h↘0

ϵ↘0 h↘0

= ∫ [ρ(b, x) − ρ(a, x)] dx, ℝn

which yields (2.14).

4.3 Proof of Proposition 2.4 To deal with (2.15), we need a more precise trace formula. For θ ∈ C0∞ (ℝ), we set 1 θ̆h (τ) := ∫ eitτ/h θ(t) dt. 2πh A slight change in the proof of (4.14) actually yields tr[f (H(h))θ̆h (t − H(h))] 1 1 −1 1 ) 𝜕z Eeff χ̂L(dz)]. ≡ tr[− ∫ 𝜕z ̃f (z)θ̆h (t − z)(Eeff π

(4.19)

From now on, we assume that the support of f ̃ is located near t = a or b. Let Σt = {(x󸀠 , ξ 󸀠 ) ∈ ℝ2n ; det(eeff (x 󸀠 , ξ 󸀠 , t)) = 0}, and let χ ∈ C0∞ (ℝ2n ) be equal to one near Σt . Since eeff (x 󸀠 , ξ 󸀠 , t) is invertible near supp(1 − χ), the arguments used in (4.14) actually give tr[f (H(h))θ̆h (t − H(h))] 1 1 −1 1 ≡ tr[− ∫ 𝜕z ̃f (z)θ̆h (t − z)χ w (Eeff ) 𝜕z Eeff χ̂L(dz)]. π

(4.20)

Elliptic differential operators with periodic and oscillating decaying coefficients | 17

If we choose χ in the form ∑j χj (x, ξ ) where each χj has its support in a small neighborhood of some point of Σt , then by assumption (H) we may assume that det(eeff (x 󸀠 , ξ 󸀠 , t)) = 0 if and only if there is a unique integer j such that λj (x 󸀠 , ξ 󸀠 ) = t. On the other hand, assumption (H) implies that the energy t = a, b is not a critical value of 1 the principal symbol eeff (ξ 󸀠 , −x󸀠 , z) of Eeff . Therefore, it follows from Theorem 1.8 in [8] and (4.19) that ∞

tr[f (H(h))θ̆h (t − H(h))] ∼ ∑ βj hj−n j=0

(h ↘ 0).

(4.21)

Hence, Proposition 2.4 is a consequence of (4.21) and Theorem V-13 in [28].

4.4 Proof of Proposition 2.5 From now on, we write [aj ]1j=0 = a1 − a0 . For simplicity we may assume that q(x, y, η) = V(x, y). The proof of the general case is quite similar. Assuming, for the moment, that the following proposition holds. Proposition 4.3. Under assumption (2.16), the operator [f (H(h)) − f (H)],

f ∈ C0∞ (ℝ),

is of trace class. Moreover, if f ̃ satisfies (4.7), (4.8), and (4.9), then tr[f (H(h)) − f (H)] = tr[f (ℍ1 (h)) − f (ℍ0 (h))] = tr(−

(4.22)

1 j −1 j 1 ∫ 𝜕̃f (z)[(Eeff ) 𝜕z Eeff ]j=0 L(dz)). π ℂ

Now, as in the proof of Proposition 2.2, equation (4.22) gives Proposition 2.5. Thus, the proof is completed by showing Proposition 4.3. For that, we need the following lemmas. Lemma 4.4. We have [E+j ]j=0 = E 1 VE+0 ,

1

(4.23)

[E−j ]j=0 = E−0 VE 1 ,

1

(4.24)

1

(4.25)

1

(4.26)

and j

[Eeff ]j=0 = E−1 VE+0 . If, in addition, (2.21) holds then j

[Eeff (k, r; z, h)]j=0 ∈ Sδ (𝕋∗ × ℝn ; ℋN ).

18 | M. Dimassi Proof. Equations (4.23)–(4.25) are consequences of the following facts: 1

[ℰ j (z, h)]j=0 = ℰ 1 (z, h)[𝒫 0 (z, h) − 𝒫 1 (z, h)]ℰ 0 (z, h)

= −ℰ 0 (z, h)[𝒫 1 (z, h) − 𝒫 0 (z, h)]ℰ 1 (z, h), V 0

1

[𝒫 j (z, h)]j=0 = (

0 ), 0

while (4.26) follows from (4.25) and Theorem 5.7. Proposition 4.5. Assume (2.16). The following operators: VE+0 : L2 (𝕋∗ ; ℂN ) → 𝕃0

(4.27)

E−0 V : 𝕃0 → L2 (𝕋∗ ; ℂN ),

(4.28)

and

are of trace class. Proof. We conclude from the equality 𝒫 j (z, h)∗ = 𝒫 j (z, h) that ℰ j (z, h)∗ = ℰ j (z, h), hence E−0 (z, h)∗ = E+0 (z, h),

j = 0, 1.

(4.29)

On the other hand, since (E−0 V)∗ = VE+0 (z, h), (4.28) follows from (4.27). Thus, we need δ

only to consider (4.27). Consider the operator A = (Id − h2 Δ𝕋∗ )− 2 on L2 (𝕋∗ ; ℂN ). Set B = VE+0 (z, h), C = B∗ B and D = A−1 CA−1 . Under assumption (2.16) it follows from Theorems 5.7 and 5.6, as well as Proposition 3.2, that D ∈ S0 (𝕋∗k×ℝn ; ℋN ). Hence, by r

Theorem 5.2 we deduce that D is a bounded operator from L2 (𝕋∗ ; ℂN ) into L2 (𝕋∗ ; ℂN ). Therefore, since C is positive (in the sense of operators), we get 0 ≤ C = ADA ≤ ‖D‖A2 , which implies 1

0 ≤ C 2 ≤ √‖D‖A. Theorem 5.3 implies that A : L2 (𝕋∗ ; ℂN ) → L2 (𝕋∗ ; ℂN ) is of trace class, since δ > n. Hence the above inequality, as well as the fact that C = BB∗ , implies that B is of trace class. This ends the proof of the lemma. Lemma 4.6. Assume (2.16), and let z ∈ Ω with t Im(z) ≠ 0. The operator j

1

[E+j (Eeff ) E−j ]j=0 −1

is of trace class from L2 (ℝn ) to L2 (ℝn ). Moreover, j

1

j

j

1

tr([E+j (Eeff ) E−j ]j=0 ) = tr([(Eeff ) 𝜕z Eeff ]j=0 ). −1

−1

Notice that the right-hand side of (4.30) is defined on L2 (𝕋∗ ; ℂN ).

(4.30)

Elliptic differential operators with periodic and oscillating decaying coefficients | 19

Proof. An obvious computation shows that j

1

1

1 [E+j (Eeff ) E−j ]j=0 = [([E+j ]j=0 )(Eeff ) E−1 ] −1

−1

0 + [E+0 (Eeff )

−1

1 ([E−j ]j=0 )]

(4.31) 1 − [E+0 (Eeff )

−1

j 1 0 −1 1 ([Eeff ]j=0 )(Eeff ) E− ].

Clearly, by Lemma 4.4 and Proposition 4.5, the right-hand side of the above equality is of trace class. Thus, using identity (3.16) and the cyclicity of the trace, we get j

1

1 1 tr([E+j (Eeff ) E−j ]j=0 ) = tr((Eeff ) (𝜕z Eeff − E−1 E+0 )) −1

−1

(4.32)

0 0 0 1 + tr((Eeff ) (E−1 E+0 − 𝜕z Eeff )) − tr((Eeff ) E−1 E+0 − (Eeff ) E−1 E+0 ), −1

−1

−1

which yields Lemma 4.6. Now let us return to the proof of Proposition 4.3. By (4.10), we have f (ℍ1 (h)) − f (ℍ0 (h)) 1 −1 −1 = − ∫ 𝜕̃f (z)[(z − ℍ1 (h)) − (z − ℍ0 (h)) ]L(dz), π ℂ

which, together with (3.14), yields f (ℍ1 (h)) − f (ℍ0 (h)) =

1 1 ∫ 𝜕̃f (z)[E j ]j=0 L(dz) π

(4.33)

ℂ

−

1 1 j −1 ∫ 𝜕̃f (z)[E+j (Eeff ) E−j ]j=0 L(dz). π ℂ

Since E j is analytic near supp(̃f ), the first integral in (4.33) vanishes. Thus, f (ℍ1 (h)) − f (ℍ0 (h)) = −

1 1 j −1 ∫ 𝜕̃f (z)[E+j (Eeff ) E−j ]j=0 L(dz). π ℂ

According to Lemma 4.6, the operator [f (ℍ1 (h)) − f (ℍ0 (h))] is of trace class. Now the second equality in (4.22) follows from (4.30). The first one is a simple consequence of the fact that ℍ1 (h) and ℍ0 (h) are unitarily equivalent to H(h) and H, respectively.

4.5 Proof of Theorem 2.6 If t 󳨃→ ξ (t; h) is a monotonic function then the proof is exactly the same as that of Theorem 2.3. For the general case, using a Robert’s trick [29], we may write ξ (t; h) = ξ1 (t; h) − ξ2 (t; h) where μ 󳨃→ ξi (t; h), i = 1, 2 are monotonic, and then repeat the above arguments to ξ1 (t; h) and ξ2 (t; h)

20 | M. Dimassi

4.6 Proof of Proposition 2.7 1 In this subsection we will just be concerned with E 1 , Eeff , and E±1 . We denote them ∗ briefly by E, Eeff , and E± . Fix λ0 ∈ μm (𝕋 ), and let Ω be a small complex neighborhood of λ0 . Under assumptions (H1)–(H2), we may choose Eeff (x, ξ , z; h) in Proposition 3.2 scalar-valued with the following properties: (a)

eeff (x, ξ , z) = z − λm (x, ξ ),

∀z ∈ Ω.

(b) For all N ∈ ℕ, there exists FN such that Eeff (x, ξ , z; h) = eeff (ξ , −x, z) N

+ ∑ hl Ej,eff (x, ξ , z) + hN+1 FN (x, ξ , z; h). l=1

(c) The functions El,eff (x, ξ , z), FN (x, ξ , z; h) are analytic with respect to (x, ξ , λ0 ) in a neighborhood of E ∗ × ℝn × {λ0 }. Here, the proof is quite similar to that in [9]. So we omit the details. Fix t0 > 0. For t ∈ D(t0 ) := {t ∈ ℂ; |t| < t0 }, we put νt (x) = x − t∇μm (x), and we let Jt (x) = det[Dνt (x)] be the Jacobian of νt (x). Since μm (x) is bounded with all its derivatives, we may choose t0 small enough so that νt is invertible for all t ∈ D(t0 ). For u in the Schwartz space S(ℝn ), we define 1

𝒰t u(x) = {Jt2 (x)(ℱh u)(νt (x))},

where ℱh u(ξ ) = ∫ e

−ix⋅ξ /h

u(x) dx.

ℝn

Clearly, for real t, 𝒰t extends to a unitary operator on L2 (ℝn ). Now, for t ∈ D(t0 ) with t0 small enough, let us define w,t w Eeff (x, hDx , z; h) := 𝒰t Eeff (x, hDx , z; h)𝒰t−1 .

(4.34)

The following properties may be proved in much the same way as Theorem 2.3 and Corollary 2.5 in [9]: t t (a) Eeff (x, ξ , z; h) ∼ ∑l≥0 El,eff (x, ξ , z)hl in Sδ1 (𝕋∗ × ℝn ). w,t w (b) For t ∈ ]−t0 , t0 [, the operator Eeff (x, hDx , z; h) is unitarily equivalent to Eeff (x, hDx , z; h).

Elliptic differential operators with periodic and oscillating decaying coefficients | 21

w,t (x, hDx , z; h) is well defined from L2 (𝕋∗ ) into L2 (𝕋∗ ). (c) The operator Eeff 𝜕ν

t (d) E0,eff (x, ξ , z) = z − λm ((1 − tM(x))−1 ξ , νt (x)), where M(x) = ( 𝜕xj (x))1≤i,j≤n . i

w,t (x, hDx , z; h))−1 has a meromorphic extension (e) If t = i Im t with Im t > 0, then (Eeff from D(ϵ) = {z ∈ ℂ; Im z > 0, |z − λ0 | < ϵ} to 𝒟η,ϵ := {z ∈ ℂ; Im z > −ηh, |z − λ0 | < ϵ} provided that ϵ and η are small enough. (f) z ∈ 𝒟η,ϵ with ℑz ≤ 0 is a resonances of H(h) with multiplicity k if and only if z0 is w,t (x, hDx , z; h))−1 with multiplicity k. a pole of (Eeff

w,t According to the above properties, we are left with proving that (Eeff ((x, hDx , z; h)))−1 has exactly N poles e1 (h), . . . , eN (h) in D(λ0 , C0 h) (repeated with their algebraic mult tiplicity). Assumption (2.22) tells us that Eeff (x, ξ , λ0 ; h) is elliptic except at (x, ξ ) = w,t (x0 , ξ0 ). Thus, to describe the spectrum of Eeff (x, hDx , z; h) near λ0 , we need to study t the operator χ w (x, hDx )Eeff (x, ξ , z; h) where χ = 1 near (x0 , ξ0 ) and χ = 0 outside a small neighborhood of (x0 , ξ0 ). On the other hand, near (x0 , ξ0 ) we can use the WKB method near a local minimum (see [13], Chapter 3 and the end of Chapters 4 and 14), to construct z1 (h), . . . , zN (h) satisfying (2.23) and g1 (x, h), . . . , gN (x, h) such that t,w (x, hDx , zi (h); h)gj (x, h) = 𝒪(h∞ ). Eeff

Now Proposition 2.7 is a simple consequence of the above equality. For the details, we refer to [9, Section 5].

5 Appendix For the convenience of the reader, we repeat the relevant material on operator-valued h-pseudodifferential calculus from [13] without proofs.

5.1 Review on semiclassical analysis In this appendix, X denotes either 𝕋∗ × ℝn or ℝ2n . Recall that S(𝕋∗ × ℝn ℋN ) is a subset of P ∈ S(ℝ2n ; ℋN ), Γ∗ -periodic with respect to x. We denote Y = Πx X (i. e., Y = ℝn ) (resp, 𝕋∗ ). Theorem 5.1 (Composition formula). For A1 , A2 ∈ S(X; ℋN ), the operator Bw (y, hDy ; h) = w Aw 1 (y, hDy ) ∘ A2 (y, hDy ) is an h-pseudodifferential one, and its symbol has a complete asymptotic expansion in powers of h: ∞

B(x, ξ ; h) ∼ ∑ bl (x, ξ )hl l=0

in S(X; ℋN ).

22 | M. Dimassi Theorem 5.2 (L2 -boundedness). If A = A(y, η; h) ∈ S(X; ℋN ), then Aw (y, hDy ; h) is bounded from L2 (Y; ℂN ) into L2 (Y; ℂN ), and there is an h-independent constant C such that 󵄩 󵄩󵄩 w 󵄩󵄩A (y, hDy ; h)󵄩󵄩󵄩 ≤ C. β

Theorem 5.3. Let A = A(y, η; h) ∈ S(X; ℋN ) satisfy 𝜕yα 𝜕η A ∈ L1 (X), for all |α|+|β| ≤ 2n+2. The operator Aw (y, hDy ; h) is of trace class, and we have tr(Aw (y, hDy ; h)) =

1 ̂ η; h)) dy dη. ∫ ∫ tr(A(y, (2πh)n Y

Theorem 5.4. Suppose A = A(y, η; h) ∈ S(X; ℋN ) is such that 󵄨󵄨 󵄨 󵄨󵄨det A(y, η; h)󵄨󵄨󵄨 ≥ C, where C is an h-independent constant. The operator Aw (y, hDy ; h) : L2 (Y) → L2 (Y), is invertible for h small enough.

5.2 Operator-valued h-pseudodifferential operator Let 𝒜𝒳 be a family of Hilbert spaces depending on 𝒳 = (x, ξ ) ∈ ℝ2n and satisfying the following properties: 𝒜𝒳 = 𝒜𝒴

as vector spaces for all

2n

𝒳,𝒴 ∈ ℝ ,

(5.1)

∃N0 , C0 > 0 such that for all u ∈ 𝒜0 and all 𝒳 , 𝒴 ∈ ℝ2n , we have ‖u‖𝒜𝒳 ≤ C⟨𝒳 − 𝒴 ⟩N0 ‖u‖𝒜𝒴 .

(5.2)

Given ℬ𝒳 , 𝒳 ∈ ℝ2n a second family satisfying (5.1) and (5.2), we say that P ∈ C ∞ (ℝ2n ; ℒ(𝒜0 , ℬ0 )) belongs to S0 (ℝ2n ; ℒ(𝒜𝒳 , ℬ𝒳 )) if ∀α ∈ ℕ2n , there exist a constant Cα such that 󵄩󵄩 α 󵄩󵄩 2n 󵄩󵄩𝜕𝒳 P 󵄩󵄩ℒ(𝒜𝒳 ;ℬ𝒳 ) ≤ Cα ,

(5.3)

uniformly on 𝒳 ∈ ℝ2n . To P we associate the Weyl-operator P w (x, hDx ) given by (3.7). As in the scalar case, the following results hold: Theorem 5.5. Let 𝒜𝒳 , ℬ𝒳 satisfy (5.1), (5.2), and let P ∈ S0 (ℝ2n ; ℒ(𝒜𝒳 , ℬ𝒳 )). The operw n n ator Opw h (P) = P (x, hDx ) is uniformly continuous from 𝒮 (ℝ ; 𝒜0 ) into 𝒮 (ℝ ; ℬ0 ).

Elliptic differential operators with periodic and oscillating decaying coefficients | 23

Theorem 5.6. Suppose that 𝒜𝒳 = 𝒜0 , ℬ𝒳 = ℬ0 , ∀𝒳 ∈ ℝ2n , and let P ∈ S0 (ℝ2n ; ℒ(𝒜0 , ℬ0 )), i. e., 󵄩󵄩 α 󵄩󵄩 󵄩󵄩𝜕𝒳 P 󵄩󵄩ℒ(𝒜0 ;ℬ0 ) ≤ Cα ,

∀𝒳 ∈ ℝ2n .

2 n 2 n The operator Opw h (p) is uniformly bounded from L (ℝ ; 𝒜0 ) into L (ℝ ; ℬ0 ).

Now let 𝒞X be a third family as 𝒜𝒳 and ℬ𝒳 . Theorem 5.7. If P ∈ S0 (ℝ2n ; ℒ(ℬ𝒳 , 𝒞𝒳 )) and Q ∈ S0 (ℝ2n ; ℒ(𝒜𝒳 , ℬ𝒳 )), then there exists R ∈ S0 (ℝ2n ; ℒ(𝒜𝒳 , 𝒞𝒳 )) such that w w Opw h (P) ∘ Oph (Q) = Oph (R),

with R = exp (

ih 󵄨 σ(Dx , Dξ ; Dy , Dη ))(P(x, ξ )Q(y, η))󵄨󵄨󵄨 . 2 x=y,ξ =η

Here σ is the standard symplectic 2-form. In particular: ∞

k

1 ih 󵄨 ( σ(Dx , Dξ ; Dy , Dη )) P(x, ξ )Q(y, η)󵄨󵄨󵄨x=y,ξ =η . k! 2 k=0

R∼ ∑

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Hannes Gernandt and Carsten Trunk

Locally finite extensions and Gesztesy–Šeba realizations for the Dirac operator on a metric graph Abstract: We study extensions of direct sums of symmetric operators S = ⨁n∈ℕ Sn . In general there is no natural boundary triplet for S∗ even if there is one for every Sn∗ , n ∈ ℕ. We consider a subclass of extensions of S which can be described in terms of the boundary triplets of Sn∗ and investigate the self-adjointness, the semiboundedness from below and the discreteness of the spectrum. Sufficient conditions for these properties are obtained from recent results on weighted discrete Laplacians. The results are applied to Laplace and Dirac operators on metric graphs with point interactions at the vertices. In particular, we allow graphs with arbitrarily small edge length. Keywords: Direct sums, quantum graphs, Dirac operator, discrete Laplacian MSC 2010: Primary 47B25, 81Q35, Secondary 34B45, 34L05

1 Introduction We consider direct sum operators S = ⨁n∈ℕ Sn in a direct sum Hilbert space ℋ = ⨁n∈ℕ ℋn associated to a family of closed densely defined symmetric operators {Sn }n∈ℕ , where Sn is defined in the Hilbert space ℋn . It is easy to see that S is closed and symmetric. Furthermore, if Sn has self-adjoint extensions for all n ∈ ℕ, then also S has self-adjoint extensions. The direct sum operator S can be viewed as a diagonal operator matrix with infinitely many entries. Its self-adjoint extensions are no longer diagonal. Here we are interested in the spectrum and related properties. Setting ℋn = {0} for all but two or three entries, we end up with a 2 × 2 (3 × 3, respectively) operator matrix, see the books [30] and [14]. For the description of the extensions of closed symmetric operators and their spectral properties, we use boundary triplets and their associated Weyl functions, see [8, 9, 13, 18]. A boundary triplet {𝒢 , Γ0 , Γ1 } consists of a Hilbert space 𝒢 and a surjection (Γ0 , Γ1 )T : dom S∗ → 𝒢 × 𝒢 that satisfies an abstract Green identity, cf. (3.1) below. Here the closed extensions of S correspond one-to-one to the closed linear subspaces Θ ⊆ 𝒢 × 𝒢 and the extension of S is given by Hannes Gernandt, Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany, e-mail: [email protected] Carsten Trunk, Institut für Mathematik, TU Ilmenau, Postfach 100565, D-98684 Ilmenau, Germany, e-mail: [email protected] https://doi.org/10.1515/9783110598193-002

26 | H. Gernandt and C. Trunk SΘ := {f ∈ dom S∗ | (Γ0 f , Γ1 f ) ∈ Θ}.

(1.1)

In order to apply this approach to quantum graphs, we will write the extension (1.1) of S in a different, more suitable way: given a closed subspace 𝒢op of 𝒢 and a closed operator L with dom L ⊆ 𝒢op , a specific closed extension of S is given by SL = {f ∈ dom S∗ | LΓ0 f = P𝒢op Γ1 f , Γ0 f ∈ 𝒢op }.

(1.2)

To illustrate the above abstract concept, we will briefly show how (1.2) looks like for a δ-type point interaction on a graph G with countable sets of vertices V and edges E and with the edge length function ℓ : E → (0, ∞). Consider ℋ = L2 (G) = ⨁e∈E L2 (0, ℓ(e)) with the operator S = ⨁ Se , e∈E

dom Se = W02,2 (0, ℓ(e)),

Se := −

d2 , dxe2

(1.3)

where Se is the minimal operator on the edge e associated with the differential expression above and W02,2 (0, ℓ(e)) denotes the usual second order Sobolev space with boundary values equal to zero. The operator S in (1.3) is symmetric with the adjoint S∗ defined on W 2,2 (G) := ⨁e∈E W 2,2 (0, ℓ(e)). A point interaction of δ-type on a graph is an extension Hα of S. It is introduced for finite graphs in [3, 4] and for infinite graphs in [10]. The domain of Hα can be specified with a real-valued sequence (α(v))v∈V by dom Hα := {(ψe )e∈E ∈ W 2,2 (G) ∩ C(G) | ∑ sgn(e, t)ψ󸀠e (tℓ(e)) = α(v)ψ(v), v ∈ V},

(1.4)

(e,t)∈Iv

where C(G) is the set of continuous functions on G viewed as a metric space, ψ(v) is the evaluation of ψ at the vertex v and Iv is the set of pairs (e, t) with e ∈ E, t = 0, 1. We have (e, 0) ∈ Iv if v is an initial vertex of the directed edge e and in this case we set sgn(e, 0) := 1. Furthermore, we have (e, 1) ∈ Iv if v is a terminal vertex of the directed edge e and we set sgn(e, 1) := −1. We show how (1.4) can be written in the form (1.2). First, we need a boundary triplet for S∗ . It is well known [28, Example 15.3] that a boundary triplet {𝒢e , Γe0 , Γe1 } for Se∗ is given by 2

𝒢e := ℂ ,

Γ(e) 0 ψe := (

ψe (0+)

), ψe (ℓ(e)−)

Γ(e) 1 ψe := (

ψ󸀠e (0+) −ψ󸀠e (ℓ(e)−)

).

(1.5)

If 0 < infe∈E ℓ(e) < supe∈E ℓ(e) < ∞, then it follows from [19] that a boundary triplet for S∗ is given by the direct sum of the triplets (1.5), (e) {𝒢 , Γ0 , Γ1 } := {⨁ 𝒢e , ⨁ Γ(e) 0 , ⨁ Γ1 }. e∈E

e∈E

e∈E

(1.6)

Locally finite extensions and infinite metric graphs | 27

Each entry of an element of 𝒢 corresponds to a vertex of the decoupled graph, i. e., the elements of 𝒢 are sequences (x(e,t) )(e,t)∈I with I := E × {0, 1}. For ψ ∈ W 2,2 (G), we write Γ0 ψ := (Γ0(e,t) ψ)(e,t)∈I = (ψe (tℓ(e)))(e,t)∈I ,

Γ1 ψ := (Γ1(e,t) ψ)(e,t)∈I = (sgn(e, t)ψ󸀠e (tℓ(e)))(e,t)∈I .

Using this boundary triplet, the condition ψ ∈ C(G) in (1.4) is equivalent to (Γ(e,t) 0 ψe )(e,t)∈I ∈ 𝒢v := span{1v }, v

1v := (1, . . . , 1) ∈ ℂ|Iv | ,

for all v ∈ V. Let deg v := |Iv | be the degree of v ∈ V. Here and in the following we make the (crucial) assumption that the graphs are locally finite, i. e., deg v < ∞

for all v ∈ V.

The expressions in the equality in (1.4) are equivalent to 1 ((Γ(e,t) ψe )(e,t)∈I , 1v )ℂdeg v 1v v ‖1v ‖2 1 1 = ∑ sgn(e, t)ψ󸀠e (tℓ(e)) ⋅ 1v deg v (e,t)∈I v α(v) (e,t) = (Γ ψe )(e,t)∈I v deg v 0

P𝒢v (Γ1(e,t) ψe )(e,t)∈I = v

Let ιv be the natural embedding of elements of 𝒢v in the sequence space 𝒢 . For the operator L := ⨁ Lv v∈V

with Lv ιv 1v :=

α(v) ι 1 , deg v v v

dom Lv = ιv span{1v }

on 𝒢𝒱 := ⨁v∈V ιv 𝒢v , we have SL = Hα in the case 0 < infe∈E ℓ(e) < supe∈E ℓ(e) < ∞. In Proposition 3.1, we show that the extension SL of S is self-adjoint, semibounded from below and has discrete spectrum if and only if the operator L has this property. In our point interaction example, the operator L is just an infinite diagonal matrix, therefore the above mentioned spectral properties translate easily to Hα , see [10]. If infe∈E ℓ(e) = 0, then there is no natural candidate for a boundary triplet associated to S∗ since the operators in (1.6) are in general not defined on dom S∗ . However, it was shown in [19] that the triplet (1.6) is a so-called boundary relation in the sense of [7]. To obtain a boundary triplet for S∗ from (1.6), a regularization technique has been applied in [5, 10, 19, 23, 24]. Here we apply in Theorem 4.1 below the technique from [5] for operators where there exists λ0 ∈ ℝ and ε > 0 such that (λ0 − ε, λ0 +

28 | H. Gernandt and C. Trunk ∞ ̃ (n) ∗ ̃(n) ). Then a (regularized) boundary triplet {𝒢̃, ⨁∞ ε) ∈ ⋂∞ n=0 Γ0 , ⨁n=0 Γ1 } n=0 ρ(Sn |ker Γ(n) 0 is given by

𝒢̃ := 𝒢 ,

√󵄩󵄩󵄩Mn󸀠 (λ0 )󵄩󵄩󵄩󵄩Γ(n) Γ̃(n) 0 := 󵄩 0 ,

Γ̃(n) 1 :=

(n) Γ(n) 1 − Mn (λ0 )Γ0

√‖Mn󸀠 (λ0 )‖

,

(1.7)

(n) where Mn is the Weyl function of the boundary triplet {𝒢n , Γ(n) 0 , Γ1 }. Again, one can ̃ (now with respect to the regularized represent extensions of S in terms of an operator L ̃ ̃ triplet {𝒢 , Γ0 , Γ1 } from (1.7)) in the form of (1.2),

̃ Γ̃0 f = P ̃ Γ̃1 f , Γ̃0 f ∈ 𝒢̃op }, SL̃ = {f ∈ dom S∗ | L 𝒢 op

(1.8)

̃ is defined on some subspace 𝒢̃op of 𝒢 . Whereas in the example above the where L ̃ has a operator L is just an (infinite) diagonal operator, now, in general, the operator L more complex structure. ̃ from above, that describes the extensions with respect to the reguThe operator L larized boundary mappings, is studied in [5, 10, 19]. In [19] Schrödinger operators with point interactions on the real line are considered. In this case, roughly speaking, the operator L in (1.8) for a point interaction if (1.6) is a boundary triplet, is a diagonal op̃ is a Jacobi operator and therefore a correspondence of erator, whereas the operator L extensions describing such interactions and Jacobi operators is made in [19]. In particular, criteria for self-adjointness, semiboundedness from below, and discreteness of the spectrum are obtained from corresponding criteria for Jacobi operators. Later, in [5] the ideas of [19] were extended to the case of Dirac operators with point interactions on the real line, so-called Gesztesy–Šeba realizations, see [12]. Recently, in [10] the regularization is applied to quantum graphs and Laplacians with point inter̃ in (1.8) is a discrete Laplacians on a actions are studied. In this case, the operator L weighted ℓ2 -space, see [16, 17, 11] and the references therein. Here we consider a more general class of extensions of symmetric direct sum oploc erators S = ⨁∞ n=0 Sn , namely locally finite extensions SL . It turns out that the operator ̃ from above is also a weighted discrete Laplacian. The locally finite extensions of S L are such that they extend the quantum graph examples to more general structures. In particular, the symmetric operators Sn may have an arbitrary, but finite defect indices. We study properties of the extensions SLloc like self-adjointness, semiboundedness from below, and discreteness of the spectrum in terms of the associated weighted dis̃ to the extension Sloc . We show that self-adjointness, semiboundedcrete Laplacian L L ̃ implies the same property for ness from below, and discreteness of the spectrum of L loc loc SL . Sufficient conditions for such properties for SL were obtained recently in [11, 17]. In the case where (1.6) is not a boundary triplet, some recent approaches [26, 29] without using the regularization technique lead to a parametrization of the selfadjoint extensions of S, but without explicit criteria for the above mentioned properties (like self-adjointness, semiboundedness from below, and discreteness of the spectrum).

Locally finite extensions and infinite metric graphs | 29

Moreover, the boundary triplet approach to quantum graphs was previously applied in numerous works, see, e. g., [2, 10, 20, 21, 25, 27]. In [2, 20, 21] finite graphs are considered. Graphs with an infinite number of edges but with a finite vertex degree were considered in [25], under the assumption that ℓ(e) = 1 for all e ∈ E, and assuming that infe∈E ℓ(e) > 0 in [27]. The study of the operators SL was carried out in [2] for star-graphs and for quantum graphs satisfying infe∈E ℓ(e) > 0 in [22]. Here the locally finite extensions SLloc provide a natural generalization to describe self-adjoint extensions for quantum graphs with infe∈E ℓ(e) = 0. The paper is organized as follows: First, we recall linear relations in a Hilbert space and boundary triplets. From the boundary triplet theory, we collect some results on the properties of the extension SL given by (1.2) which can be described terms of the operator L and the Weyl function of an underlying boundary triplet for S∗ . In Section 4 we introduce locally finite extension SLloc and construct an associated discrete Laplacian DL such that, roughly speaking, SLloc = SDL holds in the sense of (1.8) ̃ = DL . From this relation, we obtain conditions for the self-adjointness, lower with L semiboundedness, and discreteness of the spectrum of SLloc . These conditions only depend on the matrices Lv , the subspaces 𝒢v , and the decoupled Weyl functions Mn . In Section 5 we apply our results on locally finite extensions to point interactions on infinite metric graphs with the negative Laplacian on each edge. Finally, as a second application, we consider in Section 6 infinite metric graphs with Dirac operators on the edges and point interactions at the vertices.

2 Linear relations in Hilbert spaces Let (ℋ, (⋅, ⋅)ℋ ) be a separable Hilbert space. A (closed) linear relation in ℋ is a (closed) subspace of ℋ × ℋ and the set of all closed linear relations in ℋ is denoted by 𝒞̃(ℋ). For a linear operator T defined in ℋ with values in ℋ, the graph of T is a linear relation in ℋ. The set of all closed linear operators in ℋ is denoted by 𝒞 (ℋ). For the subspace of bounded linear operators defined on ℋ, we write ℒ(ℋ). The domain, range, kernel, multivalued part, and inverse of a linear relation Θ in ℋ are respectively given by dom Θ := {f ∈ ℋ | (f , f 󸀠 ) ∈ Θ for some f 󸀠 ∈ ℋ}, ran Θ := {f 󸀠 ∈ ℋ | (f , f 󸀠 ) ∈ Θ for some f ∈ ℋ}, ker Θ := {f ∈ ℋ | (f , 0) ∈ Θ}, mul Θ := {f 󸀠 ∈ ℋ | (0, f 󸀠 ) ∈ Θ}, Θ−1 := {(f 󸀠 , f ) ∈ ℋ2 | (f , f 󸀠 ) ∈ Θ}. Recall that the (operator-like) sum of two linear relations Θ1 and Θ2 in ℋ is given by Θ1 + Θ2 := {(f , f1󸀠 + f2󸀠 ) ∈ ℋ × ℋ | (f , f1󸀠 ) ∈ Θ1 , (f , f2󸀠 ) ∈ Θ2 }.

30 | H. Gernandt and C. Trunk Let Θ be a closed linear relation in ℋ. The set of all λ ∈ ℂ such that (Θ−λ)−1 is the graph of an operator from ℒ(ℋ) is called the resolvent set ρ(Θ) of Θ. The complement of ρ(Θ) in ℂ is the spectrum σ(Θ) of Θ. The adjoint Θ∗ of a linear relation Θ in ℋ is defined as Θ∗ := {(g, g 󸀠 ) ∈ ℋ2 | (f 󸀠 , g)ℋ = (f , g 󸀠 )ℋ for all (f , f 󸀠 ) ∈ Θ}. A linear relation is called symmetric (self-adjoint) if Θ ⊆ Θ∗ (resp. Θ = Θ∗ ). For Θ ∈ 𝒞̃(ℋ), we have mul Θ = (dom Θ∗ ) , ⊥

mul Θ∗ = dom Θ⊥ .

Given a self-adjoint linear relation Θ, we can associate a self-adjoint operator on the Hilbert space dom Θ, see [1, Theorem 5.3]. Below, we present a somehow converse result. Proposition 2.1. Let ℋop be a closed subspace of a Hilbert space ℋ and consider a densely defined operator L from ℋop to ℋop . Define ⊥ ΘL := {(f , Lf + g) | f ∈ dom L, g ∈ ℋop } ⊆ ℋ × ℋ.

(2.1)

Then the following holds: (a) We have Θ∗L = ΘL∗ . If L is closable, we have ΘL = ΘL . (b) ΘL is closed (symmetric, self-adjoint) if and only if L is closed (resp. symmetric, selfadjoint). ̃ with Θ ⊆ Θ ̃ ⊆ Θ∗ are of the form Θ̃ , where (c) If L is symmetric then all extensions Θ L L L ̃ is an extension of L. L (d) If L is self-adjoint, then ρ(L) = ρ(ΘL ) and for all λ ∈ ρ(L), (L − λ)−1 0

(ΘL − λ)−1 = (

0 ⊥ ) ∈ ℒ(ℋop ⊕ ℋop ). 0

⊥ , we Proof. Let (f , g) ∈ Θ∗L . Then for all (f 󸀠 , Lf 󸀠 + g 󸀠 ) ∈ ΘL with f 󸀠 ∈ dom L and g 󸀠 ∈ ℋop have

(g, f 󸀠 ) = (f , Lf 󸀠 + g 󸀠 ). Choosing f 󸀠 = 0, we obtain f ∈ ℋop . Therefore, we conclude from (2.2) that (g, f 󸀠 ) = (f , Lf 󸀠 ) for all f 󸀠 ∈ dom L. This implies that f ∈ dom L∗ and Pℋop g = L∗ f . Hence, (f , g) = (f , Pℋop g + Pℋ⊥op g) = (f , L∗ f + Pℋ⊥op g) ∈ ΘL∗ .

(2.2)

Locally finite extensions and infinite metric graphs | 31

⊥ . Then we Assume conversely that (f , L∗ f + g) ∈ ΘL∗ with f ∈ dom L∗ and some g ∈ ℋop 󸀠 󸀠 󸀠 󸀠 󸀠 ⊥ have for all (f , Lf + g ) ∈ ΘL with f ∈ dom L and g ∈ ℋop ,

(f 󸀠 , L∗ f + g) = (f 󸀠 , L∗ f ) = (Lf 󸀠 , f ) = (Lf 󸀠 + g 󸀠 , f ) and therefore (f , L∗ f + g) ∈ Θ∗L . Thus we have seen that Θ∗L = ΘL∗ . Let (f , Lf + g) ∈ ΘL with f ∈ dom L. Then there is a sequence ((fn , Lfn ))n∈ℕ which converges in ℋ2 to (f , Lf ). But then (fn , Lfn + g) ∈ ΘL converges in ℋ2 to (f , Lf + g), as n → ∞, which implies (f , Lf + g) ∈ ΘL . Conversely, let (f , g 󸀠 ) ∈ ΘL . Then there exists a sequence (fn , Lfn + gn ) ∈ ΘL with ⊥ ⊥ ⊥ gn ∈ ℋop and fn ∈ dom L which converges to (f , g 󸀠 ). As ℋ2 = (ℋop × ℋop ) ⊕ (ℋop × ℋop ), 󸀠 󸀠 we have (fn , Lfn ) → (f , Pℋop g ) and (0, gn ) → (0, Pℋ⊥op g ), as n → ∞. Therefore f ∈ dom L with Lf = Pℋop g 󸀠 . Hence

(f , g 󸀠 ) = (f , Lf + Pℋ⊥op g 󸀠 ) ∈ ΘL . ̃ be an extension of Θ Assertion (b) is a consequence of (a). We show (c). Let Θ L ̃ with ΘL ⊆ Θ ⊆ ΘL∗ . Obviously, ̃ ⊆ dom Θ ∗ = dom L∗ . dom L = dom ΘL ⊆ dom Θ L ̃ Then L ̃ ⊂ Θ ∗ , every element (f , g 󸀠 ) ∈ Θ ̃ ̃ := L∗ | dom Θ. ̃ is an extension of L. As Θ Set L L ∗ ̃ ⊂ dom L and has a representation satisfies f ∈ dom Θ (f , g 󸀠 ) = (f , L∗ f + g) ⊥ ̃ (f , g 󸀠 ) ∈ Θ̃ follows. Hence, Θ ̃ ⊂ Θ̃ . The ̃ for f ∈ dom Θ, for some g ∈ ℋop . As L∗ f = Lf L L converse inclusion is obvious and (c) is shown. For the last statement, observe that we have for all λ ∈ ℂ, ⊥ (ΘL − λ)−1 = {((L − λ)f + g, f ) | f ∈ dom L, g ∈ ℋop }.

From this (d) follows easily.

3 Extension theory of symmetric operators with boundary triplets We review the boundary triplet theory following [8], see also [18]. Definition 3.1. For a densely defined symmetric operator A ∈ 𝒞 (ℋ) in a Hilbert space ℋ, we say that {𝒢 , Γ0 , Γ1 } is a boundary triplet for A∗ if (𝒢 , (⋅, ⋅)𝒢 ) is a Hilbert space, (Γ0 , Γ1 )⊤ : dom A∗ → 𝒢 2 is surjective, and the following abstract Green identity holds: (A∗ f , g)ℋ − (f , A∗ g)ℋ = (Γ1 f , Γ0 g)𝒢 − (Γ0 f , Γ1 g)𝒢 .

(3.1)

32 | H. Gernandt and C. Trunk Boundary triplets are a standard tool to describe all closed extensions of a given symmetric operator. For a densely defined symmetric operator A ∈ 𝒞 (ℋ), we fix a boundary triplet {𝒢 , Γ0 , Γ1 } for A∗ . The extension AΘ of A corresponding to a parameter Θ ∈ 𝒞̃(𝒢 ) is defined as dom AΘ := {f ∈ dom A∗ | (Γ0 f , Γ1 f ) ∈ Θ},

AΘ f := A∗ f .

The correspondence between the closed linear relations Θ ∈ 𝒞̃(𝒢 ) and the closed extensions AΘ of A is bijective (see, e. g., [8]). The following two special self-adjoint extensions of A will play a prominent role: A0 := A{0}×𝒢 = A∗ |ker Γ0

and A1 := A𝒢×{0} = A∗ |ker Γ1 .

In [8] a correspondence of properties between Θ ∈ 𝒞̃(𝒢 ) and AΘ ∈ 𝒞 (𝒢 ) was established using the concept of the γ-field and the Weyl function given by γ : ρ(A0 ) → ℒ(ℋ, 𝒢 ), γ(λ) := (Γ0 |𝒩λ )−1 , M : ρ(A0 ) → ℒ(𝒢 ),

𝒩λ (A) := {f ∈ dom A | A f = λf }, ∗

∗

M(λ) := Γ1 γ(λ).

Here we prefer the following description of the extensions. Let L be a densely defined operator on a subspace 𝒢op of 𝒢 mapping into 𝒢op . We consider the relation ΘL from (2.1) and the associated extension AL := AΘL and therefore dom AL = {f ∈ dom A∗ | Γ0 f ∈ dom L, LΓ0 f = Pdom L Γ1 f },

(3.2)

where Pdom L is the orthogonal projection onto 𝒢op = dom L since L is assumed to be densely defined in 𝒢op . Proposition 2.1 and some well known results on the relationship between Θ ∈ 𝒞̃(𝒢 ) and AΘ ∈ 𝒞 (ℋ) from [8, 19] lead to the next statement. Here we use the notation Sp (ℋ) with p ∈ (0, ∞] for the two sided Schatten–von Neumann ideal and we denote by n± (A) := dim 𝒩±i (A) the defect numbers of a symmetric densely defined linear operator A. Proposition 3.1. Let A be a densely defined symmetric operator in ℋ with boundary triplet {𝒢 , Γ0 , Γ1 } for A∗ and let L be a densely defined operator in a subspace 𝒢op of 𝒢 then the following holds: (a) AL is self-adjoint (symmetric) if and only if L is self-adjoint (resp. symmetric). (b) AL = AL , AL∗ = A∗L , and n± (AL ) = n± (L). (c) If L is symmetric, then there is a bijective correspondence between the extensions of L and the extensions of AL . (d) For λ ∈ ρ(A0 ), we have λ ∈ ρ(AL ) if and only if 0 ∈ ρ(ΘL − M(λ)). In this case the Krein resolvent formula holds (AL − λ)−1 − (A0 − λ)−1 = γ(λ)(ΘL − M(λ)) γ(λ)∗ . −1

Locally finite extensions and infinite metric graphs | 33

(e) Let (A0 −λ0 )−1 ∈ Sp (ℋ) for some λ0 ∈ ρ(A0 ) and p ∈ [1, ∞]. Then (AL −λ)−1 ∈ Sp (ℋ) if and only if (L − λ)−1 ∈ Sp (𝒢 ) for λ ∈ ρ(L). Let A be a densely defined symmetric operator which is semi-bounded from below, i. e., A ≥ γ for some γ ∈ ℝ. Then there is a distinguished, in some sense maximal, semibounded self-adjoint extension AF ≥ γ, which is called the Friedrichs extension of A, see, e. g., [28, Section 10.4]. Given a boundary triplet {𝒢 , Γ0 , Γ1 } of A∗ with Weyl function M such that A0 equals the Friedrichs extension AF , we use the notation M(λ) 󴁂󴀱 −∞ for λ → −∞ to indicate that for any γ > 0 there exists λγ with −M(λγ ) ≥ γ. We collect some results on nonnegative extensions from [8, 9], see also [28]. Proposition 3.2. Consider a densely defined symmetric operator A ∈ 𝒞 (ℋ), a boundary triplet {𝒢 , Γ0 , Γ1 } for A∗ with A0 = AF ≥ γ, and a self-adjoint operator L ∈ 𝒞̃(𝒢op ) on a subspace 𝒢op of 𝒢 . Then the following holds: (a) L − P𝒢op M(λ0 )|𝒢op ≥ 0 for λ0 < γ implies AL ≥ λ0 . (b) If M(λ) 󴁂󴀱 −∞ for λ → −∞ then AL is semibounded from below if and only if L is semibounded from below. In the lemma below, we describe the change of a boundary triplet {𝒢 , Γ0 , Γ1 } under unitary transformations of the space 𝒢 . Lemma 3.1. Let A ∈ 𝒞 (ℋ) be a densely defined symmetric operator with a boundary triplet {𝒢 , Γ0 , Γ1 } for A∗ and consider a unitary operator U : 𝒢 → 𝒢̂. Then {𝒢̂, UΓ0 , UΓ1 } is a boundary triplet for A∗ with Weyl function λ 󳨃→ UM(λ)U ∗ on ρ(A∗ |ker Γ0 ). Furthermore ̂ := ULU ∗ as the extension AL given by (3.2) can be written with L ̂ LUΓ ̂ 0f = P dom AL = {f ∈ dom A∗ | UΓ0 f ∈ dom L,

̂ dom L

UΓ1 f }.

Proof. Since U is unitary, the mapping f 󳨃→ (UΓ0 f , UΓ1 f ) from dom A∗ into 𝒢̂2 is onto and the abstract Green identity (3.1) holds. Hence {𝒢̂, UΓ0 , UΓ1 } is a boundary triplet for A∗ with A∗ |ker Γ0 = A∗ |ker UΓ0 and Weyl function λ 󳨃→ UM(λ)U ∗ which is defined for all λ ∈ ρ(A∗ |ker Γ0 ). Given that f ∈ dom AL , we have Γ0 f ∈ dom L and LΓ0 f = Pdom L Γ1 f , which is equivalent to UΓ0 f ∈ U dom L,

ULU ∗ UΓ0 f = UPdom L U ∗ UΓ1 f .

(3.3)

Furthermore, it is easy to see that ̂ = dom ULU ∗ = dom LU ∗ = U dom L. dom L

(3.4)

Moreover, UPdom L U ∗ is an orthogonal projection satisfying UPdom L U ∗ = PUdom L = PU dom L = P

̂ dom L

.

Rewriting (3.3) with (3.4) and (3.5) completes the proof of the lemma.

(3.5)

34 | H. Gernandt and C. Trunk

4 Locally finite extensions of direct sums of symmetric operators In this section, we introduce direct sum operators and their locally finite extensions. Throughout this section we consider a family of Hilbert spaces {ℋn }n∈ℕ with inner product (⋅, ⋅)ℋn and densely defined symmetric operators Sn ∈ 𝒞 (ℋn ) with boundary (n) ∗ triplets {𝒢n , Γ(n) 0 , Γ1 } for Sn such that dim 𝒢n < ∞, n ∈ ℕ. We introduce the direct sum Hilbert space ℋ, ∞

ℋ := ⨁ ℋn := {x = (xn )n∈ℕ : xn ∈ ℋn , (x, x)ℋ < ∞} n=0

with inner product ∞

((xn )n∈ℕ , (yn )n∈ℕ )ℋ := ∑ (xn , yn )ℋn . n=0

Acting on ℋ, we introduce the direct sum operator S := ⨁∞ n=0 Sn via ∞

󵄨 dom S := {(fn )n∈ℕ 󵄨󵄨󵄨 fn ∈ dom Sn , ∑ ‖Sn fn ‖2ℋn < ∞}, n=0

S(fn )n∈ℕ := (Sn fn )n∈ℕ . The case of a finite-dimensional direct sum Hilbert space ℋ is obtained by setting ℋn := {0} and Sn := 0 for all n ≥ N and some N ∈ ℕ. It is easy to see that S is densely defined, closed, and with the adjoint ∞

∗

∞

(⨁ Sn ) = ⨁ Sn∗ . n=0

n=0

Since Sn ⊆ Sn∗ for all n ∈ ℕ, it is easy to see that S is symmetric with n± (S) = ∑∞ n=0 n± (Sn ). To describe the extensions of S, the natural candidate for a boundary triplet for S∗ is given by 𝒢 := ⨁∞ n=0 𝒢n with the boundary mappings Γi , i = 1, 2, ∞

󵄩󵄩 (n) 󵄩󵄩2 dom Γi := {(fn )n∈ℕ : fn ∈ dom Γ(n) 󵄩Γi fn 󵄩󵄩 < ∞}, i , ∑󵄩 n=0

Γi (fn )n∈ℕ :=

(Γ(n) i fn )n∈ℕ ,

which can also be written in the form ∞

𝒢 := ⨁ 𝒢n , n=0

∞

Γ0 := ⨁ Γ(n) 0 , n=0

∞

Γ1 := ⨁ Γ(n) 1 . n=0

(4.1)

Locally finite extensions and infinite metric graphs | 35

In general, the operators Γ0 and Γ1 are only defined on a subspace of dom S∗ such that (4.1) is not a boundary triplet for S∗ . However, it was shown in [19] that the triplet {𝒢 , Γ0 , Γ1 } given by (4.1) forms a single-valued boundary relation in the sense of [7]. We use a particular regularization from [5] for the direct sum triplet (4.1) for operators with a common real point in the resolvent set, i. e., we assume that for Sn0 := Sn∗ |ker Γ(n) there exist λ0 ∈ ℝ and ε > 0 such that (λ0 − ε, λ0 + ε) ⊆ ⋂n∈ℕ ρ(Sn0 ). In the 0

theorem below we use [5, Theorem 2.12] to provide a boundary triplet for the direct sum operator S∗ . Theorem 4.1. Let {Sn }n∈ℕ be a family of densely defined symmetric linear operators Sn ∈

(n) ∗ 𝒞 (ℋn ) with boundary triplets {𝒢n , Γ(n) 0 , Γ1 } for Sn and Weyl functions Mn and (λ0 − ε, λ0 + ∞ ̃ (n) ∞ ̃ (n) ∞ ε) ⊆ ⋂n=0 ρ(Sn0 ) for some ε > 0 and λ0 ∈ ℝ. Then {⨁∞ n=0 𝒢n , ⨁n=0 Γ0 , ⨁n=0 Γ1 } with −1

√󵄩󵄩󵄩Mn󸀠 (λ0 )󵄩󵄩󵄩󵄩Γ(n) Γ̃(n) 0 := 󵄩 0 ,

(n) √󵄩󵄩󵄩Mn󸀠 (λ0 )󵄩󵄩󵄩󵄩 (Γ(n) Γ̃(n) 1 := 󵄩 1 − Mn (λ0 )Γ0 )

(4.2)

∗ ̃ is a boundary triplet for S∗ = ⨁∞ n=0 Sn . The Weyl function M of this triplet is given by ∞ ∗ ̃n (λ) with ̃ : ρ(S |ker Γ ) → ℒ(𝒢 ), λ 󳨃→ ⨁ M M n=0

0

̃n := M

fore

1

‖Mn󸀠 (λ0 )‖

(Mn − Mn (λ0 )).

(4.3)

The construction of this regularization implies that Sn∗ |ker Γ(n) = Sn∗ |ker Γ̃(n) and there0

∞

S∗ |ker Γ̃ = ⨁ Sn0 . 0

n=0

0

(4.4)

The remainder of this section is devoted to locally finite extensions. We assume that the Hilbert space 𝒢 is given as the direct sum Hilbert space ∞

∞

n=0

n=0

𝒢 = ⨁ 𝒢n = ⨁ ℂ

dn

with dn < ∞ and 𝒢n = ℂdn .

The elements of 𝒢 are sequences of the form x = (xi )i∈I , where I := {(n, d) | n ∈ ℕ, d = 1, . . . , dn }. In the following we will consider a partition of I into subsets Iv , where v is an element of a countable index set V such that the following conditions are fulfilled: (i) |Iv | < ∞; (ii) Iv ∩ Iw = 0 for all v, w ∈ V with v ≠ w; (iii) ⋃v∈V Iv = I.

36 | H. Gernandt and C. Trunk dn Since Γ(n) f ∈ ℂdn , the sequence (Γ(n) f ) , i = 0, 1 is an element of ×∞ n=0 ℂ , but i n i n n∈ℕ ∞ dn not necessarily of the Hilbert space ⨁n=0 ℂ . Thus, we can view it as sequence (Γ(n,d) fn )(n,d)∈I , where i

Γ(n,d) fn := (Γ(n) i i fn )d ,

1 ≤ d ≤ dn ,

i = 0, 1

is the dth entry of Γ(n) f . With this we introduce for f ∈ dom S∗ , i n Γvi f := (Γ(n,t) fn )(n,t)∈I , i v

i = 0, 1.

Before we continue with the definition of locally finite extensions, we illustrate the definitions from above with the quantum graph example from the introduction. Example 4.1. Consider the densely defined symmetric operators S1 , . . . , SN with domains dom Sn := W02,2 (0, ℓ(en )), n = 1, . . . , N with Sn ψn := −ψ󸀠󸀠 n . Then a boundary triplet for Sn∗ with n = 1, . . . , N is given by {ℂ2 , (ψn (0+), ψn (ℓ(en )−)) , (ψ󸀠n (0+), −ψ󸀠n (ℓ(en )−)) }. ⊤

⊤

Hence dn = 2 for all n and therefore I = {1, . . . , N} × {1, 2}. Consider the index set V = {v1 , . . . , vN+1 }. We introduce Ivi := {(i, 1)} for i = 1, . . . , N and IvN+1 := {(i, 2) : i = 1, . . . , N}. It is easy to see that the conditions (i)–(iii) from above are satisfied. For each index i = 1, . . . , N + 1, there is an edge associated with it and the sets Ivi describe which edges are glued together at the vertex vi , which leads to a graph. In this simple example, all vertices vi , i = 1, . . . , N correspond to singleton sets Ivi , i. e., only one vertex leads to vi , whereas in vN+1 we have N vertices. Hence the underlying graph is a star-graph with N + 1 vertices and N edges. Furthermore, we have v

v

Γ0n (ψj )Nj=1 = ψn (0+), Γ1 n (ψj )Nj=1 = ψ󸀠n (0+),

v Γ0N+1 (ψj )Nj=1 v Γ1 N+1 (ψj )Nj=1

n = 1, . . . , N,

= (ψ1 (ℓ(e1 )−), ψ2 (ℓ(e2 )−), . . . , ψN (ℓ(eN )−)) , ⊤

⊤

= (−ψ󸀠1 (ℓ(e1 )−), −ψ󸀠2 (ℓ(e2 )−), . . . , −ψ󸀠N (ℓ(eN )−)) .

Obviously, one easily can construct examples with infinitely many vertices and edges. Observe, as we only consider locally finite extensions, that |Iv | < ∞ always holds, which means, in the cases of graph-like constructions, that on each edge there are only finitely many vertices. Example 4.2. Here we give an example for a star-graph with finite edges and vertices but with infinite edge length. Consider the densely defined symmetric operators S1 , . . . , SN from Example 4.1 and, in addition, dom SN+1 := W02,2 (0, ∞) with ∗ SN+1 ψN+1 := −ψ󸀠󸀠 N+1 . Then a boundary triplet for Sn with n = 1, . . . , N is given as in

Locally finite extensions and infinite metric graphs | 37

∗ Example 4.1 and a triplet for SN+1 is given by {ℂ, ψN+1 (0+), ψ󸀠N+1 (0+)}, see, e. g., [28, Example 15.5]. Hence dn = 2 for all n = 1, . . . , N but dN+1 = 1, and therefore

I = ({1, . . . , N} × {1, 2}) ∪ {(N + 1, 1)}. Consider the index set V = {v1 , . . . , vN+1 }. We introduce Ivi := {(i, 1)} for i = 1, . . . , N, IvN+1 := {(i, 2) : i = 1, . . . , N} ∪ {(N + 1, 1)}. As above we have a star-graph, but with one vertex less, as the edge corresponding to N + 1 is a semiaxis, v

v

N+1 󸀠 n Γ0n (ψj )N+1 j=1 = ψn (0+), Γ1 (ψj )j=1 = ψn (0+),

v Γ0N+1 (ψj )N+1 j=1 vN+1 Γ1 (ψj )N+1 j=1

n = 1, . . . , N,

= (ψ1 (ℓ(e1 )−), ψ2 (ℓ(e2 )−), . . . , ψN (ℓ(eN )−), ψN+1 (0+)) , ⊤

= (−ψ󸀠1 (ℓ(e1 )−), −ψ󸀠2 (ℓ(e2 )−), . . . , −ψ󸀠N (ℓ(eN )−), ψ󸀠N+1 (0+)) . ⊤

Similarly, one can construct graphs with infinitely many vertices and edges. Moreover, we stress that we are able to allow dn > 2, leading to structures which no longer allow an interpretation as a graph. Now let 𝒢v be a subspace of ℂ|Iv | and consider the Hermitian matrix Lv : 𝒢v → 𝒢v . We introduce the locally finite extension SLloc of S by dom SLloc := {f ∈ dom S∗ | Lv Γv0 f = P𝒢v Γv1 f , Γv0 f ∈ 𝒢v , v ∈ V}, SLloc f := S∗ f .

(4.5)

It is shown in Proposition 4.1 below that SLloc is the adjoint of the operator SLmin ⊆ S∗ with dom SLmin := {f ∈ dom SLloc | supp(Γv0 f )v∈V , supp(P𝒢v Γv1 f )v∈V finite}, where we used the support of a sequence x = (xi )i∈I ∈ ℂI given by supp x := {i ∈ I | xi ≠ 0}. For its proof we need a variant of the abstract Green identity (3.1). Lemma 4.1. If f , g ∈ dom S∗ then (S∗ f , g) − (f , S∗ g) = ∑ (Γv1 f , Γv0 g) − (Γv0 f , Γv1 g). v∈V

(4.6)

Furthermore, given v ∈ V, y0 ∈ 𝒢v , and y1 ∈ 𝒢v⊥ , there exists g = (gn )n∈ℕ ∈ dom SLmin with finite support such that the following equations hold: Γv0 g = y0 ,

Γv1 g = y1 + Lv y0 ,

w Γw 0 g = Γ1 g = 0,

for all w ∈ V \ {v}.

(4.7)

38 | H. Gernandt and C. Trunk Proof. First, we show that for all f = (fn )n∈ℕ , g = (gn )n∈ℕ ∈ dom S∗ , the sum ∗ ∑∞ n=0 (Sn fn , gn ) converges absolutely. From Cauchy–Bunyakowsky and Hölder inequality, we have ∞

∞

n=0

n=0

󵄩 󵄩 󵄩 󵄩 󵄨 󵄨 ∑ 󵄨󵄨󵄨(Sn∗ fn , gn )󵄨󵄨󵄨 ≤ ∑ 󵄩󵄩󵄩Sn∗ fn 󵄩󵄩󵄩‖gn ‖ ≤ 󵄩󵄩󵄩S∗ f 󵄩󵄩󵄩‖g‖ < ∞.

Next, using the abstract Green identity (3.1) for the operators Sn∗ and changing the order of summation leads to ∞

(S∗ f , g) − (f , S∗ g) = ∑ (Sn∗ fn , gn ) − (fn , Sn∗ gn ) n=0 ∞

(n) (n) (n) = ∑ (Γ(n) 1 fn , Γ0 gn ) − (Γ0 fn , Γ1 gn ) n=0

= ∑ (Γv1 f , Γv0 g) − (Γv0 f , Γv1 g), v∈V

where the last equality follows from ⋃v∈V Iv = I. For the proof of the second assertion, we construct g = (gn )n∈ℕ ∈ dom SLmin satisfying the equations (4.7). Consider n ∈ ℕ and the set Iv . Given that (n, d) ∉ Iv for all d = 1, . . . , dn , we set gn := 0. For (n, d) ∈ Iv , for some d = 1, . . . , dn , the surjectivity of (n) ⊤ ∗ (Γ(n) 0 , Γ1 ) : dom Sn → 𝒢n × 𝒢n for all n ∈ ℕ implies that we can choose gn such that the first and second equation in (4.7) hold. From the construction we also have that the lower system of equations in (4.7) hold. Next, we show that SLloc is the adjoint of SLmin . Proposition 4.1. We have SLloc = (SLmin )∗ , in particular SLloc is closed. Proof. If f ∈ (SLmin )∗ then we have from (4.6) for all g ∈ dom SLmin , 0 = (S∗ f , g) − (f , S∗ g) = ∑ (Γv1 f , Γv0 g) − (Γv0 f , Γv1 g). v∈V

(4.8)

For this equation we use (4.7) from Lemma 4.1 with y0 = 0 and y1 ∈ 𝒢v⊥ , which leads to (Γv0 f , y1 ) = 0. Since y1 was arbitrary, we conclude that Γv0 f ∈ 𝒢v for all v ∈ V. Choose g ∈ dom SLmin that solves (4.7) for y1 = 0 and arbitrary y0 ∈ 𝒢v . With (4.8) this leads to 0 = (Γv1 f , y0 ) − (Γv0 f , Lv y0 ) = (P𝒢v Γv1 f − Lv Γv0 f , y0 ). Since y0 ∈ 𝒢v was arbitrary, we see P𝒢v Γv1 f = Lv Γv0 for all v ∈ V, proving f ∈ dom SLloc .

Locally finite extensions and infinite metric graphs | 39

Assume conversely that f ∈ dom SLloc . For all g ∈ dom SLmin , we have ∑ (Γv1 f , Γv0 g) − (Γv0 f , Γv1 g) = ∑ (P𝒢v Γv1 f , Γv0 g) − (Γv0 f , P𝒢v Γv1 g)

v∈V

v∈V

= ∑ (Lv Γv0 f , Γv0 g) − (Γv0 f , Lv Γv0 g) v∈V

= 0, which, together with (4.6), implies f ∈ dom(SLmin )∗ . We prove the main theorem of this section that allows us to describe the extension ̂ for a countable index set V. ̂ For this we use the notation SLloc with operators on ℓ2 (V) ̂ := {(fv ) ̂ ∈ ℓ2 (V) ̂ | supp f finite}. C(V) v∈V Furthermore, for the subspaces 𝒢v of ℂ|Iv | we use the canonical embedding ιv : 𝒢v → ⨁ℂdn , n∈ℕ

(x(n,d) )(n,d)∈Iv 󳨃→ (y(n,d) )(n,d)∈I ,

x(n,d) , if (n, d) ∈ Iv , y(n,d) := { 0, otherwise. Therefore ιv (𝒢v ) is a subspace of 𝒢 , and we have an orthogonal sum decomposition 𝒢𝒱 := ⨁ ιv 𝒢v .

(4.9)

v∈V

In the following, we consider an orthogonal basis {bw }w∈V̂ of the subspace 𝒢𝒱 , which has the property that each bw is an element of an orthogonal basis for some 𝒢v and ̂ is a countable set of indices. In the theorem below we will make use of the unitary V ̂ given by bw 󳨃→ ‖bw ‖ew . operator U : 𝒢𝒱 → ℓ2 (V) Theorem 4.2. Let {Sn }n∈ℕ be a family of densely defined symmetric linear operators Sn ∈ (n) ∗ 𝒞 (ℋn ) with boundary triplets {𝒢n , Γ(n) 0 , Γ1 } for Sn and Weyl functions Mn and (λ0 − ε, λ0 + loc ε) ⊆ ⋂∞ n=0 ρ(Sn0 ) for some ε > 0 and λ0 ∈ ℝ. Consider SL with Hermitian matrices Lv , subspaces 𝒢v , 𝒢𝒱 given by (4.9) with orthogonal basis {bw }w∈V̂ and the operator L = ⨁v∈V Lv on 𝒢𝒱 . Then the following holds: ̂ with dom Lmin = C(V) ̂ given as an infinite matrix opera(a) The operator Lmin in ℓ2 (V) tor, Lmin := (

((L − ⨁n∈ℕ Mn (λ0 ))bv , bw ) ) , ‖Rbv ‖‖Rbw ‖ ̂ v,w∈V

∞

󵄩 󵄩 R := ⨁ √󵄩󵄩󵄩Mn󸀠 (λ0 )󵄩󵄩󵄩Iℂdn , n=0

̂ satisfies Smin = SL and Sloc = SL∗ . with dom R := U −1 C(V), L L min min

(b) We have n± (SLmin ) = n± (Lmin ) and there is a bijective correspondence between the self-adjoint extensions of Lmin and the self-adjoint extensions of SLmin .

40 | H. Gernandt and C. Trunk ̃ (c) Assume that ⨁∞ n=0 Sn0 = SF ≥ γ with γ > 0 and that M given by (4.3) satisfies ̃ ̃ be a self-adjoint extension of Lmin which is semiM(λ) 󴁂󴀱 −∞ for λ → −∞. Let L bounded from below then SL̃ is semibounded from below. Proof. For the proof of (a), we use the regularized boundary triplet {𝒢 , Γ̃0 , Γ̃1 } defined in (4.2) for f = (fn )n∈ℕ ∈ dom S∗ as 󵄩 󵄩 Γ̃0 f = (√󵄩󵄩󵄩Mn󸀠 (λ0 )󵄩󵄩󵄩Γ(n) 0 fn )n∈ℕ ,

󵄩 󵄩−1/2 (n) Γ̃1 f = (󵄩󵄩󵄩Mn󸀠 (λ0 )󵄩󵄩󵄩 (Γ(n) 1 − Mn (λ0 )Γ0 )fn )n∈ℕ .

−1 ̂ ̃ Consider M := ⨁∞ n=0 Mn (λ0 ) with dom M := U C(V) and let Lmin be given by

̃ min := RU −1 C(V), ̂ dom L

−1 −1 ̃ min f := P L ran R R (L − M)R f .

We show that −1 ̂ ̃ min Γ̃0 f = P ̃ ̃ dom SLmin = {f ∈ dom S∗ | L ran R Γ1 f , Γ0 f ∈ RU C(V),

supp(P𝒢v Γv1 )v∈V finite}.

(4.10)

Let f ∈ dom S∗ be in the set on the right-hand side of (4.10). Obviously, supp Γ0 f is finite and, rewriting the conditions on the right-hand side of (4.10), we obtain Pran R R−1 (L − M)R−1 R(Γ(n) 0 fn )n∈ℕ ̃ ̃ ̃ Γ1 f = Lmin Γ0 f = P ran R

󵄩 󵄩−1/2 (n) = Pran R (󵄩󵄩󵄩Mn (λ0 )󵄩󵄩󵄩 (Γ(n) 1 − Mn (λ0 )Γ0 )fn )n∈ℕ ,

and therefore 󵄩󵄩 󵄩󵄩−1/2 (n) Pran R R−1 L(Γ(n) 󵄩Mn (λ0 )󵄩󵄩 Γ1 fn )n∈ℕ . 0 fn )n∈ℕ = Pran R (󵄩

(4.11)

̃ Note that (‖Mn (λ0 )‖−1/2 Γ(n) 1 fn )n∈ℕ ∈ 𝒢 , since Γ1 f ∈ 𝒢 and supp Γ0 f is finite. The definition of R implies that {Rbw }w∈V̂ is an orthogonal basis of ran R. Furthermore, we have ̂ from (4.11) that for all w ∈ V, 󵄩󵄩 󵄩󵄩−1/2 (n) (Pran R R−1 L(Γ(n) 󵄩Mn (λ0 )󵄩󵄩 Γ1 fn )n∈ℕ , Rbw ), 0 fn )n∈ℕ , Rbw ) = (Pran R (󵄩 which is equivalent to (n) v ((Lv Γv0 f )v∈V , bw ) = (L(Γ(n) 0 fn )n∈ℕ , bw ) = ((Γ1 fn )n∈ℕ , bw ) = (P𝒢v (Γ1 f )v∈V , bw )

̂ Note that (Γ(n) fn )n∈ℕ and L(Γ(n) fn )n∈ℕ are in general not in 𝒢 , but the for all w ∈ V. 1 0 formal scalar product of these sequences with bv exists because the support of bv is

Locally finite extensions and infinite metric graphs | 41

finite. Since for each v ∈ V there exists a subset of {bw }w∈V̂ which is an orthogonal basis for 𝒢v , we see that Lv Γv0 f = P𝒢v Γv1 f for all v ∈ V and all f in the set of the right-hand side of (4.10). Moreover, Γ̃0 f ∈ ̂ hence Γ0 f ∈ U −1 C(V) ̂ and, by construction, Γv f ∈ 𝒢v follows. Thus we have RU −1 C(V), 0 proven that f ∈ dom SLmin . Assume conversely that f ∈ dom SLmin . Then we have that for finitely many v ∈ V, Lv Γv0 f = P𝒢v Γv1 ,

Γv0 f ∈ 𝒢v ,

̂ and supp(P𝒢 Γv f )v∈V is and Γv0 f = P𝒢v Γv1 f = 0 otherwise. Obviously, Γ̃0 f ∈ RU −1 C(V) v 1 finite. Furthermore, it is also clear from the calculations in the first part of the proof, ̂ that for all w ∈ V, 󵄩󵄩 󵄩󵄩−1/2 (n) (Pran R R−1 L(Γ(n) 󵄩Mn (λ0 )󵄩󵄩 Γ1 fn )n∈ℕ , Rbw ) 0 fn )n∈ℕ , Rbw ) = (Pran R (󵄩 holds. Since span {Rbw }w∈V̂ is dense in ran R, we have ̃ min Γ̃0 f = P ̃ L ran R Γ1 f . Thus identity (4.10) holds. We apply Lemma 3.1 to obtain a different representation of SL̃ in terms of the min ̂ 0 , UΓ ̂ 1 } where Û : ran R → ℓ2 (V) ̂ is given by Rbw 󳨃→ ‖Rbw ‖ew boundary triplet {Û 𝒢 , UΓ ̃ min Û ∗ which is given by and with the operator Lmin = Û L ̃ min Rbv , Rbw ) (Pran R R−1 (L − M)R−1 Rbv , Rbw ) (L = ‖Rbv ‖‖Rbw ‖ ‖Rbv ‖‖Rbw ‖ =

(R−1 (L − M)bv , Pran R Rbw )

‖Rbv ‖‖Rbw ‖ ((L − ⨁n∈ℕ Mn (λ0 ))bv , bw ) = (Lmin )v,w . = ‖Rbv ‖‖Rbw ‖

Assertion (b) follows immediately from (a) and Proposition 3.1. An application of Proposition 3.2 (b) yields (c). Under the assumption that the direct sum triplet (4.1) is a boundary triplet for S∗ , −1 we have that ⨁∞ n=0 Mn (λ0 ), R, and R are bounded, and we obtain the following special case of Theorem 4.2. For quantum graphs with edge length bounded from below, this result was also obtained in [22]. Corollary 4.1. If the triplet {𝒢 , Γ0 , Γ1 } given by (4.1) is a boundary triplet for S∗ , then SLloc has the following properties:

42 | H. Gernandt and C. Trunk (a) SLloc is self-adjoint. loc (b) If S∗ |ker Γ0 = SF ≥ γ with γ ≥ 0 and ⨁∞ n=0 Mn (λ) 󴁂󴀱 −∞, as λ → −∞, then SL is semibounded from below if and only if there exists C > −∞ with (Lv x, x) ≥ C‖x‖2 for all x ∈ 𝒢v and all v ∈ V. Proof. Since SLloc is closed, it remains to show by Theorem 4.2 that Lmin is essentially self-adjoint. Every Lv is unitarily equivalent to a diagonal matrix and therefore the op(Lbv ,bw ) erator ( ‖Rb ‖‖Rb ) ̂ is unitarily equivalent to a densely defined multiplication operv w ‖ v,w∈V 2 ̂ ator on ℓ (V), and hence essentially self-adjoint. Since {𝒢 , Γ0 , Γ1 } is a boundary triplet, [5, Theorem 2.12] implies that the operators R, R−1 , and ⨁∞ n=0 Mn (λ0 ) are bounded. Therefore Lmin is just a bounded and symmetric perturbation of an essentially selfadjoint operator and hence essentially self-adjoint according to the Kato–Rellich theorem [15, Theorem V.4.4]. Assertion (b) is a consequence of the boundedness of R, R−1 , and of ⨁∞ n=0 Mn (λ0 ) and follows from Theorem 4.2 (a). Since (4.1) is in general not a boundary triplet, we use the results of [11, 17] to provide conditions on the self-adjointness of SLloc and the discreteness of the spectrum of all self-adjoint extensions in the theorem below. For this we associate with SLloc the formal discrete Laplacian DL on the weighted space ̂ m) := {(xv ) ̂ ∈ ℂV̂ | ∑ m(v)|xv |2 < ∞}, ℓ2 (V, v∈V ̂ v∈V

m(v) := ‖Rbv ‖2 ,

(4.12)

where bv is an element of an orthogonal basis of the subspace 𝒢𝒱 defined in (4.9) and ̂ m) is given by the scalar product in ℓ2 (V, (x, y)m := ∑ m(v)xv yv . ̂ v∈V

̂ via We define an operator DL with domain dom DL := C(V) (DL f )v :=

1 ( ∑ b(v, w)(fv − fw ) + c(v)fv ), ‖Rbv ‖2 ̂ w∈V

∞

b(v, w) := ((⨁ Mn (λ0 ) − L)bv , bw ), n=0

v ≠ w,

∞

b(v, v) := 0,

c(v) := ((L − ⨁ Mn (λ0 ))bv , bv ) − ∑ b(v, w). n=0

̂ w∈V

(4.13)

The elements of {bv }v∈V̂ have finite support, and if bv1 and bv2 are elements of a basis for 𝒢w1 and 𝒢w2 with w1 ≠ w2 then supp bv1 ∩ supp bv2 = 0. Also, the support of ̂ (⨁∞ n=0 Mn (λ0 ) − L)bv (considered as a sequence) is finite. Hence, for fixed w ∈ V we

Locally finite extensions and infinite metric graphs | 43

̂ As in [10, 16] we consider the weighted have b(v, w) ≠ 0 for only finitely many v ∈ V. degree ̂ → (0, ∞), Deg : V

v 󳨃→

1 ∑ b(v, w). ‖Rbv ‖2 ̂ w∈V

(4.14)

Theorem 4.3. Consider the operator SLloc and the associated discrete Laplacian (4.13). ̂ Then the following holds: Assume that b(v, w) ≥ 0 holds for all v, w ∈ V. loc (a) The operator SL is self-adjoint if one of the following conditions holds: c(v) ̂ with (i) Assume that infv∈V̂ ‖Rb > −∞ and that for all sequences {vn }n∈ℕ in V ‖2 v

2 b(vn , vn+1 ) > 0 for all n ∈ ℕ we have ∑∞ n=1 ‖Rbvn ‖ = ∞. (ii) The weighted degree Deg is bounded. (b) All self-adjoint extensions of SLmin are in one-to-one correspondence with the selfadjoint extensions of DL . ̂ if the (c) All self-adjoint extensions Ŝ of SLmin satisfy (Ŝ − λ)−1 ∈ S1 (ℋ) for some λ ∈ ρ(S) following conditions hold: ̂ there exists k ∈ ℕ and v0 , . . . , vk such that v0 = v, vk = w, and (i) For all v, w ∈ V b(vi , vi+1 ) > 0 for all i = 0, . . . , k − 1. (ii) Let (⨁n∈ℕ Sn0 − λ)−1 ∈ S1 (ℋ) for λ ∈ ρ(⨁n∈ℕ Sn0 ). c(v) (iii) Let ∑v,w∈V,b(v,w)=0̸ b(v, w)−1 < ∞, ∑v∈V̂ ‖Rbv ‖2 < ∞, infv∈V̂ ‖Rb 2 > −∞. v‖ ̃ (d) Assume that ⨁n∈ℕ Sn0 = SF ≥ γ for γ ≥ 0 and that M(λ) 󴁂󴀱 −∞ for λ → −∞ and c(v) min infv∈V̂ ‖Rb > −∞ then all self-adjoint extensions of S are semibounded from 2 L v‖ below.

Proof. First, we prove the results for c(v) ≥ 0. To prove (a), we use that by Proposition 4.1 the operator SLloc is closed. It remains to show by Theorem 4.2 (b) and Proposition 3.1 (a) that the operator given by (Lmin )v,w :=

((L − ⨁n∈ℕ Mn (λ0 ))bv , bw ) ‖Rbv ‖‖Rbw ‖

̂ A straightforward calculation shows that Lmin is is essentially self-adjoint on C(V). ̂ m) → ℓ2 (V), ̂ (xv ) ̂ 󳨃→ (‖Rbv ‖xv ) ̂ to the operator DL . unitary equivalent via U : ℓ2 (V, v∈V v∈V ̂ ⊆ C(V) ̂ The assumption in (i) on the sequences (vn )n∈ℕ in V and the invariance DL C(V) allows us to apply [17, Theorem 6] which yields the essential self-adjointness of DL on ̂ This shows the essential self-adjointness of Smin = SL by Proposition 3.1 (a). C(V). L min Assumption (ii) implies by [16, Theorem 11] that D0 given by DL with c(v) = 0 for all ̂ is bounded. Therefore DL on C(V) ̂ is a bounded and symmetric perturbation v ∈ V c(v) ̂ of the essentially self-adjoint multiplication operator (xv )v∈V̂ 󳨃→ ( ‖Rb ̂ on C(V) 2 xv )v∈V v‖ hence essentially self-adjoint because of the Kato–Rellich theorem [15, Theorem V.4.4]. The correspondence in (b) is a consequence of Theorem 4.2 (b). Assertion (c) follows from [11, Theorem 5.1] applied to DL which shows that all self̂ m)). Note that the assumptions of adjoint extensions of DL have resolvents in S1 (ℓ2 (V,

44 | H. Gernandt and C. Trunk this Theorem 5.1 are satisfied because of ∑v∈V̂ m(v) = ∑v∈V̂ ‖Rbv ‖2 < ∞ and (i) and (iii), see also [11, Example 4.6]. Assumption (ii), namely that (⨁n∈ℕ Sn0 − λ)−1 ∈ S1 (ℋ) for λ ∈ ρ(⨁n∈ℕ Sn0 ), together with Proposition 3.1 (e), implies that (Ŝ − λ)−1 ∈ S1 (ℋ). This proves (c). ̂ with ŜL = S ̂ . Let ŜL be an extension of SLmin and ̂ DL be an extension of DL on C(V) DL It was shown in [17, p. 206] that ̂ DL has the same action as DL . For f ∈ dom ̂ DL with (f , f )m = 1, we see from b(v, w) ≥ 0 that

(̂ DL f , f )m = ∑ m(v)(̂ DL f )v fv ̂ v∈V

=

1 ∑ b(v, w)|fv − fw |2 + ∑ c(v)|fv |2 2 ̂ ̂ v,w∈V

v∈V

c(v) c(v) ≥ ∑ c(v)|fv | ≥ inf (f , f )m = inf . ̂ ‖Rbv ‖2 ̂ ‖Rbv ‖2 v∈ V v∈ V ̂ 2

v∈V

Proposition 3.2 (a) applied to the regularized boundary triplet {𝒢 , Γ̃0 , Γ̃1 } from Theorem 4.1 yields that SD̂ is semibounded from below. Here we used that, due to (4.4), we L have S∗ |ker Γ̃ = SF . 0 ̂L is the bounded Assume now that inf ̂ c(v) 2 > −∞ holds. Then the operator D v∈V ‖Rbv ‖

̂+ where we replace c(v) with its positive part c(v)+ := perturbation of an operator D L max{c(v), 0}. Therefore we can apply the previous arguments to ̂ D+L . By assumption, ̂ DL is a bounded perturbation of ̂ D+L again the Kato–Rellich theorem shows that selfadjointness is preserved which proves (a) and (c). Furthermore, (d) follows from Proposition 3.2 (b).

5 Point interactions on infinite quantum graphs As a first application we consider the δ-type point interactions example from the introduction. Let G be a graph with countable sets of vertices V and edges E, and with the edge length function ℓ : E → (0, ∞]. Consider ℋ = L2 (G) = ⨁e∈E L2 (0, ℓ(e)) with the operator S = ⨁ Se , e∈E

dom Se = W02,2 (0, ℓ(e)),

Se := −

d2 . dxe2

Note that in comparison with the example in the introduction, we allow here edges of infinite length. In this case W02,2 (0, ∞) := {f ∈ L2 (0, ∞) : f , f 󸀠 ∈ AC(0, ∞) ∩ L2 (0, ∞), f 󸀠󸀠 ∈ L2 (0, ∞), f (0) = f 󸀠 (0) = 0}, where AC(0, ∞) is the set of locally absolutely continuous functions. We assume that the graph is locally finite, which means that each vertex is contained in only finitely many edges. Furthermore, we assume

Locally finite extensions and infinite metric graphs | 45

that the graph has no self-loops which means that no edge e ∈ E starts and ends at the same vertex. This is not a restriction since we can place an additional vertex with a Kirchhoff interface condition on these edges. For a real-valued sequence {α(v)}v∈V , we consider the operator from (1.4) given by dom Hα := {(ψe )e∈E ∈ W 2,2 (G) ∩ C(G) | ∑ sgn(e, t)ψ󸀠e (tℓ(e)) = α(v)ψ(v), v ∈ V}, (e,t)∈Iv

which describes δ-type point interactions at the vertices of the graph G. For graphs without edges of infinite length, these extensions were already studied in [10]. The calculations from the introduction imply that Hα is a locally finite extension SLloc . Here we use for the edges e ∈ E with ℓ(e) < ∞ the boundary triplet from (1.5) 2

𝒢e = ℂ ,

ψe (0+) Γ(e) ), 0 ψe = ( ψe (ℓ(e)−)

ψ󸀠e (0+) Γ(e) ψ = ( ) e 1 −ψ󸀠e (ℓ(e)−)

2

π with the Weyl function which is for λ ∈ ℂ \ [ ℓ(e) 2 , ∞) given by

Me (λ) :=

√λ

sin(ℓ(e)√λ)

− cos(ℓ(e)√λ) 1

1 ). − cos(ℓ(e)√λ)

(

(5.1)

For e ∈ E with ℓ(e) = ∞, we have the boundary triplet 𝒢e := ℂ,

Γ(e) 0 ψe := ψe (0+),

󸀠 Γ(e) 1 ψe := ψe (ℓ(e)−)

and the Weyl function Me (λ) := i√λ,

λ ∈ ℂ \ [0, ∞),

(5.2)

where √z satisfies Im √z > 0. Introduce the set Iv with (e, 0) ∈ Iv if e ∈ E and e has v as initial vertex and (e, 1) ∈ Iv if e ∈ E and e has v as terminal vertex. In particular, edges with ℓ(e) = ∞ have only one initial vertex, and we interpret the evaluation ψ󸀠e (tℓ(e)) in the definition of Hα as ψ󸀠e (0). The vectors bv ∈ 𝒢 are given by

(bv )(e,t)

1, if (e, 0) ∈ Iv , { { { := {−1, if (e, 1) ∈ Iv , { { {0, if (e, t) ∉ Iv ,

and 𝒢v := span{1v },

1v := ((bv )(e,t) )(e,t)∈I , v

and

Lv 1v :=

α(v) 1 . deg v v

(5.3)

46 | H. Gernandt and C. Trunk First we consider the case that infe∈E ℓ(e) > 0. In this case the direct sum triplet (e) {⨁e∈E 𝒢e , ⨁e∈E Γ(e) 0 , ⨁e∈E Γ1 } is a boundary triplet, and as an application of Corollary 4.1 we obtain the result below. Note that it is well known, see, e. g., [28, Example 10.4] that the Friedrichs extension (Se )F for ℓ(e) < ∞ is given by dom(Se )F = {f ∈ W 2,2 (0, ℓ(e)) : f (0) = f (ℓ(e)) = 0}, i. e., it is equal to the Dirichlet extension of Se and therefore (Se )F ≥ 0. From the definition of the Weyl functions Me in (5.1) and (5.2), we can easily see that M(λ) = ⨁e∈E Me (λ) 󴁂󴀱 −∞ as λ → −∞. The result below is now a consequence of Corollary 4.1. Corollary 5.1. If infe∈E ℓ(e) > 0 then Hα is self-adjoint. Furthermore, Hα is semibounded from below if and only if inf

v∈V

α(v) > −∞. deg v

In the case infe∈E ℓ(e) = 0, the direct sum triplet is not a boundary triplet and we apply Theorem 4.3. We apply the regularization at λ0 = 0 at all edges of finite length. For this a short computation shows that the Weyl functions Me of the boundary triplet (e) {ℂ2 , Γ(e) 0 , Γ1 } satisfy Me (0) =

−1 1 ( ℓ(e) 1

1 ), −1

− ℓ(e) Me󸀠 (0) = ( ℓ(e)3 6

ℓ(e) 6 ). − ℓ(e) 3

Define R := ⨁ Re , e∈E

{√‖Me󸀠 (0)‖( 01 01 ), if ℓ(e) < ∞, Re := { 1, if ℓ(e) = ∞ {

with ‖Me󸀠 (0)‖ = ℓ(e) . We have already seen in the introduction that Hα can be viewed as 2 a locally finite extension SLloc from (4.5) with (5.3). This implies that Hα can be described by a discrete Laplacian DL on ℓ2 (V, m) given by (4.12) and (4.13) with m(v) = ‖Rbv ‖2 =

1 ∑ ℓ(e) + ℰv , 2 e=vw

where ℰv is the number of edges e ∈ E with ℓ(e) = ∞ that have v ∈ V as initial vertex and the summation ∑e=vw is over all edges e ∈ E that have v as an initial or terminal vertex. The discrete Laplacian DL is then given by (DL f )v =

1 ( ∑ b(v, w)(fv − fw ) + c(v)fv ) m(v) w∈V

Locally finite extensions and infinite metric graphs | 47

where b(v, w) and c(v) are, according to (4.13), for v ≠ w given by b(v, w) := ((⨁ Me (0) − L)bv , bw ) = (⨁ Me (0)bv , bw ) e∈E

e∈E

ℓ(e)

if e = vw ∈ E,

0

if e = vw ∉ E,

−1

={

α(v) where by the definition of L we have Lbv = deg b for all v ∈ V and we used that for v v w ≠ v the sequences bv and bw have disjoint support, which implies

(Lbv , bw ) = (

α(v) b , b ) = 0. deg v v w

Also, we see for v ∈ V that c(v) := ((L − ⨁ Me (0))bv , bv ) − e∈E

∑

w∈V,w=v̸

(⨁ Me (0)bv , bw ) e∈E

= (Lbv , bv ) + ∑ ℓ(e)−1 − ∑ ℓ(e)−1 e=vw

e=vw

= (Lv 1v , 1v ) = α(v). In the corollary below, we summarize some properties of the extension Hα as a consequence of Theorem 4.3. This result was also obtained in [10] under the assumption that ℰv = 0 for all v ∈ V. Corollary 5.2. Let G be a locally finite graph with supe∈E,ℓ(e) −∞. ℰv + ∑e=vw ℓ(e)

Then the following holds: (a) If for every infinite path (vi )i∈ℕ we have ∑i∈ℕ ℰvi + ∑e=vi w ℓ(e) = ∞ then Hα is selfadjoint and semibounded from below. (b) If G is connected with ∑e∈E ℓ(e) < ∞ then all self-adjoint extensions of Hα have purely discrete spectrum. Proof. Assertion (a) follows from Theorem 4.3 (a), parts (i) and (d). To prove (b), we verify conditions (i)–(iii) from Theorem 4.3 (c). Since b(v, w) = ℓ(e)−1 if and only if e ∈ E, condition (i) is equivalent to the connectedness of G. That condition (ii) is fulfilled π2 k2 for λ = 0 ∈ ρ(⨁e∈E Se0 ) follows from σ(Se0 ) = { ℓ(e) 2 : k ≥ 1} and spectral mapping that ∑

λ∈σ((⨁e∈E Se0 )−1 )

λ= ∑

∑

e∈E λ∈σ((⨁e∈E Se0 )−1 )

ℓ(e)2 1 = ∑ ℓ(e)2 < ∞, 2 2 6 e∈E k=1 π k e∈E ∞

λ= ∑∑

48 | H. Gernandt and C. Trunk and therefore (⨁e∈E Se0 )−1 ∈ S1 (L2 (G)). Furthermore, the first and third estimate in (iii) trivially hold. The second estimate in (iii) is also satisfied, as ∑ ‖Rbv ‖2 = ∑ ∑ ℓ(e) = 2 ∑ ℓ(e) < ∞. v∈V e=vw

v∈V

e∈E

Therefore we can apply Theorem 4.3 (c), which proves assertion (b).

6 Gesztesy–Šeba realizations of Dirac operators on metric graphs As a second application, we introduce in this section the Gesztesy–Šeba realization of Dirac operators on a locally finite graph given by a set of vertices V and a set of edges E. On each edge e ∈ E with finite length ℓ(e), we consider the Dirac operator c2 /2 De := ( −ic dxd

e

−ic dxd

e

−c2 /2

),

dom De := H01 (0, ℓ(e)) ⊗ ℂ2 ,

where c denotes the speed of light. It was shown in [5, Lemma 3.1] that a boundary triplet for D∗e is given by 2

Γ̂ (e) 0 (

𝒢e := ℂ ,

icψe,2 (0+) ψe,1 ) := ( ) Γ̂ (e) 1 ( ψe,2 ψe,1 (ℓ(e)−)

ψe,1 (0+) ) := ( ), ψe,2 icψe,2 (ℓ(e)−) ψe,1

with the Weyl function for λ ∈ ρ(D∗e |ker Γ̂(e) ) given by 0

M̂ e (λ) :=

ck1 (λ) sin(ℓ(e)k(λ)) 1 ( cos(ℓ(e)k(λ)) 1

1 (ck1 (λ)) sin(ℓ(e)k(λ)) −1

),

where we abbreviate 2

k(λ) := c−1 √λ2 − (c2 /2) , with √⋅ such that k(x) > 0 for x >

c2 . 2

k1 (λ) :=

ck(λ) λ − c2 /2 =√ 2 λ + c /2 λ + c2 /2

Under the assumption that supe∈E ℓ(e) < ∞, 2

2

it was shown in [5, Equation (3.56)] that for some ε > 0 we have ( c2 − ε, c2 + ε) ⊆ ⋂e∈E ρ(D∗e |ker Γ̂(e) ) and 0

0 c2 M̂ e ( ) = ( 1 2

1 ), ℓ(e)

ℓ(e) c2 M̂ e󸀠 ( ) = ( 2 ℓ(e) 2 2

ℓ(e)2 2 ℓ(e) c2

+

ℓ(e)3 3

).

(6.1)

Locally finite extensions and infinite metric graphs | 49

To describe a point interaction on a graph, we consider the boundary triplet for D∗e given by a unitary transformation Γ(e) 0 ( (e) ) Γ1

W00 := [ W10

1 W01 0 ] ( (e) ) = ( 0 W11 Γ̂ 1 0 Γ̂ (e) 0

0 0 0 −i

ψe,1 (0+) 0 icψe,2 (ℓ(e)−) i )( ) 0 icψe,2 (0+) 0 ψe,1 (ℓ(e)−)

0 0 1 0

with W00 , W01 , W10 , W11 ∈ ℂ2×2 , and therefore Γ(e) 0 (

ψe,1

ψe,1 (0+) )=( ), ψe,2 iψe,1 (ℓ(e)−)

Γ(e) 1 (

ψe,1

icψe,2 (0+) ) := ( ). ψe,2 cψe,2 (ℓ(e)−)

It was shown in [6] that such a unitary transformation leads to a boundary triplet with the Weyl function given by Me (λ) = (W10 + W11 M̂ e (λ))(W00 + W01 M̂ e (λ))

−1

=

ck1 (λ) cos(ℓ(e)k(λ)) ( i sin(ℓ(e)k(λ))

−i ) − cos(ℓ(e)k(λ))

(6.2)

for all λ ∈ ρ(D∗e |ker Γ̂(e) ) ∩ ρ(D∗e |ker Γ(e) ). 0

0

Introduce a set Iv with (e, 0) ∈ Iv if e ∈ E and e has v as initial vertex and (e, 1) ∈ Iv if e ∈ E and e has v as terminal vertex. The vectors bv ∈ 𝒢 are given by (bv )(e,t)

1, if (e, 0) ∈ Iv , { { { := {i, if (e, 1) ∈ Iv , { { {0, if (e, t) ∉ Iv .

Let (α(v))v∈V be a real sequence. The operator GSα is given by dom GSα := {(ψ1 , ψ2 )⊤ ∈ ⨁ D∗e | ψ1 ∈ C(G), e∈E

ic ∑ sgn(e, t)ψe,2 (tℓ(e)) = α(v)ψ1 (v), v ∈ V}, (e,t)∈Iv

where C(G) is the set of continuous functions on G viewed as a metric space and ψ1 (v) is the value of ψ1 at the vertex v. We follow here [5] and call this operator Gesztesy– Šeba realization. If supe∈E ℓ(e) < ∞, it can easily be seen from (6.2) that for some ε > 0 we have 2

2

( c2 − ε, c2 + ε) ⊆ ⋂e∈E ρ(D∗e |ker Γ(e) ) and that 0

Me (

1 c2 1 )= ( 2 ℓ(e) i

−i ). 1

50 | H. Gernandt and C. Trunk 2

We also see from (6.2) and (6.1) with T := (W00 + W01 M̂ e ( c2 ))−1 that Me󸀠 (

c2 c2 c2 c2 ) = W11 M̂ e󸀠 ( )T − (W10 + W11 M̂ e ( ))TW01 M̂ e󸀠 ( )T 2 2 2 2 = ( iℓ(e)

1 ℓ(e)c2

2

−

+

i ℓ(e)c2

ℓ(e) 3

−

iℓ(e) 3

− iℓ(e) + 2

i + iℓ(e) 3 ℓ(e)c2 ), ℓ(e) 1 + 3 ℓ(e)c2

and this implies 2 2 󵄩 󵄩󵄩 1 1 1 ℓ(e) 󵄩󵄩 󸀠 c 󵄩󵄩󵄩 󸀠 c ≥ + . 󵄩󵄩Me ( )󵄩󵄩 ≥ (1, 0)Me ( ) ( ) = 󵄩󵄩 0 2 󵄩󵄩 2 3 ℓ(e)c2 ℓ(e)c2

(6.3)

Furthermore, we define 𝒢v := span{1v },

1v := ((bv )(e,t) )(e,t)∈I , v

and

Lv 1v :=

α(v) 1 . deg v v

We have, according to (4.13), for v ≠ w that b(v, w) := ((⨁ Me ( e∈E

ℓ(e)−1 ={ 0

c2 c2 ) − L)bv , bw ) = (⨁ Me ( )bv , bw ) 2 2 e∈E

if e = vw ∈ E, if e = vw ∉ E,

and we see for v ∈ V that c(v) := ((L − ⨁ Me ( e∈E

c2 c2 ))bv , bv ) − ∑ (⨁ Me ( )bv , bw ) 2 2 e∈E w∈V,w=v̸

= (Lv 1v , 1v ) = α(v). As an application of Theorem 4.3, we have the following result on the selfadjointness of the Gesztesy–Šeba realizations. Proposition 6.1. Consider a locally finite graph with set of vertices V and set of edges E and let {α(v)}v∈V be a real-valued sequence. Then the operator GSα is a locally finite extension of ⨁e∈E De and if supe∈E ℓ(e) < ∞ then GSα is self-adjoint. Proof. We show that GSα is a locally finite extension of ⨁e∈E De . Let I := E ×{0, 1}, then Γ0 ψ = (Γ0(e,t) (ψe,1 , ψe,2 )e∈E )(e,t)∈I = (it ψe,1 (tℓ(e)))(e,t)∈I ,

Γ1 ψ = (Γ1(e,t) (ψe,1 , ψe,2 )e∈E )(e,t)∈I = (ci1−t ψe,1 (tℓ(e)))(e,t)∈I ,

and therefore Γv0 (ψe,1 , ψe,2 )e∈E := (it ψe,1 (tℓ(e)))(e,t)∈I , v

Locally finite extensions and infinite metric graphs | 51

Γv1 (ψe,1 , ψe,2 )e∈E := (ci1−t ψe,2 (tℓ(e)))(e,t)∈I . v

Since 𝒢v = span{1v } for all v ∈ V, we see that ψ1 ∈ C(G) is equivalent to the condition Γv0 (ψe,1 , ψe,2 )e∈E ∈ 𝒢v for all v ∈ V. Moreover, it is easy to see that the sum condition in the definition of dom GSα is equivalent to 1 (Γv (ψ , ψ ) , 1 )1 ‖1v ‖2 1 e,1 e,2 e∈E v v 1 = ∑ (b ) ci1−t ψe,2 (tℓ(e))1v deg v (e,t)∈I v (e,t)

P𝒢v Γv1 (ψe,1 , ψe,2 )e∈E =

v

ic = ∑ sgn(e, t)ψe,2 (tℓ(e))1v deg v (e,t)∈I v

α(v) = ψ (v)1v = Lv Γv0 (ψe,1 , ψe,2 )e∈E . deg v 1

Thus, we have seen that GSα is a locally finite extension of ⨁e∈E De . For supe∈E ℓ(e) < ∞, the assumptions of Theorem 4.2 are fulfilled. To see that GSα is self-adjoint, we apply Theorem 4.3 (a). The estimate (6.3) implies that the weighted degree (4.14) satisfies Deg(v) =

∑w∈V b(v, w) ∑w∈V b(v, w) ℓ(e)−1 ∑ = ≤ e=vw 1 = c2 < ∞ 2 2 ‖Rbv ‖ ∑e=vw c2 ℓ(e) ∑e=vw ‖Me󸀠 ( c2 )‖

for all v ∈ V, where the summation ∑e=vw is taken over all edges e that contain v as a vertex. Hence, according to Theorem 4.3, GSα is self-adjoint.

7 Conclusions In this note, we have considered locally finite extensions SLloc of infinite direct sums of symmetric operators S. We discussed two examples in Sections 5 and 6 which demonstrate that these extensions appear naturally in the study of infinite metric graphs with differential operators on the edges and boundary and interface conditions at the vertices. The well-known boundary triplet approach was applied to study the properties of SLloc like self-adjointness, lower semiboundedness, and discreteness of the spectrum. Here we formulated these conditions in a different and for our purposes more natural way in terms of extension SL of S given by (3.2) and an operator L. As an application of the main results on locally finite extensions from Section 4, we recovered some well-known results for point interactions on infinite quantum graphs from [10]. As a second application, we introduced and studied Gesztesy–Šeba realizations of the Dirac operator on infinite metric graphs. This demonstrates that our general approach via locally finite extensions can easily be utilized to study extensions of more general types of operators on infinite metric graphs.

52 | H. Gernandt and C. Trunk

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E. Savaş

Double lacunary statistically convergent sequences of weight g Abstract: The main aim of this paper is to present double lacunary statistical convergence of weight t. Additionally, we prove some interesting inclusion theorems. Keywords: Statistical convergence, double statistical convergence of weight t, double lacunary statistical convergence MSC 2010: Primary 40B05, Secondary 40C05

1 Introduction The idea of convergence of a real sequence was extended to statistical convergence by Fast [3] (see also Schoenberg [19]) as follows: We say that a sequence y is statistically convergent to a number ξ if for each ε > 0, 1󵄨 󵄨 lim 󵄨󵄨󵄨{i < ρ : |xi − ξ | ≥ ε}󵄨󵄨󵄨 = 0. ρ ρ For some interesting investigations about statistical convergence, one may consider the papers of Fridy [4], Cakalli [2], Miller [6], Maddox [5], Šalát [10] and others. Recently, in [15], Savaş considered the notion of statistical convergence of order κ, 0 < κ < 1. In this paper we present the idea of double lacunary statistical convergence of weight t, where t : [0, ∞) × [0, ∞) → [0, ∞) is such that t(yij ) → ∞ whenever yij → ∞. The set of all such functions will be denoted by T. Let w2 be the class of all double sequences. A double sequence y = (yij ) has a Pringsheim limit ξ denoted by P − lim y = ξ provided that given ε > 0 there exists N ∈ ℕ such that |yij − ξ | < ε whenever i, j ≥ N. We shall describe such a y more briefly as “P-convergent” (see [9]). We denote by c2 , the space of all P-convergent sequences. A double sequence y = (yij ) is bounded if ‖y‖ = supi,j≥0 |yij | < ∞. Let l2∞ and c2∞ be the sets of all bounded double sequences and all bounded and convergent double sequences, respectively. The statistical convergence was extended to double sequences by Mursaleen and Edely [7]. Several investigations on double sequences can be found in [1, 8, 11, 14, 16, 18]. E. Savaş, Usak University, Department of Mathematics, Usak, Turkey, e-mail: [email protected] https://doi.org/10.1515/9783110598193-003

56 | E. Savaş Mursaleen and Edely [7] presented the following definition: We say that a double sequence y = (yij ) is statistically convergent to ξ if, for each ε > 0, P − lim p,q

1 󵄨󵄨 󵄨 󵄨{(i, j) : i ≤ p and j ≤ q : |yij − ξ | ≥ ε}󵄨󵄨󵄨 = 0. pq 󵄨

Definition 1.1. A double sequence αu,v = {(ku , lv )} is called double lacunary if there exist two increasing sequences of integers such that k0 = 0,

hu = ku − ku−1 → ∞ as u → ∞

and l0 = 0,

h̄ v = lv − lv−1 → ∞

as v → ∞.

Let us denote ku,v = ku lv , huv = hu h̄ v and let αu,v be determined by Iu,v = {(i, j) :

ku−1 < k ≤ ku and lv−1 < l ≤ lv }, qu =

ku , ku−1

q̄ v =

lv , lv−1

and qu,v = qu q̄ v .

Let E ⊆ N × N have double lacunary density δ2α (E) if P − lim u,v

1 󵄨󵄨 󵄨 󵄨{(i, j) ∈ Iu,v : (i, j) ∈ E}󵄨󵄨󵄨 huv 󵄨

exists. R. F. Patterson and E. Savas [8] considered the double lacunary statistical convergence as follows: Definition 1.2. Let αu,v be a double lacunary sequence. A double sequence y is Sα2 -convergent to ξ if, for every ε > 0, P − lim u,v

1 󵄨󵄨 󵄨 󵄨{(i, j) ∈ Iu,v : |yij − ξ | ≥ ε}󵄨󵄨󵄨 = 0. huv 󵄨

In this situation, we write Sα2 − limi,j yi,j = ξ . Some more investigations and applications of double lacunary and double sequences can be found in [12, 13, 14, 17].

2 Main results We now consider Definition 2.1. Let t ∈ T. We say that a sequence y = (yij ) ∈ w2 has double lacunary statistical convergence of weight t if there is a real number ξ such that P − lim u,v

1 󵄨󵄨 󵄨 󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 = 0. t(huv ) 󵄨

Double lacunary statistically convergent sequences of weight g

| 57

In case y = (yij ) has double lacunary statistical convergence of weight t to ξ , we write Sαt u,v − lim yij = ξ . We denote the set of all Sαt u,v -statistically convergent sequences of weight t by Sαt u,v .

Remark 2.1. Taking t(huv ) = hκuv , 0 < κ ≤ 1, we get the definition of double lacunary statistical convergence of order κ [18]. If α = 2rs and t(ij) = (ij)κ then we get the definition of double lacunary statistical convergence of order κ. For t(huv ) = huv , we obtain the definition of double lacunary statistical convergence [8]. Definition 2.2. Let r be a positive real number and let t ∈ T. We say that a double sequence y is strongly double lacunary r-summable of weight t if there is a number ξ such that P − lim uv

1 ∑ |y − ξ |r = 0. t(huv ) (i,j)∈I ij uv

We denote the class of all strongly double lacunary r-summable sequences of weight t by wαt u,v (r). Remark 2.2. When we consider t(huv ) = hκuv , 0 < κ ≤ 1, the above notion coincides with the notion of a strong double lacunary r-summable sequence of order κ. Theorem 2.1. Let t ∈ T and y = (yij ), z = (zij ) be sequences.

(i) If Sαt u,v − lim yij = ξ1 and γ ∈ ℂ, then Sαt u,v − lim γxij = γξ1 .

(ii) If Sαt u,v − lim yij = ξ1 and Sαt u,v − lim zij = ξ2 then Sαt u,v − lim(yij + zij ) = ξ1 + ξ2 . Proof. (i) For γ = 0, the result is clear. Let γ ≠ 0. Now observe that 1 󵄨󵄨󵄨󵄨 ε 󵄨󵄨󵄨󵄨 1 󵄨󵄨 󵄨 }󵄨, 󵄨󵄨{(i, j) ∈ Iuv : |γyij − γξ1 | ≥ ε}󵄨󵄨󵄨 = 󵄨󵄨{(i, j) ∈ Iuv : |yij − ξ1 | ≥ 󵄨 t(huv ) t(huv ) 󵄨 |γ| 󵄨󵄨󵄨

and the result is obvious. (ii) 1 󵄨󵄨 󵄨 󵄨 󵄨 󵄨{(i, j) ∈ Iuv : 󵄨󵄨󵄨yij + zij − (ξ1 + ξ2 )󵄨󵄨󵄨 ≥ ε}󵄨󵄨󵄨 t(huv ) 󵄨 ≤

ε 󵄨󵄨󵄨 1 󵄨󵄨󵄨󵄨 ε 󵄨󵄨󵄨 1 󵄨󵄨󵄨󵄨 󵄨󵄨{(i, j) ∈ Iuv : |yij − ξ1 | ≥ }󵄨󵄨󵄨 + 󵄨󵄨{(i, j) ∈ Iuv : |zij − ξ2 | ≥ }󵄨󵄨󵄨, t(huv ) 󵄨󵄨 2 󵄨󵄨 t(huv ) 󵄨󵄨 2 󵄨󵄨

which implies the needed result. Theorem 2.2. Let t1, t2 ∈ T and suppose there are R > 0 and (u0 , v0 ) ∈ ℕ × ℕ such that t

t

t1 (huv )/t2 (huv ) ≤ R for each u, v ≥ u0 , v0 . Then Sα1u,v ⊂ Sα2u,v .

58 | E. Savaş Proof. We write 1 󵄨󵄨 1 󵄨󵄨 󵄨 t (h ) 󵄨 󵄨󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 = 1 uv ⋅ 󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 t2 (huv ) t2 (huv ) t1 (huv ) 󵄨 ≤R⋅

1 󵄨󵄨 󵄨 󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 t1 (huv ) 󵄨

t

for all u, v ≥ u0 , v0 . If y = (yij ) ∈ Sα1u,v then the result is clear for each ε > 0, and we get 1 󵄨󵄨 󵄨 󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 = 0, t2 (huv ) 󵄨 t

t

t

so y ∈ Sα2u,v . Finally, Sα1u,v ⊂ Sα2u,v . Theorem 2.3. We have Sαu,v ⊂ Sαt u,v provided that lim inf uv

Proof. Since lim infuv

we have get

t(huv ) t(kuv )

t(huv ) t(kuv )

t(huv ) > 1. t(kuv )

> 1, we take an M > 1 such that for sufficiently large u, v,

≥ M. Since yij → ξ (Sαu,v )t , for every ε > 0 and sufficiently large u, v, we

1 󵄨󵄨 1 󵄨󵄨 󵄨 󵄨 󵄨{i ≤ ku and j ≤ lv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 ≥ 󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨 t(kuv ) 󵄨 t(kuv ) 󵄨 ≥H⋅

1 󵄨󵄨 󵄨 󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ε}󵄨󵄨󵄨. t(huv ) 󵄨

Theorem 2.4. For a double lacunary double sequence α, lacunary double statistical convergence of weight t implies double statistical convergence of weight t (where t(n) ≠ n), provided that lim supu qut < ∞ and lim supv q̄ vt < ∞, where it should be noted that t is monotonically increasing. Proof. Now suppose lim supu qut < ∞ and lim supv q̄ vt < ∞. Then there is M > 0 such that qut < M and q̄ vt < M for all u and v. Suppose that yij → L(Sαt u,v ) and 󵄨 󵄨 Nu,v = 󵄨󵄨󵄨{(i, j) ∈ Iuv : |yij − ξ | ≥ ϵ}󵄨󵄨󵄨. By the definition of yij → L(Sαt u,v ), given ϵ > 0 there is (u0 , v0 ) ∈ ℕ × ℕ such that Nu,v t(huv)

< ε for all u > u0 and v > v0 . Let

P := max{Nu,v : 1 ≤ u ≤ u0 and 1 ≤ v ≤ v0 }.

Double lacunary statistically convergent sequences of weight g

| 59

Let p and q be such that ku−1 < p ≤ ku and lv−1 < q ≤ lv . Then we have 1 󵄨󵄨 1 󵄨 󵄨 󵄨󵄨 󵄨{i ≤ p&j ≤ q : |yij − ξ | ≥ ϵ}󵄨󵄨󵄨 ≤ 󵄨{i ≤ ku &j ≤ lv : |yij − ξ | ≥ ϵ}󵄨󵄨󵄨 t(pq) 󵄨 t(pu−1 qv−1 ) 󵄨 =

≤

u,v

1

t(ku−1 lv−1 )

{ ∑ Np,q } p,q=1,1

u,v Pu0 v0 1 + { ∑ t(ku−1 lv−1 ) t(ku−1 lv−1 ) p,q=u +1,v 0

≤

u,v

∑

Np,q t(hpq ) t(hpq )

p,q=u0 +1,v0 +1

u,v

∑

p,q=u0 +1,v0 +1

t(hpq )}

u,v Pu0 v0 + ε{ ∑ t(ku−1 lv−1 ) p,q=u +1,v 0

≤

}

Np,q Pu0 v0 1 + ( sup ) t(ku−1 lv−1 ) t(ku−1 lv−1 ) p,q≥u0 ,v0 t(hpq ) ×{

≤

Np,q }

Pu0 v0 1 + t(ku−1 lv−1 ) t(ku−1 lv−1 ) ×{

≤

0 +1

0 +1

t(hp,q )}

Pu0 v0 + εM 2 . t(ku−1 lv−1 )

Theorem 2.5. Let r be a positive real number and t1 , t2 ∈ T. Then there are R > 0 and t (u0 , v0 ) ∈ ℕ × ℕ such that t1 (huv )/t2 (huv ) ≤ R for all u, v ≥ u0 , v0 , and then wα1u,v (r) ⊆ t

wα2u,v (r).

t

Proof. Let y = (yij ) ∈ wα1u,v (r). Then for a positive real number r, we write t (h ) 1 1 1 ≤R⋅ ∑ |y − ξ |r = 1 uv ⋅ ∑ |y − ξ |r t2 (huv ) (i,j)∈I ij t2 (huv ) t1 (huv ) t1 (huv ) (k,l)∈I ijl uv

t

uv

t

and get that wα1u,v (r) ⊆ wα2u,v (r). Theorem 2.6. Let g1 , g2 ∈ G. Then there are R > 0 and (u0 , v0 ) ∈ ℕ × ℕ such that t1 (huv )/t2 (huv ) ≤ R for all u ≥ u0 , v ≥ v0 . Also let 0 < r < ∞. If a sequence y = (yij ) t

t

belongs to wα1u,v (r) and converges to ξ , then it is Sα2u,v -statistically convergent of weight t2 t

t

to ξ , i. e., wα1u,v (r) ⊂ Sα2u,v .

60 | E. Savaş Proof. For a double sequence y = (yij ) and ε > 0, write ∑ |yij − ξ |r =

(i,j)∈Iuv

≥

∑ (i,j)∈Iuv |yij −ξ |≥ϵ

∑ (i,j)∈Iuv |yij −ξ |≥ϵ

|yij − ξ |r +

∑ (i,j)∈Iuv |yij −ξ | 0. The NEFs F(ℛa ) generated by the measure ℛa , a ∈ Ξ0 , are called the Riesz NEFs; they are characterized by invariance under the action of the triangular group T. R. Zine, Sfax University, Laboratory of Probability and Statistics, Faculty of sciences of Sfax, Sfax, Tunisia https://doi.org/10.1515/9783110598193-004

64 | R. Zine Our work is an investigation in different directions of the probabilistic properties of the Riesz probability distribution. A Euclidean Jordan algebra 𝕍 is a finite-dimensional Euclidean space of dimension n, with a scalar product ⟨X, Y⟩ and a bilinear map (X, Y) 󳨃󳨀→ XY

𝕍 × 𝕍 󳨀→ 𝕍, which verifies some specific properties.

Definition 1.1. A Euclidean Jordan algebra is said to be simple if it does not contain a nontrivial ideal. Definition 1.2. Let 𝕍 a Euclidean Jordan algebra. We define the Jordan squares by 2

𝒱 = {X ; X ∈ 𝕍}.

Its interior is a symmetric cone and it is denoted by 𝒱 . Various properties of 𝒱 may be found in the literature (see, for example, [3]) which coincide with those for symmetric positive definite matrices on the space of real symmetric matrices. Definition 1.3. We define 𝒦 = {k ∈ 𝒢 ; k(e) = e}, where 𝒢 is the connected component of the identity in the group G(𝒱 ) of linear automorphisms of 𝒱 . For any X in the Euclidean space, we define the linear map by 𝕃(X) : 𝕍 󳨀→ 𝕍;

Y 󳨃󳨀→ XY

and the quadratic representation of 𝕍 by ℙ(X) = 2𝕃(X)2 − 𝕃(X 2 ).

(1.1)

For (X, Y, Z) ∈ 𝕍, let {XYZ} denote the Jordan triple product of 𝕍 defined by {XYZ} = ℙ(X + Z)Y − ℙ(X)Y − ℙ(Z)Y. An element c of 𝕍\{0} is said to be a primitive idempotent if c2 = c and it cannot be written as the sum of two nonzero orthogonal idempotents. The only possible eigenvalues of 𝕃(c) are 0, 21 , and 1. We define the Peirce decomposition of 𝕍 with respect to c by 1 𝕍 = 𝕍(c, 0) ⊕ 𝕍(c, ) ⊕ 𝕍(c, 1), 2 where 𝕍(c, 0), 𝕍(c, 21 ), and 𝕍(c, 1) are the corresponding eigenspaces. We say that c1 , . . . , cr is a complete system of orthogonal primitive idempotents, if ∑rj=1 cj = e and cj cl = 0, if j ≠ l. The size r of such a frame is a constant called the rank of 𝕍.

Riesz probability distributions on symmetric cones | 65

Letting (ck )1≤k≤r be a fixed Jordan frame, the space 𝕍 decomposes in the following orthogonal direct sum: 𝕍 = ⨁ 𝕍kl , k≤l

where 𝕍kl = {

𝕍(ck , 1) = ℝck 𝕍(ck , 21 )

∩

𝕍(cl , 21 )

if k = l, if k ≠ l.

If the dimension of 𝕍kl = d, for k ≠ l, the dimension n of 𝕍 verifies the relation n=r+

r(r − 1) d. 2

We denote by Pj the orthogonal projection onto 𝕍(j) = 𝕍(c1 + ⋅ ⋅ ⋅ + cj , 1), det(j) is the determinant in 𝕍(j) and, for X in 𝕍, Δj (X) = det(j) (Pj (X)). Then Δj is called the principal minor of order j with respect to the Jordan frame (ck )1≤k≤r . Notice that Δr (X) = det(X). For a = (a1 , . . . , ar ) ∈ ℝr , and X in 𝒱 , we write Δa (X) = Δ1 (X)a1 −a2 Δ2 (X)a2 −a3 ⋅ ⋅ ⋅ Δr (X)ar , where Δa is called a generalized power function. Similarly, we define Pj∗ to be the orthogonal projection onto 𝕎(j) = 𝕍(cr−j+1 + ⋅ ⋅ ⋅ + cr , 1) and Δ∗j (X) = det∗j (Pj∗ (X)), where det∗j is the determinant in the subalgebra 𝕎(j) . Additionally, for a = (a1 , . . . , ar ) ∈ ℝr , we let Δ∗a (X) = Δ∗1 (X)a1 −a2 Δ∗2 (X)a2 −a3 ⋅ ⋅ ⋅ Δ∗r (X)ar . In 1994, Faraut and Korányi proved the following result: Δa (X −1 ) = Δ∗−a∗ (X), where a∗ = (ar , . . . , a1 ).

2 Properties of Riesz probability distributions At the beginning of this section, we define the set Ξ of elements a on ℝr by ε(v) = 0 if v = 0

and ε(v) = 1 if v > 0.

For v1 , v2 , . . . , vr ≥ 0, we define a1 = v1 and ak = vk + d2 (ε(v1 ) + ⋅ ⋅ ⋅ + ε(vk−1 )), for 2 ≤ k ≤ r. We denote by Lμ (θ) = ∫𝕍 e⟨θ,U⟩ μ(dU) the Laplace transform of a measure μ on 𝕍. The next result can be found in Gindikin [4] and Faraut and Korányi [3].

66 | R. Zine Theorem 2.1. For θ ∈ −𝒱 , Δa (−θ−1 ) is the Laplace transform of a positive measure ℛa if and only if a ∈ Ξ. According to the position of a in Ξ, the measures ℛa are divided into two classes: the first is class of measures concentrated on the symmetric cone 𝒱 and the second is concentrated on the boundary 𝜕𝒱 of 𝒱 . For further details, we refer to [5] and [6]. The following theorem describes the Riesz measures which are absolutely continuous with respect to the Lebesgue measure on 𝒱 . Theorem 2.2. Let a = (a1 , . . . , ar ) in Ξ be such that for all j, aj > (j − 1) d2 . Then ℛa (dX) =

where Γ𝒱 (a) = (2π)

n−r 2

1 Δ n (X)1𝒱 (X)dX, Γ𝒱 (a) a− r

∏rk=1 Γ(ak − (k − 1) d2 ).

Letting a = (a1 , . . . , ar ) be in Ξ and Σ in 𝒱 , we consider the Riesz probability distribution ℛr (a, Σ) given by ℛr (a, Σ)(dX) =

e−⟨Σ,X⟩ ℛ (dX). Δa (Σ−1 ) a

For θ ∈ Σ − 𝒱 , we have Lℛr (a,Σ) (θ) =

Δa ((Σ − θ)−1 ) . Δa (Σ−1 )

(2.1)

If a = (q, . . . , q) is such that q ∈ Λ = { d2 , . . . , d2 (r − 1)} ∪ ] d2 (r − 1), +∞[, then ℛr (a, Σ) is nothing but the Wishart probability distribution with shape parameter q and scale parameter Σ. For X and Y in 𝕍, we note X◻Y = 𝕃(XY) + [𝕃(X), 𝕃(Y)] = 𝕃(XY) + 𝕃(X)𝕃(Y) − 𝕃(Y)𝕃(X).

(2.2)

Definition 2.1. Let c be an idempotent in 𝕍 and z in 𝕍(c, 21 ). The Frobenius transformation τc (z) in 𝒢 is given by τc (z) = exp(2z◻c). We define the triangular group corresponding to a Jordan frame c1 , . . . , cr by (1)

𝒯 = {τc1 (z ) ⋅ ⋅ ⋅ τcr−1 (z

(r−1)

r

r

j=1

k=l+1

)ℙ(∑ bj cj ), bj > 0, z (l) ∈ ⨁ 𝕍lk }.

Now we will use another parametrization of the symmetric cone 𝒱 of 𝕍. For r

tU = τc1 (z (1) ) ⋅ ⋅ ⋅ τcr−1 (z (r−1) )ℙ(∑ ui ci ) i=1

(2.3)

Riesz probability distributions on symmetric cones | 67

with zil =

uil , ui

i < l and z (l) = ∑rk=l+1 zlk , we obtain a bijection U 󳨃󳨀→ tU (e) from r

𝕍+ = {U = ∑ ui ci + ∑ uil , ui > 0 and uil ∈ 𝕍il } i=1

i 0, we define Jλ : D(J) ⊆ (D(ℰ ), ℰλ1/2 ) → ℋaux by Jλ u := Ju. Let ker J ⊥λ be the ℰλ -orthogonal complement of ker Jλ and let Pλ the ℰλ -orthogonal projection onto ker J ⊥λ . For λ > 0, we define a quadratic form ℰλ̌ in ℋaux by D(ℰλ̌ ) := ran J,

ℰλ̌ [Ju] := ℰλ [Pλ u]

for all u ∈ D(J).

(2.1)

Note that ℰλ̌ is indeed well-defined: if u, v ∈ D(J) and Ju = Jv then u − v ∈ ker J, and therefore Pλ (u − v) = 0, i. e., Pλ u = Pλ v. It is easy to see that ℰλ̌ is a closed, densely defined positive form. Let Ȟ λ be the positive self-adjoint operator associated with ℰλ̌ . Theorem 2.1 (Dirichlet principle, see [7, Theorem 2.1 and Lemma 2.2]). Let λ > 0, u ∈ D(J). Then (a) Pλ u ∈ D(J) and Jλ Pλ u = Jλ u. (b) ℰλ̌ [Ju] = inf{ℰλ [v] : v ∈ D(J), Jv = Ju}. Proof. Note that closed operators have closed kernels; thus ker Jλ is closed. Since u − Pλ u⊥ℰλ Pλ u, we have u − Pλ u ∈ ker Jλ . In particular, Pλ u ∈ D(J) and Jλ u = Jλ Pλ u. This proves (a). For (b), on the one hand, we have inf{ℰλ [v] : v ∈ D(J), Jv = Ju} ≤ ℰλ [Pλ u] = ℰλ̌ [Ju]. On the other hand, we get ℰλ [v] ≥ ℰλ [Pλ v]

for all v ∈ D(J),

since Pλ is a ℰλ -orthogonal projection. For v ∈ D(J) such that Jv = Ju, we obtain Pλ v = Pλ u and thus inf{ℰλ [v] : v ∈ D(J), Jv = Ju} ≥ ℰλ [Pλ u] = ℰλ̌ [Ju].

A note on traces of forms with applications to the Bessel operator

| 77

Let us now add the second ingredient to describe the trace form, namely the decomposition of a form into regular and singular part. To explain this, let Q be a densely defined positive quadratic form in ℋaux . Then Q can be uniquely decomposed into Q = Qreg +Qsing such that Qreg is the largest positive densely defined closable quadratic form dominated by Q. In particular, if Q is closable then Qreg = Q. The form Qreg is called the regular part of Q. See [19, 16] for more details on this decomposition. Theorem 2.2 (compare with [7, Theorem 2.4]). There exists a positive self-adjoint operator Ȟ in ℋaux such that Ȟ λ → Ȟ in strong resolvent sense. Furthermore, Ȟ is associated with the closure of the regular part of ℰ0̌ defined by D(ℰ0̌ ) := ran J,

ℰ0̌ [Ju] := lim ℰλ̌ [Ju] λ↓0

for all u ∈ D(J).

Proof. Since ℰλ ≤ ℰμ for 0 < λ ≤ μ, the Dirichlet principle directly yields ℰλ̌ ≤ ℰμ̌ for 0 < λ ≤ μ. Hence, as λ → 0, the family (ℰλ̌ ) is monotonically decreasing (and bounded below by 0). Thus, [14, Theorem VIII.3.11] yields existence of a positive self-adjoint operator Ȟ in ℋaux such that Ȟ λ → Ȟ in strong resolvent sense. Note that ℰ0̌ is densely defined (as J has dense range) and positive. By [19, Theorem 3.2], Ȟ is associated with the closure of the regular part of ℰ0̌ . We define ℰ ̌ to be the closure of (ℰ0̌ )reg , i. e., ℰ ̌ := (ℰ0̌ )reg such that ℰ ̌ is associated with H.̌ We will call ℰ ̌ the trace of ℰ with respect to J. Since Mosco-convergence of forms is equivalent to strong resolvent convergence of the respective operators (see, e. g., [16]), we have ℰλ̌ → ℰ ̌ in the sense of Mosco as λ ↓ 0. Note that ℰ ̌ = (ℰ0̌ )reg . Let us quote that from the definition of the regular part we ̌ Hence the domain of ℰ ̌ is the closure of ran J w. r. t. √ℰ [̌ ⋅ ]+‖ ⋅ ‖2 . have ran J ⊆ D(ℰ ). aux

̌ But even Remark 2.3. From the definition of the regular part, we have ran J ⊆ D(ℰ ). more is true. Indeed, ran J is a core for ℰ ,̌ i. e., D(ℰ )̌ is the closure of ran J w. r. t. √ℰ [̌ ⋅ ]+‖ ⋅ ‖2aux and ℰ |̌ ran J = ℰ .̌ Example 2.4. This simple example will show that ℰ ̌ is very sensitive to the choice of J. Let B be the unit ball in ℝd , ℋ := ℋaux := L2 (B), D(ℰ ) := H 1 (B), and 󵄨2

ℰ [u] := ∫󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 dx

󵄨

for all u ∈ H 1 (B).

B

We first choose J to be the identity operator on H 1 (B). In this case we obtain ℰ ̌ = ℰ , which is the classical Dirichlet form with Neumann boundary conditions. However, for the choice J: D(J) := H01 (B) → H 1 (B) given by Ju := u, we obtain ℰ ̌ = ℰ |H 1 (B) , which 0 is the classical Dirichlet form with Dirichlet boundary conditions. Here is an example which shows that the singular part of ℰ0̌ might be nontrivial.

78 | A. BenAmor et al. Example 2.5. It is well known that every element from the Sobolev space H 1 (0, 1) has a continuous representative on [0, 1]. Thus we may and shall assume that elements from H 1 (0, 1) are continuous on [0, 1]. Define ℰ in ℋ := H 1 (0, 1) by D(ℰ ) := H 1 (0, 1),

2

ℰ [u] := u(0)

for all u ∈ D(ℰ ).

Let ℋaux := L2 (0, 1) and J: H 1 (0, 1) → L2 (0, 1), Ju := u. Owing to Sobolev inequality, ℰ is bounded. Obviously, J is also bounded and ker J = {0}. Thus 2

ℰ0̌ [u] = ℰ [u] = u(0)

for all u ∈ H 1 (0, 1).

However, it is known that the latter quadratic form is not closable in L2 (0, 1). The following result states that different forms may have the same trace. Proposition 2.6. Assume that Q := ℰ |D(J) is closed. Then Q̌ = ℰ .̌ Hence ℰ and Q have the same trace. Proof. As for every u ∈ D(J) and λ > 0, we have Pλ u ∈ D(J) and Qλ [u] = ℰλ [u], we get Q̌ λ = ℰλ̌ for all λ > 0. Hence by the very construction of the trace, we obtain Q̌ = ℰ .̌ Example 2.7. Let ℰ and ℋ as in Example 2.4. We have the ℰ -orthogonal decomposition H 1 (B) = H01 (B) ⊕ Har(B) where Har(B) is the space of harmonic functions from H 1 (B). Let σ be the surface measure on Γ := 𝜕B. Consider J: Har(B) → L2 (Γ, dσ),

u 󳨃→ u|Γ .

Then D(J) is a proper subset of D(ℰ ) and J is closed. Hence ℰ |Har(B) is closed as well. Obviously, ker J = {0}. Hence, applying Green’s formula, we get 2

ℰ0̌ [Ju] = ∫ |∇u| dx = ∫ B

Γ

𝜕u u dσ 𝜕ν

for all u ∈ Har(B).

It is well known that ran J = H 1/2 (Γ). Let ψ ∈ H 1/2 (Γ) and u be the harmonic extension of ψ on B. By elliptic regularity considerations, we obtain D(ℰ )̌ = H 1/2 (Γ),

̌ ℰ [ψ] =∫ Γ

𝜕u ψ dσ. 𝜕ν

Hence, as expected, ℰ ̌ is that quadratic form related to the Dirichlet-to-Neumann operator.

A note on traces of forms with applications to the Bessel operator

| 79

Let us now drop the assumptions that ℰ is closed and that J1 is closed. We will reduce this situation to that discussed above. In order to do this, we define the form ℰ J in ℋ0 := D(J), the closure of D(J) in ℋ, by D(ℰ J ) := D(J), J

2

ℰ [u] := ℰ [u] + ‖Ju‖aux

for all u ∈ D(ℰ J ).

Then ℰ J is densely defined in ℋ0 and positive. Let 𝒟 be the completion of D(J) w. r. t. ℰ J ; in particular, 𝒟 is a Hilbert space and D(J) is dense in 𝒟. By the Cauchy–Schwarz inequality, ℰ J admits a continuous extension to 𝒟, which we will call Q. Note that Q is positive again, and Q is closed in the Hilbert space (𝒟, Q1/2 ). Since ‖Ju‖aux ≤ ℰ J [u]1/2 for all u ∈ D(J), we observe that J admits a continuous extension to (𝒟, Q1/2 ) as a contraction with dense range. We will denote this extension again by J. Thus, we are back to the situation considered above, having a closed form and a densely defined closed operator. Hence, we can construct the trace form of Q in this case by the method described at the beginning of this section.

3 Example: a discretization of the Bessel operator Let us consider the measure m defined on (0, ∞) by dm(x) = 2x4 dx. We denote by ∞

2 4

𝒟0 := {u: (0, ∞) → ℝ : u ∈ ACloc ((0, ∞)), ∫ (u (x)) x dx < ∞}, 󸀠

0

where ACloc ((0, ∞)) is the space of locally absolutely continuous functions on (0, ∞), and ℰ the Dirichlet form in L2 ([0, ∞), m) defined by ∞

2

2 4

𝒟 := 𝒟0 ∩ L ([0, ∞), m), ℰ [u] := ∫ (u (x)) x dx 󸀠

for all u ∈ 𝒟.

0

We define the Bessel operator ℒ with parameter ℒ := −

3 2

on the half-line by

1 d2 2 d − . 2 2 dx x dx

It is well known that ℰ is a regular strongly local Dirichlet form in L2 ([0, ∞), m) (hence in particular closed and densely defined). Moreover, the positive self-adjoint operator associated with the form ℰ via Kato’s representation theorem, which we denote by L,

80 | A. BenAmor et al. is given by D(L) = {u ∈ 𝒟 : u󸀠 ∈ ACloc ((0, ∞)), lim x 4 u󸀠 (x) = 0, x↓0

1 󸀠󸀠 2 󸀠 2 ℒu = − u − u ∈ L ((0, ∞), m)}, 2 x Lu = ℒu for all u ∈ D(L). Let (ak )k∈ℕ be a sequence of real positive numbers. We fix the atomic measure μ = ∑ ak δk . k∈ℕ

Thus, the support of μ is the set of natural numbers ℕ. Let us define the operator J from 𝒟 to L2 ((0, ∞), μ) = ℓ2 (ℕ, (ak )) by D(J) := {u ∈ 𝒟 : ∑ ak u(k)2 < ∞}, Ju := u|ℕ

k∈ℕ

for all u ∈ D(J).

Note that J has dense range. Obviously, ker J = {u ∈ 𝒟 : u(k) = 0 for all k ∈ ℕ}. Theorem 3.1. For u = (uk ) ∈ ℓ2 (ℕ, (ak )), set 3k 3 (k + 1)3 (u − uk )2 ∈ [0, ∞]. 2 − 3k + 1 k+1 3k k=1 ∞

Q[u] := ∑

Then the trace form ℰ ̌ of ℰ with respect to J is a closed restriction of Q. In particular, if ∑k∈ℕ ak = ∞ then D(ℰ )̌ = {u = (uk ) ∈ ℓ2 (ℕ, (ak )) : Q[u] < ∞}, ̌ = Q[u] for all u ∈ D(ℰ ). ̌ ℰ [u] Proof. For every u ∈ 𝒟, and λ > 0, let Pλ u be the orthogonal projection in the Dirichlet space (𝒟, ℰλ ) onto the ℰλ -orthogonal complement of ker J. Then Pλ u is the unique solution in 𝒟 of the boundary value problem 1 2 − (Pλ u)󸀠󸀠 − (Pλ u)󸀠 + λPλ u = 0 in (0, ∞) \ ℕ, 2 x Pλ u = u on ℕ, which is equivalent to the modified Bessel equation (Pλ u)󸀠󸀠 +

4 (P u)󸀠 − 2λPλ u = 0, x λ

which has the general solution given by (see [1, p. 362, Eq. (9.1.52)], or [13, p. 369]) Pλ u(x) = x−3/2 Ak I3/2 (x√2λ) + x −3/2 Bk K3/2 (x √2λ),

A note on traces of forms with applications to the Bessel operator

| 81

for all x ∈ [k, k + 1], k ∈ ℕ, where Ak , Bk are real constants to be determined later and I3/2 , K3/2 are two modified spherical Bessel functions of fractional order 32 given respectively by I3/2 (x√2λ) = √

sinh(x √2λ) 2 (cosh(x√2λ) − ) π x √2λ (x√2λ)

and K3/2 (x√2λ) = √

π

2x √2λ

(1 +

1

x√2λ

)e−x

√2λ

.

In the following we shall determine the trace form ℰλ̌ which is defined by D(ℰλ̌ ) := ran J,

ℰλ̌ (Ju, Jv) := ℰλ (Pλ u, Pλ v)

for all u, v ∈ 𝒟.

An integration by parts yields ∞

∞

2 4

2

4

ℰλ̌ [Ju] = ℰλ [Pλ u] = ∫ ((Pλ u) (x)) x dx + λ ∫ (Pλ u(x)) 2x dx 󸀠

0

0 k+1

k+1

2

2

= ∑ ( ∫ ((Pλ u)󸀠 (x)) x4 dx + λ ∫ (Pλ u(x)) 2x 4 dx) k∈ℕ0

k

k k+1

1 2 = ∑ ∫ (− (Pλ u)󸀠󸀠 (x)(Pλ u)(x) − (Pλ u)󸀠 (x)(Pλ u)(x) 2 x k∈ℕ 0

k

2

+ λ((Pλ u)(x)) )2x4 dx + ∑ (Pλ u)󸀠 (x)(Pλ u)(x) x 4 |k+1 k k∈ℕ0

= ∑ ((Pλ u) ((k + 1) )u(k + 1)(k + 1)4 − (Pλ u)󸀠 (k + )u(k)k 4 ) 󸀠

−

k∈ℕ

+ (Pλ u)󸀠 (1− )u(1), where (Pλ u)󸀠 (k + ) and (Pλ u)󸀠 ((k + 1)− ) are the right derivative at k and the left derivative at (k+1), respectively. Taking the boundary conditions into account, we derive for each k ∈ ℕ, (Pλ u)󸀠 (k + ) = √2λ (Ak (Pλ u)󸀠 ((k + 1)− ) = √2λ (Ak

I5/2 (k √2λ)

− Bk

k 3/2 I5/2 ((k + 1)√2λ) (k + 1)3/2

K5/2 (k √2λ)

), k 3/2 K5/2 ((k + 1)√2λ)

− Bk

(k + 1)3/2

),

82 | A. BenAmor et al. where K3/2 ((k + 1)√2λ) K3/2 (k √2λ) 1 (u(k) − u(k + 1) ), det(Mk ) (k + 1)3/2 k 3/2 I3/2 (k √2λ) I3/2 ((k + 1)√2λ) 1 (u(k + 1) Bk = − u(k) ), 3/2 det(Mk ) k (k + 1)3/2

Ak =

and

Mk =

I3/2 (k √2λ) k 3/2 ( I3/2 ((k+1)√2λ) (k+1)3/2

which yields det(Mk ) =

K3/2 (k √2λ) k 3/2 ), K3/2 ((k+1)√2λ) (k+1)3/2

((1 − 2λk(k + 1)) sinh(√2λ) − √2λ cosh(√2λ)) . (2λ)3/2 k 3 (k + 1)3

For k = 0, owing to the singularity of K3/2 at 0, we get B0 = 0 and (Pλ u)(x) = A0 x−3/2 I3/2 (x √2λ)

for all x ∈ [0, 1].

We continue our computation and achieve 4

4

ℰλ̌ [Ju] = ∑ ((Pλ u) ((k + 1) )u(k + 1)(k + 1) − (Pλ u) (k )u(k)k ) 󸀠

−

󸀠

+

k∈ℕ0

= ∑ ((Pλ u)󸀠 ((k + 1)− )u(k + 1)(k + 1)4 − (Pλ u)󸀠 (k + )u(k)k 4 ) k∈ℕ

+ (Pλ u)󸀠 (1− )u(1)

= ∑ ((Pλ u)󸀠 ((k + 1)− )u(k + 1)(k + 1)4 − (Pλ u)󸀠 (k + )u(k)k 4 ) k∈ℕ

+

u(1)2 √2λ I5/2 (√2λ) . I3/2 (√2λ)

Let us recall the asymptotic behavior at 0+ for the modified Bessel functions of the first kind, which is given by Iν (x) ∼

ν

x 1 ( ) Γ(ν + 1) 2

as x → 0

for every ν > −1. Therefore, by letting λ ↓ 0, using monotone convergence theorem for interchanging the limit and the series, and taking into account the asymptotic behavior of (Pλ u)(k) and (Pλ u)󸀠 (k + ) near λ = 0, a lengthy computation leads to ∞

lim ℰλ̌ [Ju] = ∑ 3

λ→0

k=1 ∞

k 3 (k + 1)3 2 (u(k + 1) − u(k)) (k + 1)3 − k 3

3k 3 (k + 1)3 2 (u(k + 1) − u(k)) . 2 3k − 3k + 1 k=1

=∑

A straightforward computation shows that the limit form is closable.

A note on traces of forms with applications to the Bessel operator

| 83

Let us rewrite the form ℰ0̌ as follows: ℰ0̌ [Ju] = ∑ ∑ b(k, j) k∈ℕ k∼j

2

(u(k) − u(j)) ,

∀u ∈ 𝒟,

where b(k, j) =

3 k 3 j3 , 2 |k 3 − j3 |

if |k − j| = 1 and b(k, j) = 0 otherwise.

Hence if ∑k∈ℕ ak = ∞ and since b(k, k + 1) > 0 for all k ∈ ℕ, condition (A) from [15] is fulfilled, which yields the assertion.

Bibliography [1]

[2] [3] [4] [5]

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M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55 (U. S. Government Printing Office, Washington D. C., 1964). W. Arendt, A. F. M. ter Elst, Sectorial forms and degenerate differential operators. J. Oper. Theory 67(1), 33–72 (2012). W. Arendt, A. F. M. ter Elst, The Dirichlet-to-Neumann operator on exterior domains. Potential Anal. 43(2), 313–340 (2015). W. Arendt, A. F. M. ter Elst, J. B. Kennedy, M. Sauter, The Dirichlet-to-Neumann operator via hidden compactness. J. Funct. Anal. 266(3), 1757–1786 (2014). F. Belgacem, H. BelHadjAli, A. BenAmor, A. Thabet, Robin Laplacian in the large coupling limit: convergence and spectral asymptotic. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (4) XVIII(2), 565–591 (2015). H. BelHadjAli, A. BenAmor, J. F. Brasche, Large coupling convergence with negative perturbations. J. Math. Anal. Appl. 409(1), 582–597 (2014). H. BelHadjAli, A. BenAmor, C. Seifert, A. Thabet, On the construction and convergence of traces of forms. arXiv-Preprint: 1706.08314. L. Caffarelli, L. Sylvestre, An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(8), 1245–1260 (2007). D. Daners, Non-positivity of the semigroup generated by the Dirichlet-to-Neumann operator. Positivity 18(2), 235–256 (2014). U. Freiberg, C. Seifert, Dirichlet forms for singular diffusion in higher dimensions. J. Evol. Equ. 15(4), 869–878 (2015). M. Fukushima, Dirichlet Forms and Markov Processes. North-Holland Mathematical Library, vol. 23 (North-Holland Publishing Co./Kodansha, Ltd., Amsterdam-New York/Tokyo, 1980). M. Fukushima, Y. Ōshima, M. Takeda, Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics, vol. 19 (Walter de Gruyter & Co., Berlin, 1994). M. Jeanblanc, M. Yor, M. Chesney, Mathematical Methods for Financial Markets (Springer, London, Ltd., London, 2009). T. Kato, Perturbation Theory for Linear Operators. Classics in Mathematics (Springer, Berlin, 1995). Reprint of the 1980 edition. M. Keller, D. Lenz, Dirichlet forms and stochastic completeness of graphs and subgraphs. J. Reine Angew. Math. 666, 189–223 (2012).

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[16] U. Mosco, Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123(2), 368–421 (1994). [17] O. Post, Boundary pairs associated with quadratic forms. Math. Nachr. 289(8–9), 1052–1099 (2016). [18] C. Seifert, J. Voigt, Dirichlet forms for singular diffusion on graphs. Oper. Matrices 5(4), 723–734 (2011). [19] B. Simon, A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28(3), 377–385 (1978). [20] A. F. M. ter Elst, M. Sauter, H. Vogt, A generalisation of the form method for accretive forms and operators. J. Funct. Anal. 269(3), 705–744 (2015).

Aymen Ammar, Houcem Daoud, and Aref Jeribi

Pseudospectra of multivalued linear operators Abstract: We define the pseudospectra and the essential pseudospectra of multivalued linear operators. We investigate its characteristics and study certain properties of these pseudospectra. Keywords: Multivalued linear operator, pseudospectra, Fredholm, essential pseudospectra and semi-Fredholm MSC 2010: 47A06

1 Introduction In [18], J. M. Varah was the first who introduced the notion of pseudospectrum in the field of linear operators, and subsequently it was used by different authors such as H. Landau [15], L. N. Trefethen [17], D. Hinrichsen, A. J. Pritchard [13], and E. B. Davies [11]. Thanks also to L. N. Trefethen, who developed this concept for matrices and operators, and used it to study interesting problems in mathematical physics. The pseudospectrum of a closed densely defined linear operator A for every ε > 0 is defined by 󵄩 󵄩 1 σε (A) := σ(A) ∪ {λ ∈ ℂ : 󵄩󵄩󵄩(λ − A)−1 󵄩󵄩󵄩 > }. ε We write, by convention, ‖(λ − A)−1 ‖ = ∞ if (λ − A)−1 is unbounded or nonexistent, i. e., if λ is in the spectrum σ(T). In [6, 8, 7], A. Ammar and A. Jeribi introduced the notion of essential pseudospectra of densely defined closed linear operators in Banach space defined by σw,ε (A) :=

⋂ σε (A + K),

K∈𝒦(X)

where 𝒦(X) is the subspace of compact operators from X into X. This chapter deals with a treatment of pseudospectrum and essential pseudospectrum of multivalued linear operators. Multivalued linear operators (or linear relations) have appeared in functional analysis as a necessity for considering the adjoints of nondensely defined linear differential operators (see J. von Neumann work in [19]) and the inverses of some operators, used, for example, in the study of certain Cauchy problems associated to parabolic type equations in Banach spaces (see, for example, [12]). The Aymen Ammar, Aref Jeribi, Faculty of Sciences of Sfax, Department of Mathematics, University of Sfax, Sfax, Tunisia, e-mails: [email protected], [email protected] Houcem Daoud, University of Sousse, Sousse, Tunisia, e-mail: [email protected] https://doi.org/10.1515/9783110598193-006

86 | A. Ammar et al. theory of linear relations is an important domain of research in modern mathematics which can be found in artificial intelligence, economic theory, noncooperative games, medicine, and existence of solutions for differential inclusions (see J. P. Aubin and H. Frankowska [9], E. Klein and A. C. Thompson [14] and references therein). In [5, 4, 3, 2], T. Álvarez treated some characteristics of Fredholm relations that we need to study the notion of essential pseudospectra. Our principle aim in this chapter is to study some properties of pseudospectra and essential pseudospectra of closed multivalued linear operators in Banach spaces. This chapter is organized as follows: Section 2 contains preliminary and auxiliary properties that we used to obtain the results of the other sections. The principle aim of Section 3 is to characterize the pseudospectra of a linear relation and examine some properties. In Section 4, we study the essential pseudospectra of a closed linear relation and we establish certain results for perturbations.

2 Auxiliary results and preliminaries We used the notation and terminology of the book [10]. In what follows, X, Y, Z, . . . will denote vector spaces over 𝕂 = ℝ or ℂ. A linear relation T from X to Y is an map from a subspace 𝒟(T) of X, called the domain of T, into the collection of nonempty subsets of Y verifying the condition of linearity: T(ax1 + bx2 ) = aT(x1 ) + bT(x2 ) for all nonzero scalars a, b and x1 , x2 ∈ 𝒟(T). If T is an operator then T is said to be single-valued. We denote the class of linear relations from X to Y as ℒℛ(X, Y) and write ℒℛ(X) = ℒℛ(X, X). A linear relation (or multivalued operator) T ∈ ℒℛ(X, Y) is determined by its graph, G(T), defined by G(T) = {(x, y) ∈ X × Y : x ∈ 𝒟(T), y ∈ Tx}, so T can be identified with G(T). The inverse of T is the linear relation T −1 defined as follows: G(T −1 ) = {(y, x) ∈ Y × X : (x, y) ∈ G(T)}. We infer, for 0 ≠ N ⊂ Y, T −1 (N) = {u ∈ 𝒟(T) : N ∩ Tu ≠ 0}. We obtain, in particular, for v ∈ R(T), that T −1 v = {u ∈ 𝒟(T) : v ∈ Tu}. We denote by N(T) the subspace T −1 (0) and it is called the null space of T; T is said to be injective if N(T) = 0.

Pseudospectra of multivalued linear operators | 87

The subspace T(𝒟(T)), denoted by R(T), is called the range of T, and T is called surjective if R(T) = Y. We note that α(T) := dim N(T) and β(T) := dim Y/R(T). For T, S ∈ ℒℛ(X, Y), the relation T + S is defined by G(T + S) = {(x, y) ∈ X × Y : y = s + t: (x, t) ∈ G(T) and (x, s) ∈ G(S)}. For S in ℒℛ(X, Y) and R in ℒℛ(Y, Z) where R(S)∩D(R) ≠ 0, we define the linear relation RS as follows: G(RS) = {(x, z) ∈ X × Z : (x, y) ∈ G(S) and (y, z) ∈ G(R) for some y ∈ Y}. We denote by T the closure of a linear relation T ∈ ℒℛ(X, Y) defined by G(T) := G(T). The relation T is said to be closed if T = T, or equivalently, if its graph G(T) is closed in X × Y. Also T is called closable if T is an extension of T, i. e., if Tx = Tx

∀x ∈ 𝒟(T).

The class of all closed linear relations from X to Y will be denoted by 𝒞ℛ(X, Y), and we write 𝒞ℛ(X) = 𝒞ℛ(X, X). The completion of a linear relation T ∈ ℒℛ(X, Y) is the linear relation T̃ ∈ ̃ whose graph is the completion of the graph of T where the spaces X̃ and Ỹ ℒℛ(X,̃ Y) are respectively the completion of X and Y. Letting M be a closed linear subspace X, we denote by QXM (or simply, QM ) the Y natural quotient map with null space M and domain X; QT(0) is denoted by QT , or simply Q when T is understood. The norm of T is defined as follows: ‖Tx‖ := ‖QTx‖

(for x ∈ 𝒟(T)) and ‖T‖ := ‖QT‖.

For V and U nonempty subsets of a normed space, the distance between V and U is defined by the formula d(V, U) := inf{‖v − u‖ : v ∈ V, u ∈ U}. We write d(x, U), or d(U, x), for the distance between {x} and U. We define the minimum modulus of T by γ(T) := sup{λ : ‖Tx‖ ≥ λd(x, N(T)) for x ∈ 𝒟(T)}. Further, T is said to be open if γ(T) > 0 and T is called continuous if ‖T‖ < ∞; T is called bounded if T is continuous and 𝒟(T) = X, while T is said to be compact if QTBX is compact. The class of compact linear relations from X to Y is denoted by 𝒦ℛ(X, Y), and we write 𝒦ℛ(X) = 𝒦ℛ(X, X).

88 | A. Ammar et al. Letting X and Y be normed linear spaces, and X 󸀠 (resp. Y 󸀠 ) denote the dual space of X (resp. Y), we define the adjoint relation T 󸀠 of a linear relation T ∈ ℒℛ(X, Y) by G(T 󸀠 ) = G(−T −1 ) ⊂ Y 󸀠 × X 󸀠 . ⊥

A closed linear relation T from a Banach space X into a Banach space Y is called an upper semi-Fredholm relation if α(T) < ∞ and T has a closed range, we denote T ∈ Φ+ (X, Y) or simply Φ+ ; T is said to be a lower semi-Fredholm relation if R(T) is a closed finite-codimensional subspace of Y, we denoted by T ∈ Φ− (X, Y), or simply Φ− ; T is called semi-Fredholm (resp. Fredholm) relation if T ∈ Φ+ (X) ∪ Φ− (X) (resp. T ∈ Φ+ (X) ∩ Φ− (X)). The index of T is the quantity i(T) = α(T) − β(T). Lemma 2.1 ([10, Proposition I.2.8]). Let X and Y be linear spaces and T ∈ ℒℛ(X, Y), then, for x ∈ 𝒟(T), we have the following properties: (i) y ∈ Tx ⇐⇒ Tx = y + T(0). In particular, (ii) ∈ Tx ⇐⇒ Tx = T(0). Remark 2.2. For X and Y linear spaces and T ∈ ℒℛ(X, Y), it follows from Lemma 2.1 that N(T) = {x ∈ 𝒟(T) : Tx = T(0)}. Lemma 2.3. If T ∈ ℒℛ(X, Y) where X and Y are normed spaces, then we have (i) QT is single-valued. (ii) ‖Tx‖ = d(y, T(0)) for any y ∈ Tx. (iii) ‖Tx‖ = d(Tx, T(0)) = d(Tx, 0) (for x ∈ 𝒟(T)). (iv) ‖T‖ = supx∈BX ‖Tx‖, where BX := {x ∈ X : ‖x‖ ≤ 1}. (v) γ(T) = ‖T −1 ‖−1 . (see [10, Proposition II.1.2, Proposition II.1.4, Proposition II.1.6 and Theorem II.2.5]) Lemma 2.4. Let X and Y be normed spaces and T, S ∈ ℒℛ(X, Y) such that S(0) ⊂ T(0). Then 󵄩󵄩 󵄩 󵄩󵄩(T − S)x󵄩󵄩󵄩 ≥ ‖Tx‖ − ‖Sx‖ ∀x ∈ 𝒟(T) ∩ 𝒟(S). Proof. Since {0} ⊂ S(0), one has T(0) ⊂ T(0) − S(0), therefore T(0) ⊂ T(0) − S(0) = (T − S)(0). Moreover, since S(0) ⊂ T(0), we get T(0)−S(0) ⊂ T(0), therefore (T − S)(0) = T(0) − S(0) ⊂ T(0). Hence, (T − S)(0) = T(0). Now, for x ∈ 𝒟(T − S) = 𝒟(T) ∩ 𝒟(S), 󵄩󵄩 󵄩 󵄩 󵄩 󵄩󵄩(T − S)x󵄩󵄩󵄩 = 󵄩󵄩󵄩QT−S (T − S)x󵄩󵄩󵄩 󵄩 󵄩 = 󵄩󵄩󵄩QT (T − S)x󵄩󵄩󵄩

Pseudospectra of multivalued linear operators | 89

󵄩 󵄩 = 󵄩󵄩󵄩(QT T − QT S)x󵄩󵄩󵄩 (using [10, Proposition I.4.2 (e)]) = ‖QT Tx − QT Sx‖

≥ ‖QT Tx‖ − ‖QT Sx‖ (since QT T and QT S are single-valued) ≥ ‖Tx‖ − ‖QT Sx‖.

If y ∈ Sx, then QT y ∈ QT Sx, and, using Lemma 2.3 (ii), we obtain ‖QT Sx‖ = d(QT y, 0)

= d(y, T(0)) ≤ d(y, S(0)) (as S(0) ⊂ T(0))

≤ ‖Sx‖, which implies the desired result. Lemma 2.5 ([1, Lemma 3.5]). Let T ∈ ℒℛ(X, Y) be closed and S ∈ ℒℛ(X, Y) be continuous, where X is a normed space and Y is a Banach space, such that 𝒟(S) ⊃ 𝒟(T) and S(0) ⊂ T(0), then T + S is closed. Theorem 2.6 (A consequence of Hann–Banach theorem). Let X be a normed space, then ∀x ∈ X, x ≠ 0, there exists x󸀠 ∈ X 󸀠 such that x 󸀠 (x) = ‖x‖ and ‖x󸀠 ‖ = 1. In the sequel of this section, X will denote a normed space over the complex field ℂ. Lemma 2.7 ([10, Corollary III.7.7]). Let T in ℒℛ(X) be injective and open with dense range. Then for any relation S such that 𝒟(S) ⊃ 𝒟(T), S(0) ⊂ T(0), and ‖S‖ < γ(T), we have that T + S is injective and open with dense range. Definition 2.8. For T ∈ ℒℛ(X) and λ ∈ ℂ, we write λ − T := λIX − T. Also R(λ, T) := (λ −T)−1 is said to be the resolvent of T (corresponding to λ) and Tλ := (λ − T)̃ −1 is called the complete resolvent of T. The resolvent set of T is defined by ρ(T) := {λ ∈ ℂ : Tλ is single-valued and everywhere defined}. We call the spectrum of T the set σ(T) := ℂ\ρ(T). Remark 2.9. From [10, Definitions VI.1.1] and [10, Exercise VI.1.2], for T ∈ ℒℛ(X), we have (i) ρ(T) = {λ ∈ ℂ : λ − T is open, injective with dense range on X} ̃ = {λ ∈ ℂ : Tλ is a bounded linear operator on X}. (ii) Using [10, Proposition VI.1.11], it is proved that σ(T) = σ(T 󸀠 ).

90 | A. Ammar et al.

3 Pseudospectra of a linear relation In this section, we define the pseudospectra of a multivalued linear operator and investigate some of their properties. Definition 3.1. For T ∈ ℒℛ(X) and ε > 0, the pseudospectrum of T is defined by the set 1 σε (T) = σ(T) ∪ {λ ∈ ℂ : ‖Tλ ‖ > }. ε We define the pseudoresolvent set of T by 1 ρε (T) = ℂ\σε (T) = ρ(T) ∩ {λ ∈ ℂ : ‖Tλ ‖ ≤ }. ε Remark 3.2. For ε > 0 and T ∈ ℒℛ(X), (i) observe that if T ∈ 𝒞ℛ(X) and X is complete, then T = T,̃ and we have 󵄩 󵄩 1 σε (T) = σ(T) ∪ {λ ∈ ℂ : 󵄩󵄩󵄩(λ − T)−1 󵄩󵄩󵄩 > }. ε (ii) If ε1 < ε2 then σε1 (T) ⊂ σε2 (T). Proposition 3.3. If T ∈ ℒℛ(X) then ⋂ σε (T) = σ(T).

ε>0

Proof. Since σ(T) ⊂ σε (T) ∀ε > 0, it is obvious that σ(T) ⊂ ⋂ε>0 σε (T). For the converse, if λ ∉ σ(T) then (λ − T)̃ −1 is a bounded linear operator, hence there exists ε0 > 0 such that ‖(λ − T)̃ −1 ‖ ≤ ε1 . So λ ∉ σε0 (T), and then 0

λ ∉ ⋂ σε (T). ε>0

In the sequel of this section, ε will be strictly positive real number and X will denote a Banach space over the complex field ℂ. Lemma 3.4. Let T ∈ 𝒞ℛ(X). If λ ∉ σ(T) then 󵄩 󵄩 λ ∈ σε (T) if and only if there exists x ∈ X such that 󵄩󵄩󵄩(λ − T)x 󵄩󵄩󵄩 < ε‖x‖. Proof. Let λ ∈ σε (T) be such that λ ∉ σ(T). Then ‖(λ − T)−1 ‖ > y ∈ X, y ≠ 0, such that 1 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T) y󵄩󵄩󵄩 > ‖y‖. ε

1 ε

and hence there exists

(3.1)

Pseudospectra of multivalued linear operators | 91

Let x := (λ − T)−1 y, then y ∈ (λ − T)x. It is clear that (λ − T)(0) = λ(0) − T(0) = 0 − T(0) = T(0). Using Lemma 2.1, we get 0 ∈ T(0) and, applying Lemma 2.3, obtain 󵄩 󵄩󵄩 󵄩󵄩(λ − T)x󵄩󵄩󵄩 = d(y, (λ − T)(0)) = d(y, T(0)) ≤ ‖y‖.

(3.2)

Thus from Eqs (3.1) and (3.2), we conclude 1 ‖x‖ > ‖y‖ ≥ ε

1 󵄩󵄩 󵄩 󵄩(λ − T)x 󵄩󵄩󵄩. ε󵄩

Hence 󵄩󵄩 󵄩 󵄩󵄩(λ − T)x󵄩󵄩󵄩 < ε‖x‖. For the converse, suppose that there exists x ∈ X such that 󵄩󵄩 󵄩 󵄩󵄩(λ − T)x󵄩󵄩󵄩 < ε‖x‖. Since λ ∈ ρ(T), we get that λ − T is open and injective. Thus 󵄩 󵄩 γ(λ − T)‖x‖ ≤ 󵄩󵄩󵄩(λ − T)x󵄩󵄩󵄩 < ε‖x‖. So 0 < γ(λ − T) < ε. From Lemma 2.3, we obtain 1 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩 > . ε Theorem 3.5. Let T ∈ 𝒞ℛ(X). The following assertions are equivalent: (i) λ ∈ σε (T). (ii) There exists a continuous S ∈ ℒℛ(X), satisfying S(0) ⊂ T(0), 𝒟(S) ⊃ 𝒟(T), and ‖S‖ < ε, such that λ ∈ σ(T + S). (iii) Either λ ∈ σ(T) or ‖(λ − T)−1 ‖ > ε1 .

92 | A. Ammar et al. This theorem generalizes a theorem of E. Brian Davies [11, Theorem 9.2.13]. Proof. (i) ⇒ (ii) Let λ ∈ σε (T). If λ ∈ σ(T), we may put S = 0. Now, suppose that λ ∉ σ(T), then, by Lemma 3.4, there exists x0 ∈ X, ‖x0 ‖ = 1 such that 󵄩 󵄩󵄩 󵄩󵄩(λ − T)x0 󵄩󵄩󵄩 < ε. From Theorem 2.6, there exists x󸀠 ∈ X 󸀠 such that ‖x󸀠 ‖ = 1 and x󸀠 (x0 ) = ‖x0 ‖. Consider the linear relation S : X → X defined by S(x) := x󸀠 (x)(λ − T)x0 . It is obvious that S is everywhere defined and single-valued. Moreover, 󵄩 󵄩 󵄩 󵄩 ‖Sx‖ ≤ 󵄩󵄩󵄩x󸀠 󵄩󵄩󵄩‖x‖󵄩󵄩󵄩(λ − T)x0 󵄩󵄩󵄩, and for x ≠ 0, ‖Sx‖ 󵄩󵄩 󵄩 ≤ 󵄩󵄩(λ − T)x0 󵄩󵄩󵄩. ‖x‖ Hence 󵄩 󵄩 ‖S‖ ≤ 󵄩󵄩󵄩(λ − T)x0 󵄩󵄩󵄩. So, ‖S‖ < ε. On the other hand, (λ − (T + S))x0 = (λ − T − S)x0

= (λ − T)x0 − x 󸀠 (x0 )(λ − T)x0

= (λ − T)(0)

= (λ − (T + S))(0). Hence 0 ≠ x0 ∈ N(λ − (T + S)). Then (λ − (T + S)) is not injective, i. e., λ ∈ σ(T + S). (ii) ⇒ (iii) We derive a contradiction from assumption (iii). Suppose that λ ∉ σ(T) and ‖(λ − T)−1 ‖ ≤ ε1 . Then, by Lemma 2.3, λ ∈ ρ(T) and γ(λ − T) ≥ ε, and thus, from Remark 2.9(i), λ − T is open and injective with dense range. Moreover, 𝒟(S) ⊃ 𝒟(T) = 𝒟(λ − T), S(0) ⊂ T(0) = (λ − T)(0) and ‖S‖ < ε ≤ γ(λ − T). Using Lemma 2.7, λ − T − S is open and injective with dense range. Hence λ ∈ ρ(T + S). This is a contradiction. (iii) ⇒ (i) is trivial.

Pseudospectra of multivalued linear operators | 93

Remark 3.6. It follows, from Theorem 3.5, that for T ∈ 𝒞ℛ(X), we have σε (T) =

⋃

σ(T + S).

‖S‖ γ(T) − ε}. Proof. We will discuss three cases: Case 1. If γ(T) < ε then, by Lemma 2.3 (v), ‖T −1 ‖ > ε1 , so 0 ∈ σε (T) and there is nothing to prove. Case 2. If γ(T) = ε > 0 then T is open, thus 0 ∈ ρ(T) and, moreover, ‖T −1 ‖ = ε1 . Hence 0 ∉ σε (T) and therefore σε (T) ⊂ {λ ∈ ℂ : |λ| > 0}. Case 3. If γ(T) > ε then T is injective, open, and surjective. Now, suppose that λ ∈ ℂ, such that |λ| ≤ γ(T) − ε hence |λ| < γ(T). The use of Lemma 2.7 gives that the relation λ − T is injective, open with dense range, i. e., λ ∈ ρ(T) and for x ∈ 𝒟(T), using Lemma 2.4, we obtain 󵄩󵄩 󵄩 󵄩󵄩(λ − T)x󵄩󵄩󵄩 ≥ ‖Tx‖ − |λ|‖x‖ ≥ (γ(T) − |λ|)‖x‖, hence γ(λ − T) ≥ γ(T) − |λ| ≥ γ(T) − γ(T) + ε = ε. Then 1 󵄩󵄩 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩 ≤ . ε So λ ∈ ρε (T). Therefore σε (T) ⊂ {λ ∈ ℂ : |λ| > γ(T) − ε}. Theorem 3.8. If T ∈ 𝒞ℛ(X) then σε (T) = σε (T 󸀠 ).

94 | A. Ammar et al. Proof. Let λ ∈ ρε (T). Then ‖(λ − T)−1 ‖ ≤

1 ε

and by [10, Proposition III.4.6(c)],

1 󵄩 󵄩󵄩 󸀠 −1 󵄩 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩 = 󵄩󵄩󵄩(λ − T ) 󵄩󵄩󵄩 ≤ . ε Moreover, using Remark 2.9 (iii), λ ∈ ρ(T 󸀠 ), and then λ ∈ ρε (T 󸀠 ). Conversely, if λ ∈ ρε (T 󸀠 ), then λ ∈ ρ(T 󸀠 ) and

1 󵄩󵄩 󸀠 −1 󵄩 󵄩󵄩(λ − T ) 󵄩󵄩󵄩 ≤ . ε

Using Remark 2.9 (iii), λ ∈ ρ(T). So (λ − T)−1 is continuous and, by [10, Proposition III.4.6(c)], we have 1 󵄩󵄩 󵄩 −1 󵄩 󸀠 −1 󵄩 󵄩󵄩(λ − T) 󵄩󵄩󵄩 = 󵄩󵄩󵄩(λ − T ) 󵄩󵄩󵄩 ≤ . ε Hence λ ∈ ρε (T).

4 Essential pseudospectra of a linear relation In this section, we define the essential pseudospectra of a linear relation, investigate some of their properties, and establish some results of perturbations in the context of multivalued linear operators. In the sequel, ε will be a real strictly positive number and X, Y, . . . will denote Banach spaces over the complex field ℂ. Definition 4.1. For T in ℒℛ(X), the essential pseudospectrum of T is the set defined by σw,ε (T) =

⋂

K∈𝒦T (X)

σε (T + K),

where 𝒦T (X) := {K ∈ 𝒦ℛ(X) : K(0) ⊂ T(0) and 𝒟(K) ⊃ 𝒟(T)}, and we denote the essential pseudoresolvent set of T by ρw,ε (T) = ℂ\σw,ε (T). Theorem 4.2. Let T ∈ 𝒞ℛ(X), then the following assertions are equivalent: (i) λ ∉ σw,ε (T). (ii) For all continuous linear relations S in ℒℛ(X) such that S(0) ⊂ T(0), 𝒟(S) ⊃ 𝒟(T) and ‖S‖ < ε, T + S − λ ∈ Φ(X) and

i(T + S − λ) = 0.

Pseudospectra of multivalued linear operators | 95

(iii) For all continuous linear operators D in ℒℛ(X) such that ‖D‖ < ε and 𝒟(D) ⊃ 𝒟(T), T + D − λ ∈ Φ(X) and

i(T + D − λ) = 0.

Proof. (i) ⇒ (ii) Let λ ∉ σw,ε (T). Then there exists K ∈ 𝒦T (X) such that λ ∈ ρε (T + K). Letting S be in ℒℛ(X) such that S(0) ⊂ (T + K)(0) = T(0), 𝒟(S) ⊃ 𝒟(T + K) = 𝒟(T) ∩ 𝒟(K) = 𝒟(T) and ‖S‖ < ε, by Theorem 3.5, we have λ ∈ ρ(T + S + K). Hence T + S + K − λ is injective and open with dense range. Also K is compact then K is continuous, so S + K − λ is continuous. Moreover, (S + K − λ)(0) ⊂ T(0), so, using the fact that T is closed, from Lemma 2.5, we conclude that T + S + K − λ is closed and, from [10, Theorem III.4.2], obtain that R(T + S + K − λ) is closed. Hence R(T + S + K − λ) = X. Therefore T + S + K − λ ∈ Φ(X) and i(T + S + K − λ) = 0. It follows from [1, Lemma 3.6] that for all continuous linear relations S in ℒℛ(X) such that S(0) ⊂ T(0), 𝒟(S) ⊃ 𝒟(T), and ‖S‖ < ε, we get T + S − λ ∈ Φ(X)

and i(T + S − λ) = 0.

(ii) ⇒ (iii) Trivial. (iii) ⇒ (i) Assume that for all continuous operators D in ℒℛ(X) such that ‖D‖ < ε and 𝒟(D) ⊃ 𝒟(T), T + D − λ ∈ Φ(X) and i(T + D − λ) = 0. From [10, Proposition III.1.4(a)], we have N((T + D − λ)󸀠 ) = R(T + D − λ)⊥ . We put n = α(T + D − λ) = β(T + D − λ) and let {y1󸀠 , . . . , yn󸀠 } be a basis for N((T + D − λ)󸀠 ) and {x1 , . . . , xn } be a basis for N(T + D − λ). From [16, Theorems I.2.5, I.2.6], there are elements y1 , . . . , yn and functionals 󸀠 x1 , . . . , xn󸀠 ∈ X 󸀠 such that yj󸀠 (yk ) = δjk

and

xj󸀠 (xk ) = δjk ,

1 ≤ j, k ≤ n.

Note that δjk = 1 if j = k and δjk = 0 if j ≠ k. We define the operator (single-valued relation) K by n

Kx = ∑ xk󸀠 (x)yk , k=1

x ∈ X.

Since 𝒟(K) = X and ‖Kx‖ ≤ ‖x‖(∑nk=1 ‖xk󸀠 ‖‖yk ‖), it is clear that K is bounded. Further, since K is a finite-rank relation in X, by [10, Proposition V.1.3], K is compact. We shall prove that N(T + D − λ) ∩ N(K) = {0} and

R(T + D − λ) ∩ R(K) = {0}.

(4.1)

96 | A. Ammar et al. In fact, if x ∈ N(T + D − λ), then n

x = ∑ αk xk , k=1

therefore, xj󸀠 (x) = αj , 1 ≤ j ≤ n. Now, if x ∈ N(K) then Kx = 0, hence xj󸀠 (x) = 0, 1 ≤ j ≤ n. This proves the first inclusion in Eq. (4.1). The second relation is similar. In fact, let y ∈ R(K). Then n

y = ∑ αk yk , k=1

so yj󸀠 (y) = αj ,

1 ≤ j ≤ n.

yj󸀠 (y) = 0,

1 ≤ j ≤ n.

If y ∈ R(T + D − λ), we have

This gives the second inclusion in Eq. (4.1). On the other hand, since 𝒦T (X) =

𝒦T+D−λ (X), we obtain, from [1, Lemma 3.6],

T + D + K − λ ∈ Φ(X)

and

i(T + D + K − λ) = 0.

Let x ∈ N(T +D+K −λ), then −Kx ∈ (T +D−λ)x and hence Kx ∈ R(K)∩R(T +D−λ) = {0}. Thus, Kx = 0 and 0 ∈ (T + D − λ)x, which implies that x ∈ N(T + D − λ) ∩ N(K), and then x = 0. Therefore α(T + D + K − λ) = 0. In a similar way, we prove that R(T +D+K −λ) = X. Using Lemma 2.5, T +D+K −λ is closed and, from [10, Theorem III.4.2], we have that T + D + K − λ is open. Furthermore, it is injective with dense range, thus λ ∈ ρ(T + D + K). The proof of Theorem 3.5 ((i) ⇒ (ii)) gives that if λ ∈ σε (T + K) there exists a continuous single-valued relation D ∈ ℒℛ(X) satisfying 𝒟(D) ⊃ 𝒟(T + K) = 𝒟(T) and ‖D‖ < ε such that λ ∈ σ(T + K + D). This is absurd. Thus λ ∉ σε (T + K) and hence λ ∉ σw,ε (T). Remark 4.3. It follows, from Theorem 4.2, that for T ∈ 𝒞ℛ(X), σw,ε (T) =

⋃ ‖S‖0 σw,ε(T) = σw (T).

Pseudospectra of multivalued linear operators | 97

Proof. (i) Let 0 < ε1 < ε2 . If λ ∉ σw,ε2 (T) then, by Theorem 4.2, for all continuous linear relations S in ℒℛ(X) such that S(0) ⊂ T(0), 𝒟(S) ⊃ 𝒟(T), and ‖S‖ < ε2 , T + S − λ ∈ Φ(X)

and i(T + S − λ) = 0.

Let S ∈ ℒℛ(X) be such that S(0) ⊂ T(0), 𝒟(S) ⊃ 𝒟(T), and ‖S‖ < ε1 . Then T + S − λ ∈ Φ(X)

and i(T + S − λ) = 0.

Hence λ ∉ σε1 (T). Furthermore, if λ ∉ σw,ε1 (T), then, for all continuous linear relations S in ℒℛ(X) such that S(0) ⊂ T(0), 𝒟(S) ⊃ 𝒟(T), and ‖S‖ < ε1 , we have T + S − λ ∈ Φ(X)

and i(T + S − λ) = 0.

In particular, we put S = 0, and thus λ ∉ σw (T). (ii) σw,ε (T) =

⋂

K∈𝒦T (X)

σε (T + K) ⊂ σε (T + K)

∀K ∈ 𝒦T (X).

In particular, we put K = 0, then σw,ε (T) ⊂ σε (T). (iii) Assertion (i) gives that σw (T) ⊂ σw,ε (T) ∀ε > 0, and then σw (T) ⊂ ⋂ σw,ε (T). ε>0

For the converse, let λ ∉ σw (T). Then T − λ ∈ Φ(X) and i(T − λ) = 0. Thus, R(T − λ) is closed and, using [10, Theorem III.4.2], we have that T − λ is open; hence γ(T − λ) > 0. Consider ε0 such that 0 < ε0 ≤ γ(T − λ) and let D be a linear operator in ℒℛ(X) such that ‖D‖ < ε0 ≤ γ(T − λ) and 𝒟(D) ⊃ 𝒟(T). From [10, Theorem III.4.6(d)], γ(T − λ) = γ((T − λ)󸀠 ), and then ‖D‖ < γ((T − λ)󸀠 ). So from [10, Theorem V.5.12], we have T + D − λ ∈ Φ− (X) and by [10, Theorem III.7.4], α(T+D−λ) ≤ α(T−λ) < ∞, so, using [10, Proposition V.5.13], we obtain T+D−λ ∈ Φ+ (X), hence T + D − λ ∈ Φ(X). Moreover, by [10, Corollary V.15.7], we obtain i(T + D − λ) = 0. Then, using Theorem 4.2, λ ∉ σw,ε0 (T), and hence λ ∉ ⋂ε>0 σw,ε (T). Theorem 4.5. If T ∈ 𝒞ℛ(X) then σw,ε (T) = σw,ε (T 󸀠 ).

98 | A. Ammar et al. Proof. Since K ∈ 𝒦T (X), K is compact, therefore, by [10, Corollary V.2.3], K is continuous and, by [10, Proposition V.5.3], K 󸀠 is compact. Moreover, since 𝒟(K) ⊃ 𝒟(T), by [10, Proposition III.1.4(b)], K 󸀠 (0) = 𝒟(K)⊥ ⊂ T 󸀠 (0) = 𝒟(T)⊥ , and since K is continuous and K(0) ⊂ T(0), by [10, Proposition III.4.6(a)] and [10, Proposition III.1.4(d)], it follows that 𝒟(K 󸀠 ) = K(0)⊥ ⊃ T(0)⊥ = 𝒟(T 󸀠 ) ⊃ 𝒟(T 󸀠 ). Thus 𝒦T (X) ⊂ {K ∈ ℒℛ(X) : K is continuous and K ∈ 𝒦T 󸀠 (X )}. 󸀠

󸀠

Consider K ∈ {K ∈ ℒℛ(X) : K is continuous K 󸀠 ∈ 𝒦T 󸀠 (X 󸀠 )}, then, using [10, Proposition V.5.3], K is compact. Since K 󸀠 (0) = 𝒟(K)⊥ ⊂ T 󸀠 (0) = 𝒟(T)⊥ , by [10, Proposition III.1.4(b)], 𝒟(K) ⊃ 𝒟(T). Furthermore, using [10, Proposition III.1.4(d)], K(0) ⊂ K(0) = 𝒟(K 󸀠 )⊤ ⊂ 𝒟(T 󸀠 )⊤ = T(0) (as T is closed). Thus K ∈ 𝒦T (X). Then, it follows that 𝒦T (X) = {K ∈ ℒℛ(X) : K is continuous and K ∈ 𝒦T 󸀠 (X )}. 󸀠

󸀠

(4.2)

Moreover, for K ∈ 𝒦T (X), using Lemma 2.5, we have that T + K is closed and, by Theorem 3.8, obtain σε (T + K) = σε ((T + K)󸀠 ). Now, we have that K is continuous and 𝒟(K) ⊃ 𝒟(T). Therefore, applying [10, Proposition III.1.5(b)], we have (T + K)󸀠 = T 󸀠 + K 󸀠 , and thus σε (T + K) = σε (T 󸀠 + K 󸀠 ) ∀K ∈ 𝒦T (X). Therefore, by the use of Eq. (4.2), we have σw,ε (T) =

⋂

K∈ℒℛ(X)

σε (T 󸀠 + K 󸀠 ) ⊃

K 󸀠 ∈𝒦T 󸀠 (X 󸀠 ) K continuous

⋂

K∈ℒℛ(X)

σε (T 󸀠 + K 󸀠 ).

(4.3)

K 󸀠 ∈𝒦T 󸀠 (X 󸀠 )

Putting ℱ :=

⋂

K∈ℒℛ(X)

σε (T 󸀠 + K 󸀠 ),

K 󸀠 ∈𝒦T 󸀠 (X 󸀠 ) K continuous

we infer that ℱ = σw,ε (T) ⊃ σw,ε (T ). 󸀠

(4.4)

Now, let λ ∈ ℱ . Then for all K ∈ ℒℛ(X), K continuous and such that K 󸀠 ∈ 𝒦T 󸀠 (X), λ ∈ σε (T 󸀠 + K 󸀠 ).

Pseudospectra of multivalued linear operators | 99

Consider K ∈ ℒℛ(X) such that K 󸀠 ∈ 𝒦T 󸀠 (X). Then, by [10, Corollary V.5.15], K̃ is compact where G(K)̃ is the completion of the graph of K in X × X and, since X is a complete space, we have G(K)̃ = G(K) = G(K). Then K̃ = K is compact, thus it is continuous. ⊥ 󸀠 −1 Moreover, G(K ) = G(−K )⊥ = G(−K −1 ) = G(−K −1 )⊥ = G(K 󸀠 ). This implies that 󸀠 󸀠 K = K 󸀠 . We conclude that K is in ℒℛ(X), K is continuous, and K = K 󸀠 ∈ 𝒦T 󸀠 (X 󸀠 ), but λ ∈ ℱ , and then 󸀠

λ ∈ σε (T 󸀠 + K ) = σε (T 󸀠 + K 󸀠 ). We obtain that, if λ ∈ ℱ then for all K ∈ ℒℛ(X) such that K 󸀠 ∈ 𝒦T 󸀠 (X), λ ∈ σε (T 󸀠 + K 󸀠 ). Hence λ∈

⋂

K∈ℒℛ(X)

σε (T 󸀠 + K 󸀠 ) = σw,ε (T 󸀠 ).

K 󸀠 ∈𝒦T 󸀠 (X 󸀠 )

Then ℱ ⊂ σw,ε (T ). 󸀠

Finally, using Eq. (4.4), we deduce that ℱ = σw,ε (T) = σw,ε (T ). 󸀠

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F. Abdmouleh, T. Álvarez, A. Jeribi, On a characterization of the essential spectra of a linear relation. Preprint (2014). T. Álvarez, On almost semi-Fredholm linear relations in normed spaces. Glasg. Math. J. 47(1), 187–193 (2005). T. Álvarez, Linear relations on hereditarily indecomposable normed spaces. Bull. Aust. Math. Soc. 84(1), 49–52 (2011). T. Álvarez, R. W. Cross, D. Wilcox, Multivalued Fredholm type operators with abstract generalised inverses. J. Math. Anal. Appl. 261(1), 403–417 (2001). T. Álvarez, A. Ammar, A. Jeribi, On the essential spectra of some matrix of linear relations. Math. Methods Appl. Sci. (2013). A. Ammar, A. Jeribi, A characterization of the essential pseudospectra on a Banach space. Arab. J. Math. 2, 139–145 (2013). A. Ammar, A. Jeribi, A characterization of the essential pseudospectra and application to a transport equation. Extr. Math. 28, 95–112 (2013).

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A. Ammar, A. Jeribi, Measures of noncompactness and essential pseudospectra on Banach space. Math. Methods Appl. Sci. 37(3), 447–452 (2014). J. P. Aubin, H. Frankowska, Set-Valued Analysis (Birkhäuser, Boston, 1990). R. W. Cross, Multivalued Linear Operators (Marcel Dekker, New York, 1998). E. B. Davies, Linear Operators and Their Spectra. United States of America by (Cambridge University Press, New York, 2007). A. Favini, A. Yagi, Multivalued linear operators and degenerate evolution equations. Ann. Mat. Pura Appl. 353–384 (1993). D. Hinrichsen, A. J. Pritchard, Robust stability of linear operators on Banach spaces. J. Control Optim. 32, 1503–1541 (1994). E. Klein, A. C. Thompson, Theory of Correspondences: Including Applications in Mathematical Economics (John Wiley and Sons, 1984). H. J. Landau, On Szego’s eigenvalue distribution theorem and non-Hermitian kernels. J. Anal. Math. 28, 335–357 (1975). M. Shechter, Spectra of Partial Differential Operators (North-Holland, Amsterdam, New York, 1986). L. N. Trefethen, Pseudospectra of matrices, in Numerical Analysis 1991, Dundee, 1991. Pitman Res. Notes Math. Ser., vol. 260 (Longman Sci. Tech., Harlow, 1992), pp. 234–266. J. M. Varah, The computation of bounds for the invariant subspaces of a general matrix operator. Thesis (Ph. D.)-Stanford, University. ProQuest LLC, Ann Arbor, MI (1967). J. von Neumann, Über adjungierte Funktionaloperatoren. Ann. Math. 33, 294–310 (1932).

S. Charfi and H. Ellouz

On the frame properties of exponentials associated to analytic families of operators and application Abstract: In the present paper, we are mainly concerned with the existence of frames of exponential families associated to the operator 1

d4 d4 d4 2 (I + εK) + ε(I + εK)−1 K( 4 − ( 4 ) ) 4 dx dx dx −1

in L2 (−T, T), T > 0, where K is the integral operator with kernel being the Hankel function of the first kind and order 0. Via a specific growth inequality, we extend this problem to a theoretical one and study the existence of frame of exponentials. Keywords: Frame of exponentials, eigenvalues, elastic membrane, integro-differential operator MSC 2010: 37C75, 37B25

1 Introduction Riesz bases of complex exponentials in the space L2 (−T, T), where T > 0, have been thoroughly explored since the possibility of nonharmonic Fourier expansions was discovered by Paley and Wiener [17]. The first main idea developed is based on the fact that the trigonometric system {eint }n∈ℤ is stable under “sufficiently small” perturbations of the integers. Much improvement has been made subsequently by many mathematicians, and the basic result is the necessary and sufficient condition for a basis obtained by Pavlov [18]. So, based on the results developed in [18], the authors in [6] established the existence of a Riesz basis of a family of exponentials where the exponents coincide with the eigenvalues of the integro-differential operator introduced by Filippi et al. [11] and considered in some valuable papers [6, 7, 8, 9, 10], [13] and [12], namely 1

d4 d4 d4 2 −1 (I + εK) + ε(I + εK) K( − ( ) ) dx4 dx4 dx4 −1

(1.1)

in L2 (0, 4a) for some a > 0, where K is the integral operator with the kernel being the Hankel function of the first kind and order 0. S. Charfi, National School of Electronics and Telecommunications of Sfax, University of Sfax, Sfax, Tunisia, e-mail: [email protected] H. Ellouz, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia, e-mail: [email protected] https://doi.org/10.1515/9783110598193-007

102 | S. Charfi and H. Ellouz More precisely, they have proved that if the family of exponentials related to the d4 2 differential operator dx 4 is a Riesz basis in L (0, 4a), where the exponents coincide 4

d with the eigenvalues of dx 4 which are simple and isolated, then there exists a fixed complex number ε such that for |ε| small enough the family of exponentials related to the operator (1.1) forms a Riesz basis in L2 (0, 4a). So, what happens if we change the interval (0, 4a)? Indeed, the situation is more complicated since the Riesz basis properties are not satisfied. Indeed, if we reduce the interval (0, 4a), we have a serious problem for the completeness of the family of exponentials. However, in a separable Hilbert space ℋ, a frame can be viewed as an overcomplete set. In fact, the theory of frames is a useful tool to expand functions with respect to a system of functions which is, in general, nonorthogonal and overcomplete. Furthermore, the redundancy property of frames is important for many problems related to operators such as in (1.1). It is interesting to remember that Duffin and Schaeffer [5] actually introduced this concept in the context of systems of complex exponentials. The starting point was the possibility of nonharmonic Fourier expansions, as discovered by Paley and Wiener [17]. Much amelioration has been made subsequently [4, 12, 19]. Let us recall that a family {φn }n∈I is said to be a frame for a separable Hilbert space ℋ if there exist positive constants A, B > 0 such that

󵄨 󵄨2 A‖φ‖2 ≤ ∑ 󵄨󵄨󵄨⟨φ, φn ⟩󵄨󵄨󵄨 ≤ B‖φ‖2 , n∈I

for all φ ∈ ℋ,

where I a countable index set. The numbers A and B are called lower and upper frame bounds. Clearly, one can see that a sequence of vectors is a Riesz basis if and only if it is a frame and ω-linearly independent. Now, and in order to prove the existence of frames of exponential families related to Eq. (1.1), we extend this problem to a theoretical one and consider the following operator introduced by Nagy [16]: T(ε) = T0 + εT1 + ε2 T2 + ⋅ ⋅ ⋅ + εk Tk + ⋅ ⋅ ⋅ ,

(1.2)

where ε ∈ ℂ and T0 is a closed densely defined linear operator on a separable Hilbert space ℋ with domain 𝒟(T0 ), while T1 , T2 , . . . are linear operators on ℋ having the same domain 𝒟 ⊃ 𝒟(T0 ) and satisfying the following growth inequality given by Abdelmoumen et al. in [1]: N

‖Tk φ‖ ≤ qk−1 ∑ bi ‖T0 φ‖βi ‖φ‖1−βi i=1

for all φ ∈ 𝒟(T0 ) and k ≥ 1,

(1.3)

where q, b1 , b2 , . . . , bN are positive constants and {β1 , β2 , . . . , βN } ⊂ ]0, 1], with βi ≠ βj , for all i, j ∈ 1, . . . , N. So, based on the inequality (1.3), we extend the results developed in [3] and [6] to the notion of frames. More precisely, based on the results developed in [2], we prove

On the frame properties of exponentials associated to analytic families | 103

the existence of a sequence of complex numbers (εn )n∈ℕ∗ such that the family of exponentials {eiλn (εn )t }n∈ℕ∗ forms a frame for L2 (−T, T). Here, (λn (ε))n∈ℕ∗ are the eigenvalues of the perturbed operator T(εn ) which can be developed as the following entire series: λn (εn ) = λn + εn λn,1 + εn2 λn,2 + ⋅ ⋅ ⋅ . However, the frame of exponentials hence obtained depends on a sequence of complex numbers (εn )n∈ℕ∗ . Further, it is related to a family of operators (T(εn ))n∈ℕ∗ . In this context and in order to improve our result, we try to find a fixed complex number ε such that the associated family of exponentials is a frame in L2 (−T, T). Moreover, we improve these results by means of H-Lipschitz functions. We organize this paper as follows: In Section 2, we concentrate ourselves exclusively on proving the existence of frames of exponentials associated to the perturbed operator (1.2). In the last section, we give an application to our obtained results.

2 Main results In this section, let us consider the following eigenvalue problem: {

T0 φ + εT1 φ + ε2 T2 φ + ⋅ ⋅ ⋅ + εk Tk φ + ⋅ ⋅ ⋅ = λφ, φ ∈ 𝒟(T0 ),

where ε ∈ ℂ, T0 is a linear operator acting on a separable Hilbert space ℋ and verifying: (H1) T0 is closed with domain 𝒟(T0 ) dense in ℋ; (H2) The eigenvalues (λn )n of T0 are isolated, with multiplicity one, and ∀n ≥ 1,

| Im λn | ≤ h where h := sup | Im λn |; n

2 (H3) The family of exponentials {eiλn t }∞ 1 forms a frame in L (−T, T), for some T > 0 with frame bounds A and B; while T1 , T2 , T3 , . . . are linear operators on ℋ having the same domain 𝒟 and satisfying the hypothesis: (H4) 𝒟 ⊃ 𝒟(T0 ) and there exist q, b1 , b2 , . . . , bN which are positive constants and {β1 , β2 , . . . , βN } ⊂ ]0, 1], with βi ≠ βj , for all i, j ∈ 1, . . . , N, such that for all k ≥ 1, N

‖Tk φ‖ ≤ qk−1 ∑ bi ‖T0 φ‖βi ‖φ‖1−βi i=1

for all φ ∈ 𝒟(T0 ).

In this section, we are interested in the study of the frame properties of a family of nonharmonic exponentials associated to the above eigenvalue problem. Let us begin with the following result from [2].

104 | S. Charfi and H. Ellouz Theorem 2.1 ([2, Theorem 3.1]). Assume that assumptions (H1) and (H4) hold. (i) For |ε| < q−1 , the series ∑k≥0 εk Tk φ converges for all φ ∈ 𝒟(T0 ). If T(ε)φ denotes its limit, then T(ε) is a linear operator with domain 𝒟(T0 ). (ii) For |ε| < (q + ∑Ni=1 βi bi )−1 , the operator T(ε) is closed. In the following, we designate by λn the nth eigenvalue of the operator T0 . Since (T0 − zI)−1 is an analytic function of z, ‖(T0 − z)−1 ‖ is a continuous function of z. Hence, let 󵄩 󵄩 Mn := max󵄩󵄩󵄩(T0 − z)−1 󵄩󵄩󵄩, z∈𝒞n

󵄩 󵄩 󵄩 󵄩 Nn := max󵄩󵄩󵄩T0 (T0 − z)−1 󵄩󵄩󵄩 = max󵄩󵄩󵄩I + z(T0 − z)−1 󵄩󵄩󵄩, z∈𝒞n

z∈𝒞n

where 𝒞n = 𝒞 (λn , rn ) is the circle with center λn and radius rn = 21 dist(λn , σ(T0 ) \ {λn }). Now, we state our first result. Theorem 2.2. Under assumptions (H1)–(H4), there exists a sequence of complex numbers (εn )n∈ℕ∗ such that for |εn |
0. Let S ∈ ℒ(H) be such that S ≠ 0 and S ≠ A + D, for all D ∈ ℒ(H) with ‖D‖ ≤ ε, then ⋃ σS (A + D) ⊂ ΣS,ε (A).

(3.8)

‖D‖≤ε

Proof. Let us assume that λ ∈ ⋃‖D‖≤ε σS (A + D). Then, there exists D ∈ ℒ(X) such that ‖D‖ ≤ ε and λ ∈ σS (A+D). We derive a contradiction from the assumption that λ ∈ ρS (A) and ‖(λS − A)−1 ‖ < ε1 . For λ ∈ ρS (A), we can write λS − A − D = (λS − A)(I − (λS − A)−1 D).

(3.9)

Since 󵄩󵄩 −1 󵄩 󵄩 −1 󵄩 󵄩󵄩(λS − A) D󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(λS − A) 󵄩󵄩󵄩‖D‖ ε < ε = 1, by using (a) of Proposition 2.2, we infer that I − (λS − A)−1 D is invertible. By referring to (3.9), we conclude that λS − A − D is invertible. This is equivalent to saying that λ ∈ ρS (A + D). Remark 3.7. Since ⋃‖D‖ M − ε.

(3.15)

In view of (3.14) and (3.15), one obtains 󵄩󵄩 −2 󵄩󵄩2 −2 󵄩󵄩SA x0 󵄩󵄩 < εr .

(3.16)

Consequently, by referring to (3.16), we have −1 󵄩󵄩 −1 󵄩 󵄩 󵄩2 󵄩 1 = ‖x0 ‖2 ≤ 󵄩󵄩󵄩(SA−2 ) 󵄩󵄩󵄩󵄩󵄩󵄩SA−2 x0 󵄩󵄩󵄩 < 󵄩󵄩󵄩(SA−2 ) 󵄩󵄩󵄩εr −2 ,

which is impossible if ε > 0 is sufficiently small. This contradiction shows that ‖(λS − A)−1 ‖ < M, for all λ ∈ U.

128 | A. Ammar et al. Remark 3.9. (a) Theorem 3.10 is a generalization of [4, Proposition 6.1]. (b) In Proposition 3.8, we proved that the resolvent of an operator cannot have constant norm on an open set. Theorem 3.10. Let ε > 0 and A, S ∈ ℒ(H) be such that S is invertible, S ≠ A, and SA = AS. Then, ΣS,ε (A) ⊆ clos( ⋃ σS (A + D)), ‖D‖ ε

for all λ ∈ U,

where U is an open neighborhood of λ. This implies that there exists y such that ‖y‖ = 1 and ‖(λS − A)−1 y‖ > ε1 . Put 󵄩 󵄩−1 x = 󵄩󵄩󵄩(λS − A)−1 y󵄩󵄩󵄩 (λS − A)−1 y. Then, x ∈ H, ‖x‖ = 1, and 󵄩󵄩 󵄩 󵄩 −1 󵄩−1 󵄩󵄩(λS − A)x󵄩󵄩󵄩 = 󵄩󵄩󵄩(λS − A) y󵄩󵄩󵄩 < ε. Consequently, there exists x ∈ H such that ‖x‖ = 1 and ‖(λS − A)x‖ < ε. It follows from Hahn-Banach theorem that there exists x󸀠 ∈ X 󸀠 such that ‖x 󸀠 ‖ = 1 and x 󸀠 (x) = 1. We consider the following linear operator: D(x) := x󸀠 (x)(λS − A)x. Let us observe that 󵄩󵄩 󵄩 󵄩 󸀠󵄩 󵄩 󵄩 󵄩󵄩D(x)󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩x 󵄩󵄩󵄩‖x‖󵄩󵄩󵄩(λS − A)x󵄩󵄩󵄩 < ε‖x‖,

Some results of S-pseudospectra of bounded operators in a Hilbert space |

129

thus we have ‖D‖ < ε and D is everywhere defined. Therefore, D is bounded. Moreover, we have (λS − A − D)x0 = 0,

for ‖x0 ‖ = 1.

Hence, λ ∈ σS (A + D), and we can deduce that λ ∈ clos(⋃‖D‖ 0 and A, S ∈ ℒ(H) be such that S is invertible, S ≠ A, and SA = AS. Then, ΣS,ε (A) = clos( ⋃ σS (A + D)). ‖D‖ 0. 0

The following lemmas are useful to study the different types of decay rates. Lemma 2.1 ([2]). Assuming that the boundary conditions (1.2) are satisfied, we have the following inequalities: v2 (ξ , τ) ≤ ϑ‖vξ ‖22 ,

v2 (ξ , τ) ≤ ϑ3 ‖vξξ ‖22 ,

vξ2 (ξ , τ) ≤ ϑ‖vξξ ‖22 ,

and ‖v‖22 ≤ ϑ2 ‖vξ ‖22 ≤ ϑ4 ‖vξξ ‖22 where ‖ ⋅ ‖2 denotes the norm in L2 (0, ϑ).

∀ξ ∈ [0, ϑ],

∀ξ ∈ [0, ϑ],

134 | A. Berkani et al. Lemma 2.2. We have the following inequality: αβ ≤ ϵα2 +

β2 , 4ϵ

α, β ∈ ℝ,

ϵ > 0.

Lemma 2.3 ([3]). The energy ℰ (τ) of the system (1.1)–(1.2) given by τ

2ℰ (τ) = ρ‖vτ ‖22 + μ(1 − ∫ ψ(ς) dς)‖vξξ ‖22 + T‖vξ ‖22 + Mς vτ2 (ϑ, τ) 0

(2.2)

ϑ

+ μ ∫(ψ◊vξξ ) dξ , 0

satisfies ϑ

μ μ d 󸀠 2 ℰ (τ) = − ψ(τ)‖vξξ ‖2 + ∫(ψ ◊vξξ ) dξ dt 2 2

(2.3)

0

for all τ ≥ 0. We remark that the derivative of the energy functional ℰ (τ) is negative. Now, we need to construct a Lyapunov functional Θ verifying d Θ(τ) ≤ −c0 δ(τ)Θ(τ), dt

τ ≥ 0,

where c0 > 0 and δ(τ) is a decreasing function to obtain the arbitrary decay of Θ(τ). To pass to ℰ (τ), we will need some equivalence between ℰ (τ) and Θ(τ). To this end, we set 3

Θ(τ) = ℰ (τ) + ∑ λi Θi (τ), i=1

τ ≥ 0,

(2.4)

where λi , i = 1, . . . , 4 are positive constants, and ϑ

Θ1 (τ) = ρ ∫ vτ w dx + Mς vτ (ϑ, τ)v(ϑ, τ),

τ ≥ 0,

0 ϑ

τ

Θ2 (τ) = − ρ ∫ vτ ∫ ψ(τ − ς)(v(τ) − v(ς)) 0

0

τ

− Mς vτ (ϑ, τ) ∫ ψ(τ − ς)(v(ϑ, τ) − v(ϑ, ς)) dς, τ

0

󵄩 󵄩2 Θ3 (τ) =μ ∫ Hγ (τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς, 0

τ ≥ 0,

τ ≥ 0,

Stabilization of a marine riser system by the use of viscoelastic material | 135

where +∞

Hφ (τ) = φ(τ)−1 ∫ ψ(ς)φ(ς) dς,

τ ≥ 0,

τ

and φ(τ) will be chosen later (see (H3) below). The next proposition tells us that Θ(τ) and ℰ (τ) + Θ3 (τ) are equivalent. Proposition 2.4. There exist ρi > 0, i = 1, 2 such that ρ1 (ℰ (τ) + Θ3 (τ)) ≤ Θ(τ) ≤ ρ2 (ℰ (τ) + Θ3 (τ)),

(2.5)

for all τ ≥ 0. Proof. By using Lemmas 2.2 and 2.1, we get Θ1 (τ) ≤ or

Mς 2 Mς 2 ρ ρϑ4 ‖vτ ‖22 + ‖vξξ ‖22 + v (ϑ, τ) + v (ϑ, τ), 2 2 2 2 τ

Θ1 (τ) ≤ For Θ2 (τ), we have

Mς 2 ρ ϑ3 ‖vτ ‖22 + (ρϑ + Mς ) ‖vξξ ‖22 + v (ϑ, τ), 2 2 2 τ

τ ≥ 0,

τ ≥ 0.

(2.6)

ϑ

Mς 2 ρ ρϑ4 κ Θ2 (τ) ≤ ‖vτ ‖22 + v (ϑ, τ) ∫(ψ◊vξξ ) dξ + 2 2 2 τ +

Mς κ 2

τ

0

2

∫ ψ(τ − ς)(v(ϑ, ς) − v(ϑ, τ)) dς,

τ ≥ 0.

0

In view of Lemma 2.1, we get τ

2

ϑ

∫ ψ(τ − ς)(v(ϑ, ς) − v(ϑ, τ)) dς ≤ ϑ3 ∫(ψ◊vξξ ) dξ 0

0

Therefore, ϑ

Θ2 (τ) ≤

Mς 2 ρ κϑ3 ‖vτ ‖22 + (ρϑ + Mς ) v (ϑ, τ), ∫(ψ◊vξξ ) dξ + 2 2 2 τ

τ ≥ 0.

0

Now, from (2.2) and also using (2.6) and (2.7), we have ϑ

ρ 1 Θ(τ) ≤ (1 + λ1 + λ2 )‖vτ ‖22 + [μ + λ2 (ρϑ + Mς )κϑ3 ] ∫(ψ◊vξξ ) dξ () 2 2 τ

+

0

T 1 ‖vξ ‖22 + [μ(1 − ∫ ψ(ς) dς) + λ1 (ρϑ + Mς )ϑ3 ]‖vξξ ‖22 2 2 0

1 + Mς (1 + λ1 + λ2 )vτ2 (ϑ, τ) + λ3 μΘ3 (τ), 2

τ ≥ 0.

(2.7)

136 | A. Berkani et al. We deduce that Θ(τ) ≤ ρ2 (ℰ (τ) + Θ3 (τ)),

τ ≥ 0,

where ρ2 is a positive constant. Taking into account (2.6) and (2.7) in (2.4), we get ϑ

2Θ(τ) ≥ ρ(1 − λ1 − λ2 )‖vτ ‖22 + [μ − λ2 (ρϑ + Mς )κϑ3 ] ∫(ψ◊vξξ ) dξ 0

+ T‖vξ ‖22 + [μ(1 − κ) − λ1 (ρϑ + Mς )ϑ3 ]‖vξξ ‖22

+ Mς (1 − λ1 − λ2 )vτ2 (ϑ, τ) + 2λ3 μΘ3 (τ),

τ ≥ 0.

Therefore, Θ(τ) ≥ ρ1 (ℰ (τ) + Θ3 (τ)), where ρ1 > 0 is such that λ1 < min[1,

μ(1 − κ) ], (ρϑ + Mς )ϑ3

λ2 < min[1 − λ1 ,

μ ]. (ρϑ + Mς )κϑ3

Lemma 2.5 ([5]). We have for ψ ∈ C(0, ∞) and y ∈ C((0, ∞); ϑ2 (0, ϑ)), ϑ

τ

τ

0

0

0

τ

1 1 󵄩 󵄩2 ∫ y ∫ ψ(τ − ς)y(ς) dς dξ = (∫ ψ(ς) dς)‖y‖22 + ∫ ψ(τ − ς)󵄩󵄩󵄩y(ς)󵄩󵄩󵄩2 dς 2 2 0

ϑ

−

1 ∫(ψ◊y) dξ , 2

(2.8)

0

for all τ ≥ 0.

3 Asymptotic behavior We state and prove our main results in this section. Firstly, we fix the following notations. For every measurable set 𝒜 ⊂ ℝ+ , the probability measure ψ̂ of 𝒜 ⊂ ℝ+ is defined by 1 ψ(̂ 𝒜) = ∫ ψ(ς) dς. κ 𝒜

The nondecreasingness set of ψ is Qψ = {ς ∈ ℝ+ : ψ(ς) > 0 and ψ󸀠 (ς) = 0},

(3.1)

Stabilization of a marine riser system by the use of viscoelastic material | 137

and the nondecreasingness rate of ψ is defined by ̂ ℛψ := ψ(Q ψ ). We set ̃ht = {ς ∈ ℝ : 0 ≤ ς ≤ τ, ψ(τ − ς) > 0 and ψ (τ − ς) = 0}. 𝒬 +

󸀠

The kernel ψ(τ) is a function in C 1 (ℝ+ , ℝ+ ) satisfying (see [4]): (H1)The kernel ψ is a continuously differentiable nonnegative function such that +∞

κ := ∫ ψ(τ) dτ < 1. 0

(H2)ψ󸀠 (τ) ≤ 0, a. e. for all τ ≥ 0. (H3)There exists a nondecreasing function φ(τ) > 0 such that φ󸀠 (τ)/φ(τ) =: δ(τ) is a decreasing function satisfying the condition +∞

∫ φ(ς)ψ(ς) dς < +∞. 0

τ

Let τ∗ > 0 be such that ∫0 ∗ ψ(ς) dς = ψ∗ > 0. For the reason of simplicity, the kernels are considered continuous and continuously differentiable a. e. The main result reads as follows. Theorem 3.1. Assume that assumptions (H1)–(H3) are satisfied. If ℛψ < 1/4 and Hφ (0) < [ψ∗ (8 − κ) − 3κ]/4, then there exist positive constants C and ν such that ℰ (τ) ≤ Cφ(τ) , −ν

τ ≥ 0,

if limτ→+∞ δ(τ) = 0, and ℰ (τ) ≤ Ce

−ντ

,

τ ≥ 0,

if limτ→+∞ δ(τ) ≠ 0. Proof. By using (1.1), we obtain ϑ

ϑ

d Θ (τ) = ρ ∫ vτ2 dξ + Mς vτ (ϑ, τ)2 + Mς vττ (ϑ, τ)v(ϑ, τ) − μ ∫ vvξξξξ dξ dt 1 0

0

ϑ

τ

ϑ

+ μ ∫ v ∫ ψ(τ − ς)vξξξξ (ς) dς dξ + T ∫ vvξξ dξ , 0

0

0

τ ≥ 0.

(3.2)

138 | A. Berkani et al. Using integrating by parts and Lemma 2.5, we obtain d κ Θ (τ) = ρ‖vτ ‖22 + Mς vτ2 (ϑ, τ) − μ(1 − )‖vξξ ‖22 − T‖vξ ‖22 dt 1 2 ϑ

t

μ μ 󵄩2 󵄩 ∫(ψ◊vξξ ) dξ + ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς, 2 2

−

0

τ ≥ 0.

(3.3)

0

For Θ2 (τ), we have τ

τ

d Θ (τ) = − ρ(∫ ψ(ς) dς)‖vτ ‖22 − Mς (∫ ψ(ς) dς)vτ2 (ϑ, τ) dt 2 0

0

τ

ϑ

− ρ ∫ vτ ∫ ψ󸀠 (τ − ς)(v(τ) − v(ς)) dς dξ 0

0

τ

− Mς vτ (ϑ, τ) ∫ ψ󸀠 (τ − ς)(v(ϑ, τ) − v(ϑ, ς)) dς 0 τ

− Mς vττ ϑ, τ) ∫ ψ(τ − ς)(v(ϑ, τ) − v(ϑ, ς)) dς 0 ϑ

(3.4)

τ

+ μ ∫ vξξξξ ∫ ψ(τ − ς)(v(τ) − v(ς)) dς dξ 0

0

ϑ

τ

τ

− μ ∫(∫ ψ(τ − ς)vξξξξ (ς) dς) ∫ ψ(τ − ς)(v(τ) − v(ς)) dς dξ 0

0

0 τ

ϑ

− T ∫ vξξ ∫ ψ(τ − ς)(v(τ) − v(ς)) dς dξ . 0

0

Then τ

τ

d Θ (τ) = − ρ(∫ ψ(ς) dς)‖vτ ‖22 − Mς (∫ ψ(ς) dς)vτ2 (ϑ, τ) dt 2 0

ϑ

0

τ

− ρ ∫ vτ ∫ ψ󸀠 (τ − ς)(v(τ) − v(ς)) dς dξ 0

0

τ

− Mς vτ (ϑ, τ) ∫ ψ󸀠 (τ − ς)(v(ϑ, τ) − v(ϑ, ς)) dς 0 τ

ϑ

τ

+ μ(1 − ∫ ψ(ς)dς) ∫ vξξ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dςdξ 0

0

0

(3.5)

Stabilization of a marine riser system by the use of viscoelastic material | 139 ϑ󵄨 t

󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 − μ ∫󵄨󵄨󵄨∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς󵄨󵄨󵄨 dξ 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨0 τ

ϑ

+ T ∫ vξ ∫ ψ(τ − ς)(vξ (τ) − vξ (ς)) dς dξ ,

τ ≥ 0.

0

0

For all measurable sets 𝒜 and Q such that 𝒜 = ℝ+ \Q [4], let us estimate the terms in (3.5). We have τ

ϑ

∫ vξξ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ 0

0 ϑ

= ∫ vξξ 0

ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ

∫ 𝒜∩[0,τ]

ϑ

+ ∫ vξξ 0

ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ

∫ Q∩[0,τ]

ϑ

≤ ∫ vξξ 0

ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ

∫ 𝒜∩[0,τ]

ϑ

ψ(τ − ς) dς)‖vξξ ‖22 − ∫ vξξ

+( ∫

0

Q∩[0,τ]

∫

ψ(τ − ς)vξξ (ς) dς dξ .

Q∩[0,τ]

Clearly, for η2 > 0, we obtain ϑ

∫ vξξ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ 0

𝒜τ

≤

η2 ‖vξξ ‖22

ϑ

κ 2 + ∫ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ , 4η2 0 𝒜τ

where ℬτ = ℬ ∩ [0, τ], and also ϑ

∫ vξξ ∫ ψ(τ − ς)vξξ (ς) dς dξ 0

Qτ

1 1 󵄩 󵄩2 ≤ (∫ ψ(τ − ς) dς)‖vξξ ‖22 + ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς. 2 2 Qτ

Qτ

(3.6)

140 | A. Berkani et al. These two estimates show that (3.6) becomes ϑ

τ

∫ vξξ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ 0

0

3 1 󵄩2 󵄩 ∫ ψ(τ − ς) dς)‖vξξ ‖22 + ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς 2 2

≤ (η2 +

Qτ

Qτ

(3.7)

ϑ

κ 2 + ∫ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ , 4η2

η2 > 0,

τ ≥ 0.

0 𝒜τ

Further, we have for η3 > 0, 󵄨󵄨2 󵄨󵄨 󵄨󵄨 󵄨󵄨 ∫󵄨󵄨∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς󵄨󵄨󵄨 dξ 󵄨󵄨 󵄨󵄨 󵄨 0 󵄨0 ϑ󵄨 τ

ϑ

≤ (1 +

1 2 )κ ∫ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ η3

(3.8)

0 𝒜τ

ϑ

2

+ (1 + η3 )(∫ ψ(τ − ς) dς) ∫ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ . 0 Qτ

Qτ

Now, using Lemma 2.2 leads, for η4 > 0, to ϑ

τ

∫ vξ ∫ ψ(τ − ς)(vξ (τ) − vξ (ς)) dς dξ 0

0

≤ η4 ‖vξ ‖22 +

ϑ

κϑ 2 ∫ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ 2η4

(3.9)

0 𝒜τ

ϑ

κϑ 2 (∫ ψ(τ − ς) dς) ∫ ∫ ψ(τ − ς)(vξξ (τ) − vξξ (ς)) dς dξ . + 2η4 Qτ

0 Qτ

Now we pass to the second and forth terms on the right-hand side of (3.5). By exploiting Lemma 2.2 and assumption (H2), we get for η5 > 0, ϑ

τ

∫ vτ ∫ ψ󸀠 (τ − ς)(v(τ) − v(ς)) dx ds 0

0

(3.10)

ϑ

ϑ4 ≤ η5 ‖vτ ‖22 − ψ(0) ∫(ψ󸀠 ◊vξξ ) dξ , 4η5 0

τ ≥ 0,

Stabilization of a marine riser system by the use of viscoelastic material | 141

and

τ

vτ (ϑ, τ) ∫ ψ󸀠 (τ − ς)[v(ϑ, τ) − v(ϑ, ς)] dς 0

≤

η5 vτ2 (ϑ, τ)

(3.11)

ϑ

ϑ3 − ψ(0) ∫(ψ󸀠 ◊vξξ ) dξ , 4η5

τ ≥ 0.

0

By combining (3.7)–(3.11) and (3.5), we arrive, for all τ ≥ τ∗ , at d Θ (τ) ≤ (η5 − ψ∗ )(ρ‖vτ ‖22 + Mς vτ2 (ϑ, τ)) dt 2 μ 󵄩 󵄩2 + (1 − ψ∗ ) ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς 2 Qτ

+ μ(1 − ψ∗ )(η2 +

+ [μ(1 +

3 ∫ ψ(τ − ς) dς)‖vξξ ‖22 + Tη4 ‖vξ ‖22 2 Qτ

ϑ

1 − ψ∗ 1 2 + )]κ ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ 4η2 η3 0 𝒜τ

ϑ

+

Tϑ 2 κ ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ + [μ(1 + η3 )] 2η4 0 𝒜τ

ϑ

(3.12)

2

× (∫ ψ(τ − ς) dς) ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ 0 Qτ

Qτ

ϑ

Tϑ 2 (∫ ψ(τ − ς) dς) ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ + 2η4 Qτ

3

−

ϑ (ρϑ + Mς ) 4η5

ϑ

0 Qτ

2

ψ(0) ∫ ∫ ψ󸀠 (τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ . 0 𝒜τ

Using assumption (H3) and differentiating Θ3 (τ) [5], we can write τ

d 󵄩 󵄩2 Θ (τ) ≤ Hγ (0)μ‖vξξ ‖22 − δ(τ)μΘ3 (τ) − μ ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς, dt 3

τ ≥ 0.

(3.13)

0

Taking into account the estimates (2.3), (3.3), (3.12), and (3.13), we obtain that, for all τ ≥ τ∗ > 0, ϑ

d 2 Θ(τ) ≤ τ1 ∫ ∫ ψ󸀠 (τ − ς)(vξξ (ς) − vξξ (τ)) dςdξ + [λ1 + λ2 (η5 − ψ∗ )] dt 0 𝒜τ

1 󵄩 󵄩2 × (ρ‖vτ ‖22 + Mς vτ2 (ϑ, τ)) + λ2 μ (1 − ψ∗ ) ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς 2 Qτ

142 | A. Berkani et al. t

λ 󵄩2 󵄩 + ( 1 − λ3 )μ ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς + τ2 μ‖vξξ ‖22 2

(3.14)

0

ϑ

2

+ τ3 κ ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ 0 𝒜τ ϑ

2

+ τ4 ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dςdξ 0 Qτ

+ (λ2 η4 −

λ1 )T‖vξ ‖22

ϑ

λμ − λ3 δ(τ)μΘ3 (τ) − 1 ∫(ψ◊vξξ )dξ , 2 0

where τ1 =

ϑ3 (ρϑ + Mς ) μ − λ2 ψ(0), 2 4η5

τ2 = λ2 (1 − ψ∗ )(η1 + τ3 = λ2 [μ(1 +

κ 3 ∫ ψ(τ − ς) dς) + λ3 Hγ (0) − λ1 (1 − ), 2 2 Qτ

1 − ψ∗ 1 Tϑ + )+ ], 4η2 η3 2η4

and τ4 = λ2 {[μ(1 + η3 ) +

Tϑ ]κψ(̂ 𝒬)}. 2η4

As in [4], we define the sets as follows: 𝒜n := {ς ∈ ℝ , nψ (ς) + ψ(ς) ≤ 0}, +

󸀠

n ∈ ℕ,

and ̃nt = {ς ∈ ℝ , 0 ≤ ς ≤ τ, nψ (τ − ς) + ψ(τ − ς) ≤ 0}, 𝒜 +

󸀠

n ∈ ℕ.

Note that ⋃ 𝒜n = ℝ+ \{Qψ ∪ 𝒩ψ }, n

where 𝒩ψ represents the null set where ψ󸀠 is not defined. We set Qn := ℝ+ \𝒜n , then we ̂ ̂ obtain limn→∞ ψ(Q n ) = ψ(Qψ ) because Qn+1 ⊂ Qn for all n and ⋂n 𝒬n = Qψ ∪ 𝒩ψ . If we ̃ in (3.14), and λ = λ (ψ − ε) for small ε < ψ , then we need ̃ ,Q = Q take 𝒜 = 𝒜 τ

nτ

τ

to select λ2 as follows: λ2 ≤

nτ

μη5 , ϑ3 (ρϑ + Mς )ψ(0)

1

so that

2

∗

∗

ϑ3 (ρϑ + Mς ) μ μ − λ2 ψ(0) ≥ . 2 4η5 4

Stabilization of a marine riser system by the use of viscoelastic material | 143

We deduce that for all τ ≥ τ∗ > 0, d Θ(τ) ≤ λ2 (η5 − ε)(ρ‖vτ ‖22 + vτ2 (ϑ, τ)) + λ2 [η4 − (ψ∗ − ε)]T‖vξ ‖22 dt t

(1 − ε)λ2 󵄩2 󵄩 − λ3 ]μ ∫ ψ(τ − ς)󵄩󵄩󵄩vξξ (ς)󵄩󵄩󵄩2 dς + [λ2 (1 − ψ∗ ) +[ 2 0

3 κ × (η1 + κψ(̂ 𝒬n )) + λ3 Hγ (0) − (ψ∗ − ε)λ2 (1 − )]μ‖vξξ ‖22 2 2 ϑ

+ λ2 {[(1 + η3 ) +

(ψ − ε) Tϑ ]κψ(̂ 𝒬n ) + η6 ϑ4 κ − ∗ }μ ∫(ψ◊vξξ )dξ 2η4 μ 2

1 − ψ∗ μ 1 Tϑ + )+ + η6 ϑ4 ]κ − } + {λ2 [μ(1 + 4η2 η3 2η4 4n ϑ

(3.15)

0

2

× ∫ ∫ ψ(τ − ς)(vξξ (ς) − vξξ (τ)) dς dξ − λ3 μδ(τ)Θ3 (τ). 0𝒜 ̃nt

At this point, we set η5 = ε/2, λ3 = (1−ε) λ2 , and η4 = (ψ∗ − ε)/2, then for small ε, large 2 ̂ n and τ , we take ψ(𝒬 ) < 0.25 so that ∗

n

ψ −ε ≤0 κψ(̂ 𝒬n ) − ∗ 2 and κ 3 (1 − ψ∗ ) κψ(̂ 𝒬n ) < ϱ(ψ∗ − ε)(1 − ), 2 2

(3.16)

with ϱ=

3κ(1 − ψ∗ ) . 4ψ∗ (2 − κ)

For τ∗ large enough, we have 0 < ϱ < 1, then it suffices to take Hγ (0) satisfying the condition κ (1 − ε) Hγ (0) < (1 − ϱ)(ψ∗ − ε)(1 − ), 2 2

(3.17)

this is possible if Hγ (0)
3κ/(8 − κ). Taking into account of (3.16) and (3.17) yields 3 (1 − ε) κ (1 − ψ∗ )[η1 + κψ(̂ 𝒬n )] + Hγ (0) − [(1 − ϱ) + ϱ](ψ∗ − ε)(1 − ) < 0, 2 2 2

(3.18)

144 | A. Berkani et al. where η1 is small enough. We fix n, τ∗ and ε, then we select η3 , η6 , and λ2 small enough such that [(1 + η3 ) +

(ψ − ε) Tϑ ]κψ(̂ 𝒬n ) + η6 ϑ4 κ < ∗ 2η4 μ 2

and λ2 [μ(1 +

1 − ψ∗ μ 1 Tϑ + )+ + η6 ϑ4 ]κ < . 4η2 η3 2η4 4n

So, we deduce that d Θ(τ) ≤ −c0 ℰ (τ) − λ3 μδ(τ)Θ3 (τ), dt

τ ≥ τ∗ ,

(3.19)

for c0 > 0. ̄ 0 ) ≥ τ∗ such that δ(τ) ≤ c0 for τ ≥ τ(c ̄ 0 ). If limτ→∞ δ(τ) = 0 then there exists a τ(c Hence, by using Proposition 2.4, we obtain d Θ(τ) ≤ −c1 δ(τ)Θ(τ), dt

τ ≥ τ,̂

(3.20)

̂ we obtain for c1 > 0. An integration of (3.16) on [τ, τ], τ

̂ −c1 ∫τ̂ Θ(τ) ≤ Θ(τ)e

δ(ς) dς

,

τ ≥ τ.̂

Therefore, by virtue of inequality (2.5), we get τ

̂ −c1 ∫τ̂ α1 (ℰ (τ) + Θ3 (τ)) ≤ Θ(τ)e

δ(ς) dς

,

τ ≥ τ.̂

Now, as φ󸀠 (τ)/φ(τ) = δ(τ), we conclude that ℰ (τ) ≤ Aφ (τ), −ν

τ ≥ τ,̂

where A > 0 and ν > 0. If limτ→∞ δ(τ) ≠ 0 then there exist τ̂ ≥ τ∗ and c2 such that δ(τ) ≥ c2 for τ̂ ≥ τ∗ . Then for c3 we obtain d Θ(τ) ≤ −c3 Θ(τ), dt

τ ≥ τ,̂

this implies that ℰ (τ) ≤ Ae

where A > 0 and ν > 0.

−ντ

,

τ ≥ τ,̂

Stabilization of a marine riser system by the use of viscoelastic material | 145

Bibliography [1] R. G. Airapetyan, A. G. Ramm, A. B. Smirnova, Continuous methods for solving nonlinear ill-posed problems. Operator Theory and Applications 25, 111–138 (2000). [2] A. Berkani, Stabilization of a viscoelastic rotating Euler–Bernoulli beam. Math. Meth. Appl. Sci. 41, 1–22 (2018). https://doi.org/10.1002/mma.4793. [3] L. Seghour, A. Berkani, N-e. Tatar, F. Saedpanah, Vibration control of a viscoelastic flexible marine riser with vessel dynamics. Math. Model. Anal. 23, 433–452 (2018). [4] N-e. Tatar, Uniform decay in viscoelasticity for kernels with small non-decreasingness zones. Appl. Math. Comput. 218, 7939–7946 (2012). [5] N-e. Tatar, A new class of kernels leading to an arbitrary decay in viscoelasticity. Mediterr. J. Math. 10, 213–226 (2013).

Aymen Ammar, Aref Jeribi, and Nawrez Lazrag

Sequence of linear operators converging in the generalized sense and its essential pseudospectra Abstract: This paper studies some results about the essential pseudospectrum of a sequence of linear operators converging in the generalized sense. Keywords: Pseudospectrum, sequence of linear operators, convergence MSC 2010: 11B05, 47A53, 39B42

1 Introduction Let X and Y be Banach spaces, then X × Y is also Banach space and its norm defined in a natural way as 1 󵄩󵄩 󵄩 2 2 󵄩󵄩(x, y)󵄩󵄩󵄩 = (|x‖ + ‖y‖ ) 2 .

Let T be a linear operator which starts from its domain, 𝒟(T) ⊂ X, into its range, R(T) ⊂ Y. A linear operator T is said to be bounded if 𝒟(T) = X and ‖T‖ = sup

x∈X\{0}

‖Tx‖ = inf{M ≥ 0 : ‖Tx‖ ≤ M‖x‖, for all x ∈ X}, ‖x‖

is finite. By ℒ(X, Y) we denote the set of all bounded linear operators. Moreover, an unbounded linear operator T : 𝒟(T) ⊂ X 󳨀→ Y is said to be closed, if its graph G(T) = {(x, Tx) ∈ X × Y : x ∈ 𝒟(T)}, is closed on X × Y. By 𝒞 (X, Y) we denote the set of all closed linear operators and we denote by 𝒦(X, Y) the subspace of compact operators from X into Y. Proposition 1.1 ([7, Theorem 7.3.1]). Let T ∈ ℒ(X) be such that ‖T‖ < 1. Then, (I − T)−1 is invertible and +∞

(I − T)−1 = ∑ T k ∈ ℒ(X). k=0

Aymen Ammar, Aref Jeribi, Nawrez Lazrag, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Sfax, Tunisia, e-mails: [email protected], [email protected], [email protected] https://doi.org/10.1515/9783110598193-011

148 | A. Ammar et al.

by

The range R(T) and null space N(T) of a linear operator T are respectively defined R(T) = {Tx : x ∈ 𝒟(T)} and N(T) = {x ∈ 𝒟(T) : Tx = 0}.

Definition 1.2. Let T ∈ 𝒞 (X, Y). We say that T is a Fredholm operator if ∙ α(T) = dim N(T) < ∞, { { ∙ β(T) = dim(Y/R(T)) < ∞, { { ∙ R(T) is closed in Y. { In this case the number i(T) = α(T) − β(T) is called the index of T. By Φ(X, Y) we denote the set of all Fredholm operators from X into Y. In the case that Y = X, briefly by Φ(X) we denote Φ(X, Y). Lemma 1.3 ([8, Theorem 7.9]). Let T ∈ Φ(X, Y). If there exists an η > 0 such that for every A ∈ ℒ(X, Y) one has ‖A‖ < η, then T + A ∈ Φ(X, Y) and i(A + T) = i(T). Let T ∈ 𝒞 (X). The resolvent set of T is defined by ρ(T) = {λ ∈ ℂ : λ − T is invertible and (λ − T)−1 ∈ ℒ(X)}, and the spectrum set of T is defined by σ(T) = ℂ\ρ(T). Furthermore, the Weyl essential spectrum of the operator T ∈ 𝒞 (X) is defined by σw (T) =

⋂ σ(T + K).

K∈𝒦(X)

Proposition 1.4 ([8, Theorem 7.27]). Suppose that T is a closed linear operator on X. We have λ ∉ σw (T) if and only if

λ − T ∈ Φ(X) and i(λ − T) = 0.

In the last years, many scholars have studied the concept of pseudospectrum. We can cite as examples [2, 3, 4, 5, 9, 10, 11, 12]. The pseudospectrum of T ∈ 𝒞 (X) is defined by 󵄩 󵄩 1 σε (T) = σ(T) ∪ {λ ∈ ℂ : 󵄩󵄩󵄩(λ − T)−1 󵄩󵄩󵄩 > }, ε

for every ε > 0.

By convention λ ∈ σ(T) if and only if ‖(λ − T)−1 ‖ = ∞. This is equivalent to saying (see [3]) that σε (T) = ⋃ σ(T + D). ‖D‖ 0. Then, the following statement are equivalent: (a) λ ∉ σw,ε (T); (b) λ − T + D ∈ Φ(X) and i(λ − T + D) = 0, for all D ∈ ℒ(X) such that ‖D‖ < ε. In this paper, stimulated by the results of the generalized convergence found in [6], we establish a relationship between σw,ε (Tn ) and σw,ε (T), so that Tn converges in the generalized sense to T. The rest of this paper is organized as follows. In Section 2, some basic concepts are recalled. In Section 3, we examine the essential pseudospectra of (Tn ), which converges in the generalized sense.

2 Preliminary results The purpose of this section is to recall some basic concepts, including the gap between linear operators and generalized convergence, given by [6]. Definition 2.1. Let T, S ∈ 𝒞 (X, Y). We define the gap between T and S by δ(T, S) =

sup

1

[ inf (‖x − y‖2 + ‖Tx − Sy‖2 ) 2 ],

y∈𝒟(S) x∈𝒟(T) ‖x‖2 +‖Tx‖2 =1

and ̂ S) = max{δ(T, S), δ(S, T)}. δ(T, ̂ S), we may refer For more details related to the results regarding δ(T, S) and δ(T, to [6, pages 197, 200, 201]. ̂ S) < Lemma 2.2 ([6, Theorem IV.5.17]). Let T ∈ Φ(X). If δ(T,

γ(T)

1

(1+[γ(T)]2 ) 2

, then S ∈ Φ(X),

̂ S) < b, then α(S) ≤ α(T), and β(S) ≤ β(T). Moreover, if there exists b > 0 such that δ(T, we have i(S) = i(T). Here γ(T) = inf{

‖Tx‖ , x ∈ 𝒟(T), x ∈ ̸ N(T)}. d(x, N(T))

By convention, if 𝒟(T) ⊂ N(T), we have γ(T) = +∞.

150 | A. Ammar et al. Definition 2.3. Let (Tn ) be a sequence of closed linear operators from X into Y and T ∈ C(X, Y). We say that (Tn ) converges in the generalized sense to T, denoted by g

Tn 󳨀→ T, if the following is true: g

Tn 󳨀→ T

̂ , T) 󳨀→ 0, as n → +∞. if and only if δ(T n

In the following, we shall recall some fundamental results about convergence in the generalized sense, which will be used in the subsequent section. Proposition 2.4 ([6, Theorem IV.2.23]). Let T ∈ C(X, Y) and (Tn ) be a sequence of closed linear operators from X into Y. g

g

(a) Tn 󳨀→ T if and only if Tn + S 󳨀→ T + S, where S ∈ ℒ(X, Y). g

(b) Let T ∈ ℒ(X, Y). Then Tn 󳨀→ T if and only if Tn ∈ ℒ(X, Y) for all sufficiently large n and Tn converges to T. g

(c) Assume that Tn 󳨀→ T. Then T −1 exists and T −1 ∈ ℒ(Y, X) if and only if Tn−1 exists and Tn−1 ∈ ℒ(Y, X) for all sufficiently large n and Tn−1 converges to T −1 .

3 The main results The aim of this section is to establish a relationship between σw,ε (Tn ) and σw,ε (T), so g

that Tn 󳨀→ T.

Lemma 3.1. Assume that S ∈ ℒ(X). If λ ∈ ρ(T) is such that ‖S‖ < γ(λ − T), then λ ∈ ρ(T + S). Proof. Without loss of generality, we suppose that λ = 0. Let us assume that 0 ∈ ρ(T) and ‖S‖ < ‖T1−1 ‖ . Hence, by using Proposition 1.1, we infer that I − ST −1 is invertible and −1 n (I − ST −1 )−1 = ∑+∞ n=0 (ST ) . This implies that

1 󵄩󵄩 −1 −1 󵄩 . 󵄩󵄩(I − ST ) 󵄩󵄩󵄩 ≤ 1 − ‖S‖‖T −1 ‖ In other words, we have that T + S = (I + ST −1 )T is invertible, and (T + S)−1 = T −1 (I + ST −1 )

−1

∈ ℒ(X).

In view of the above, we conclude that 0 ∈ ρ(T + S). Lemma 3.2. Let T ∈ 𝒞 (X) and ε > 0. If λ ∈ ℂ\{0}, then λ ∈ ̸ σw,ε (T) if and only if λ−1 ∈ ̸ σw,ε (T −1 ). Proof. Let λ ∈ ̸ σw,ε (T). It follows from Proposition 1.5 that for all D ∈ ℒ(X) such that ‖D‖ < ε, λ − T + D ∈ Φ(X)

and

i(λ − T + D) = 0.

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| 151

Hence, from Lemma 1.3, we get λ − T + D − D ∈ Φ(X)

and i(λ − T + D) = i(λ − T) = 0.

This leads us to conclude that λ ∈ ̸ σw (T). So, we obtain λ−1 ∈ ̸ σw (T −1 ). Again by using Lemma 1.3, we get λ−1 − T −1 + D ∈ Φ(X)

and i(λ−1 − T −1 + D) = i(λ−1 − T −1 ) = 0.

By referring to Proposition 1.5, we conclude that λ−1 ∈ ̸ σw,ε (T −1 ). Conversely, the same logic as before leads to the required result. Theorem 3.3. Let T ∈ 𝒞 (X) be such that 0 ∈ ρ(T) and let (Tn ) be a sequence of closed g

linear operators on X. If Tn 󳨀→ T, then for every ε1 , ε2 satisfying 0 < ε1 < ε2 , there exists n0 ∈ ℕ such that σw,ε1 (T) ⊆ σw,ε2 (Tn ),

for all n ≥ n0 .

g

Proof. The fact that 0 ∈ ρ(T) and Tn 󳨀→ T implies from (c) of Proposition 2.4 that 0 ∈ ρ(Tn ) and ‖Tn−1 − T −1 ‖ 󳨀→ 0 as n 󳨀→ ∞. This implies that there exists n0 ∈ ℕ such that for all n ≥ n0 we have 󵄩󵄩 −1 −1 󵄩 󵄩󵄩Tn − T 󵄩󵄩󵄩 ≤ ε2 − ε1 .

(3.1)

Let us assume that λ ∈ ̸ σw,ε2 (Tn ). So, by using Lemma 3.2, we get λ−1 ∈ ̸ σw,ε2 (Tn−1 ) =

⋂ σε2 (Tn−1 + K).

K∈𝒦(X)

Hence, there exists K ∈ 𝒦(X) such that λ−1 ∈ ρ(Tn−1 + K) and 1 −1 󵄩 󵄩󵄩 −1 −1 󵄩󵄩(λ − (Tn + K)) 󵄩󵄩󵄩 ≤ . ε2

(3.2)

In addition, we can express the operator λ−1 − (T −1 + K) in the form λ−1 − (T −1 + K) = (λ−1 − (Tn−1 + K))[I − (λ−1 − (Tn−1 + K)) (T −1 − Tn−1 )]. −1

By combining (3.1) and (3.2), we deduce that −1 −1 −1 󵄩󵄩 −1 󵄩󵄩 −1 −1 −1 󵄩 󵄩 −1 −1 −1 󵄩 󵄩󵄩(λ − (Tn + K)) (T − Tn )󵄩󵄩󵄩 ≤ 󵄩󵄩󵄩(λ − (Tn + K)) 󵄩󵄩󵄩󵄩󵄩󵄩T − Tn 󵄩󵄩󵄩 1 ≤ (ε2 − ε1 ) ε2 ε ≤1− 1 ε2 < 1.

(3.3)

152 | A. Ammar et al. This leads us to infer from Proposition 1.1 that I − (λ−1 − (Tn−1 + K))−1 (T −1 − Tn−1 ) is invertible and [I − (λ−1 − (Tn−1 + K)) (T −1 − Tn−1 )] −1

−1

+∞

n

= ∑ ((λ−1 − (Tn−1 + K)) (T −1 − Tn−1 )) . −1

n=0

Thus, we have 1 −1 −1 󵄩󵄩 −1 −1 −1 −1 󵄩 󵄩󵄩[I − (λ − (Tn + K)) (T − Tn )] 󵄩󵄩󵄩 ≤ 1 − (1 − ε ≤ 2. ε1

ε1 ) ε2

This implies from (3.2) and (3.3) that ε 1 −1 󵄩 󵄩󵄩 −1 −1 󵄩󵄩(λ − (T + K)) 󵄩󵄩󵄩 ≤ 2 × ε1 ε2 1 = . ε1 Therefore, by (3.3) and the above estimate, we conclude that λ−1 ∈ ̸ σε1 (T −1 + K). Hence, we get λ−1 ∈ ̸ σw,ε1 (T −1 ). By using Lemma 3.2, we conclude that λ ∈ ̸ σw,ε1 (T). Consequently, there exists n0 ∈ ℕ such that σw,ε1 (T) ⊆ σw,ε2 (Tn ),

for all n ≥ n0 .

Theorem 3.4. Let (Tn ) be a sequence of closed linear operators on X and T ∈ 𝒞 (X) with g

0 ∈ ρ(T). Suppose that 𝒰 is an open subset of ℂ containing 0. If Tn 󳨀→ T, then there exists n0 ∈ ℕ such that σw,ε (Tn ) ⊆ σw,ε (T) + 𝒰 ,

for all n ≥ n0 .

(3.4)

g

Proof. Assume that 0 ∈ ρ(T) and Tn 󳨀→ T. Then, by using Proposition 2.4 (c), we infer that 0 ∈ ρ(Tn ) and ‖Tn−1 − T −1 ‖ 󳨀→ 0 as n 󳨀→ ∞. First, we show the existence of n0 ∈ ℕ such that σw,ε (Tn−1 ) ⊆ σw,ε (T −1 ) + 𝒰 ,

for all n ≥ n0 .

(3.5)

In order to prove (3.5), suppose the assertion is not valid. We assume that for any n there exists a subsequence (ωn ) such that ωn ∈ σw,ε (Tn−1 ) and ωn ∈ ̸ σw,ε (T −1 ) + 𝒰 . Since 0 ∈ 𝒰 , we obtain ωn ∈ ̸ σw,ε (T −1 ). By referring to Proposition 1.5, we deduce that there exists D ∈ ℒ(X) with ‖D‖ < ε such that ωn ∈ σw (Tn−1 + D) and ωn ∈ ̸ σw (T −1 + D).

(3.6)

Sequence of linear operators and its essential pseudo-spectra

| 153

Let λ0 ∈ ℂ be such that ‖T −1 + D‖ < |λ0 |. This implies from Proposition 1.1 that λ0 ∈ ρ(T −1 + D). The fact that λ0 − (Tn −1 + D) → λ0 − (T −1 + D) as n → ∞ implies from Proposition 2.4 (b) that g

λ0 − (Tn −1 + D) 󳨀→ λ0 − (T −1 + D). Hence, by referring back to the fact that λ0 ∈ ρ(T −1 + D) and λ0 − (T −1 + D) ∈ ℒ(X), we conclude from Proposition 2.4 (c) that there exists an N ∈ ℕ such that for all n ≥ N, we have λ0 ∈ ρ(Tn−1 + D) and −1 −1 󵄩 󵄩󵄩 −1 −1 󵄩󵄩(λ0 − (T + D)) − (λ0 − (Tn + D)) 󵄩󵄩󵄩 󳨀→ 0

as n 󳨀→ ∞.

Since σ(⋅) is upper semicontinuous at (λ0 − (T −1 + D))−1 (see [6, p. 209]), there exists k > 0 such that k −1 ≤ |ωn − λ0 |−1 . This means that (ωn ) is bounded. Therefore, we can assumed that limn󳨀→+∞ ωn = ω. It follows from (3.6) that ω ∈ ̸ σw (T −1 + D). Therefore, ω − (T −1 + D) ∈ Φ(X)

and i(ω − (T −1 + D)) = 0.

On the one hand, since ωn − (Tn−1 + D) 󳨀→ ω − (T −1 + D) as n 󳨀→ ∞, by Proposition 2.4 (b), we get −1 −1 ̂ δ(ω n − (Tn + D), ω − (T + D)) 󳨀→ 0

as n 󳨀→ ∞.

(3.7)

Since R(ω − (T −1 + D)) is closed, by using [6, Theorem IV.5.2], we can assume that δ = γ(ω − (T −1 + D)) > 0. Hence, it follows from (3.7) that there exists N1 ∈ ℕ such that −1 −1 ̂ δ(ω n − (Tn + D), ω − (T + D))
0, as follows: 𝒜ℋ := (

T K2

K1 ), TH + K3

defined with nonmaximal domain i

𝒟(𝒜ℋ ) := {( o

o

f f f )∈𝒲 ×𝒲 :( ) = ℋ( ) }, g g g

i

where ( gf ) and ( gf ) represent the outgoing and incoming fluxes related by the bounded boundary operator ℋ. Here ℋ is expressed as {

ℋ : X o × X o 󳨀→ X i × X i , f ( gf ) 󳨃󳨀→ ( 00 H H )( g ),

where H ∈ ℒ(X o , X i ) and the boundary spaces X o and X i are identified as X o := L1 (−a × (−1, 0), |v|dv) × L1 (a × (0, 1), |v|dv) and X i := L1 (−a × (0, 1), |v|dv) × L1 (a × (−1, 0), |v|dv) ∗ each closed linear operator T is defined by: T : 𝒟(T) ⊆ X, 󳨀→ X { { { 𝜕f (x, v) − σ1 (v)f (x, v) f 󳨃󳨀→ Tf : (x, v) 󳨃󳨀→ −v 𝜕x { { { 𝜕f { 𝒟(T) := {f ∈ X : v 𝜕x ∈ X} = 𝒲 . Also the streaming operator TH is defined by TH : 𝒟(TH ) ⊆ X 󳨀→ X, { { { g 󳨃󳨀→ TH g : (x, v) 󳨃󳨀→ −v 𝜕g (x, v) − σ2 (v)g(x, v), { 𝜕x { { i o { 𝒟(TH ) := {g ∈ 𝒲 : g = Hg }.

Left–right essential spectra of one-sided operator matrix and application

| 161

The bounded linear collision operators Kj , j = {1, 2, 3}, are defined on X by {

Kj : X 󳨀→ X,

1

f 󳨃󳨀→ Kj f : (x, v) 󳨃󳨀→ ∫−1 κj (x, v, v󸀠 )f (x, v󸀠 ) dv󸀠 ,

where the frequency σj (⋅) ∈ ℒ∞ (−1, 1), j = {1, 2}, the variable v may be thought of as the cosine of the angle between the velocity of particles and the x-direction and the scattering-fission kernels κk : (−a, a) × (−1, 1) × (−1, 1) 󳨀→ ℝ are assumed to be measurable. Physically, the function ( gf )(x, v) represents the density of neutrons having the position x and the direction cosine of propagation v. Clearly, this model of transport operator may be written as a one-sided operator matrix in the following form: 𝒜ℋ := (

T K2

A K1 ) := ( C TH + K3

B ), D

defined on 𝒟(𝒜ℋ ) := {(

f ) ∈ 𝒟(T) × 𝒟(TH ) : ΦX f = ΨX g} , g

(for appropriate operators ΦX and ΨX introduced in details in Section 4). Taking advantage of the results of Section 3 with a specific choice of the boundary and collision operators, we guarantee the stability of the essential spectra between the operators A 0 𝒜ℋ and 𝒬ℋ := ( 01 T +K ). Physically, A1 corresponds to the transport operator with H 3 vacuum boundary condition as A1 := T, 𝒟(A1 ) := {f ∈ 𝒟(T) : f i = 0}. Our paper is organized as follows: Section 2 is devoted to some basic definitions from the theory of operators and presents their fundamental properties. In Section 3, we describe our model of operator matrix 𝒜. Some general hypotheses on the different components of 𝒜 are introduced in details in order to characterize its essential spectra. In the end, the theoretical results are applied to a physical model of transport operators on a nuclear space.

2 Preliminaries and definitions This section contains some auxiliary notations and gives some basic properties based on the theory of Fredholm perturbation that we will need in the sequel. To this end, we consider a linear operator T with domain 𝒟(T) ⊂ X and range R(T) ⊂ Y as an operator T acting from Banach space X into Banach space Y. Sets of all closed densely defined linear operators, of all bounded linear operators, and all compact operators are denoted by 𝒞 (X, Y), ℒ(X, Y), and 𝒦(X, Y) from X into Y, respectively.

162 | S. Bouzidi and I. Walha For T ∈ 𝒞 (X), we denote the dimension of the null space 𝒩 (T) by α(T), the codimension of the range R(T) by β(T), the spectrum of T by σ(T), and the resolvent set of T by ρ(T). We denote by Φ+ (X, Y) (resp. Φ− (X, Y) and Φ(X, Y)) the set of upper semi-Fredholm (resp. lower semi-Fredholm and Fredholm) operators from X into Y. Such sets are defined as follows: Φ+ (X, Y) := {T ∈ 𝒞 (X, Y) : α(T) < ∞ and R(T) is closed}, Φ− (X, Y) := {T ∈ 𝒞 (X, Y) : β(T) < ∞ and R(T) is closed}, and Φ(X, Y) := Φ+ (X, Y) ∩ Φ− (X, Y). For a Fredholm operator T, its index is defined by the quantity i(T) = α(T) − β(T). Sets of left-Fredholm and right-Fredholm operators from X into Y are denoted respectively by Φl (X, Y) and Φr (X, Y), and are defined as Φl (X, Y) := {T ∈ Φ+ (X, Y) : R(T) is a complemented subset of Y} and Φr (X, Y) := {T ∈ Φ− (X, Y) : N(T) is a complemented subset of X}. The sets of bounded upper (resp. lower) semi-Fredholm, Fredholm, and left (resp. right) Fredholm operators from X into Y are defined as: Φb+ (X, Y) := Φ+ (X, Y) ∩ ℒ(X, Y) (resp. Φb− (X, Y) := Φ− (X, Y) ∩ ℒ(X, Y)), Φb (X, Y) := Φ(X, Y) ∩ ℒ(X, Y), and Φbl (X, Y) := Φl (X, Y) ∩ ℒ(X, Y) (resp. Φbr (X, Y) := Φr (X, Y) ∩ ℒ(X, Y)). Therefore, a relationship between these sets of Fredholm operators is formulated as follows: Φb (X, Y) ⊆ Φbl (X, Y) ⊆ Φb+ (X, Y) and

Φb (X, Y) ⊆ Φbr (X, Y) ⊆ Φb− (X, Y).

Remark 2.1. If X = Y, the sets Φb+ (X, Y), Φb− (X, Y), Φb (X, Y), Φbl (X, Y), and Φbr (X, Y) are replaced, respectively, by Φb+ (X), Φb− (X), Φb (X), Φbl (X), and Φbr (X). The theory of Fredholm perturbation played an important role in the study of some essential spectra of linear operators. Before emphasizing this importance, we shall recall some basic definitions: Definition 2.2. (a) The set of left Fredholm perturbations from X into Y is defined by ℱl (X, Y) := {F ∈ ℒ(X, Y) : T + F ∈ Φl (X, Y), ∀T ∈ Φl (X, Y)}.

Left–right essential spectra of one-sided operator matrix and application

| 163

(b) The set of right Fredholm perturbations from X into Y is defined by ℱr (X, Y) := {F ∈ ℒ(X, Y) : T + F ∈ Φr (X, Y), ∀T ∈ Φr (X, Y)}.

(c) The set of Fredholm perturbations from X into Y is defined by ℱ (X, Y) := {F ∈ ℒ(X, Y) : T + F ∈ Φ(X, Y), ∀T ∈ Φ(X, Y)}.

Remark 2.3. (a) Following Definition 2.2, the sets ℱlb (X, Y), ℱrb (X, Y), and ℱ b (X, Y) are defined by replacing Φl (X, Y), Φr (X, Y), and Φ(X, Y) with Φbl (X, Y), Φbr (X, Y), and Φb (X, Y), respectively. (b) The properties of the sets of left and right Fredholm perturbations are studied in [10]. Precisely, the authors have proved that the classes ℱlb (X) := ℱlb (X, X) and ℱrb (X) := ℱrb (X, X) are two-sided ideals of ℒ(X) containing 𝒦(X), if X = Y. (c) I. C. Gohberg et al. in [6] have proved that the class of Fredholm perturbations ℱ b (X, Y) is a closed subset of ℒ(X, Y) and when X = Y, the set ℱ b (X) := ℱ b (X, X) is a closed two-sided ideal of ℒ(X). Moreover, b

b

b

b

𝒦(X, Y) ⊆ 𝒲 (X, Y) ⊆ ℱl (X, Y) ⊆ ℱ (X, Y)

and 𝒦(X, Y) ⊆ 𝒲 (X, Y) ⊆ ℱr (X, Y) ⊆ ℱ (X, Y),

where 𝒲 (X, Y) denotes the set of weakly compact operators from X into Y, namely those bounded operators T ∈ ℒ(X, Y) which make, for every bounded B ⊂ X, T(B) relatively weakly compact in Y. Note that the set of weakly compact operators on X will be denoted by 𝒲 (X). The theory of Fredholm operators is used especially to define some notions of essential spectra for closed densely defined operators T on a Banach space X. Precisely, we are concerned in this work with some of them: σe1 (T) := {ν ∈ ℂ : νI − T ∈ ̸ Φ+ (X, Y)},

σe2 (T) := {ν ∈ ℂ : νI − T ∈ ̸ Φ− (X, Y)}, σel (T) := {ν ∈ ℂ : νI − T ∈ ̸ Φl (X, Y)},

σer (T) := {ν ∈ ℂ : νI − T ∈ ̸ Φr (X, Y)},

σew (T) := {ν ∈ ℂ : νI − T ∈ ̸ Φ(X)},

σess (T) := ℂ\{ν ∈ ℂ : νI − T ∈ Φ(X) with i(νI − T) = 0}. The sets of essential spectra will be ordered as follows: σe1 (T) ∩ σe2 (T) ⊆ σew (T) := σel (T) ∪ σer (T) ⊆ σess (T)

164 | S. Bouzidi and I. Walha and σe∗ (T) ⊆ σe∗∗ (T) ⊆ σew (T),

(2.1)

where (σe∗ (⋅), σe∗∗ (⋅)) ∈ {(σe1 (⋅), σel (⋅)), (σe2 (⋅), σer (.))}. A real progress in the theory of Fredholm perturbations of an operator matrix was developed by A. Jeribi et al. in [10] as follows: Theorem 2.4 ([10, Theorems 3.1–3.2]). Let Xi , for i ∈ {1, 2}, be a Banach space and ℙ := (Pij )1≤i,j≤2 , where Pij denotes a bounded linear operator from Xj into Xi , for 1 ≤ i, j ≤ 2. Then, we have ℙ ∈ ℰ (X1 × X2 ) ⇐⇒ Pij ∈ ℰ (Xj , Xi ), where (ℰ (X1 × X2 ), ℰ (Xj , Xi )) ∈ {

∀i, j = 1, 2,

(ℱlb (X1 ×X2 ),ℱlb (Xj ,Xi )),(ℱrb (X1 ×X2 ),ℱrb (Xj ,Xi )), (ℱ b (X1 ×X2 ),ℱ b (Xj ,Xi ))

}.

In spectral theory, the study of the invariance problems of essential spectrum of some linear operators on Banach spaces under (additive) perturbation has become one of the famous interesting problems. That is, for T ∈ 𝒞 (X), we have: σess (T) := σess (T + K),

∀K ∈ ℱ (X).

Motivated by this interesting problem, some new results have been produced by several authors. We can quote some of them [3, 8, 7]. Unfortunately, in some physical problems involving transport operators, operators arising in dynamic populations, etc., the verification on the condition of K fails to involve the class of Fredholm perturbations. So, we have merely only some information on (νI − T)−1 − (νI − T − K)−1 . A further result in this direction was proved by S. Charfi et al. in [3], which will be essential to formulate our findings, as well. Theorem 2.5 ([3]). Let T and B be two closed densely defined linear operators on a Banach space X. Then, we have the following assertions: (a) If (νI − T)−1 − (νI − B)−1 ∈ ℱlb (X), for some ν ∈ ρ(T) ∩ ρ(B), then σel (T) = σel (B). (b) If (νI − T)−1 − (νI − B)−1 ∈ ℱrb (X), for some ν ∈ ρ(T) ∩ ρ(B), then σer (T) = σer (B). Definition 2.6 ([11, Section III.5]). Let T : 𝒟(T) ⊂ X 󳨀→ Y be a closed operator and S be a closable linear operator from X into Y having the domain 𝒟(S) such that S = T, then the domain 𝒟(S) is called a core of T.

Left–right essential spectra of one-sided operator matrix and application

| 165

3 Spectral description of one-sided operator matrix In this part, we are concerned with a derivation of a precise description of spectral property of one-sided operator matrix related to its diagonal component operators which has led to significant advances in the theory of operator matrices. Let consider in X × Y (a product of Banach spaces) a one-sided operator matrix, 𝒜, given by 𝒜 := (

A C

B ) D

and defined on 𝒟(𝒜) := {(

f ) ∈ (𝒟(A) ∩ 𝒟(C)) × (𝒟(B) ∩ 𝒟(D)) : ΦX f = ΨY g} . g

The operator entries of the matrix 𝒜 have an appropriate domain and act on their corresponding spaces as follows: A : 𝒟(A) ⊂ X 󳨀→ X,

B : 𝒟(B) ⊂ Y 󳨀→ X,

C : 𝒟(C) ⊂ X 󳨀→ Y,

D : 𝒟(D) ⊂ Y 󳨀→ Y.

The linear operators ΦX : X 󳨀→ Z and ΨY : Y 󳨀→ Z, for a Banach space Z, are used to define a supplementary relation of the form ΦX f = ΨY g between the components f and g of an element of the nonmaximal domain 𝒟(𝒜). Assume that the entries of this kind of operator matrix satisfy: (ℋ1 ) A is a closable and densely defined linear operator with domain equipped with the graph norm ‖x‖A := ‖x‖ + ‖Ax‖,

x ∈ 𝒟(A),

which can be completed to a Banach space XA , and so it coincides with the domain of the closure of A, denoted by 𝒟(A), which is contained in X. (ℋ2 ) The map ΦX : XA 󳨀→ Z is bounded and has domain verifying 𝒟(A) ⊂ 𝒟(ΦX ) ⊂ XA .

(ℋ3 ) 𝒟(A) ∩ 𝒩 (ΦX ) = X and ρ(A1 ) ≠ 0, where A1 := A|𝒟(A)∩𝒩 (ΦX ) . Lemma 3.1 ([2, Lemmas 2.1–2.2]). Assume that the hypotheses (ℋ1 )–(ℋ3 ) hold. Then, for ν ∈ ρ(A1 ), we have: (a) 𝒟(A) := 𝒟(A1 ) ⊕ 𝒩 (νI − A); (b) The restriction Φν := ΦX |𝒩 (νI−A) is injective; (c) ℛ(Φν ) = ΦX (𝒩 (νI − A)) = ΦX (𝒟(A)).

166 | S. Bouzidi and I. Walha As a consequence, we have that Φν is invertible; for ν ∈ ρ(A1 ), its inverse will be used to define the operator Kν from ΦX (𝒟(A)) into 𝒩 (νI − A) as {

Kν := (Φν )−1 := (ΦX |𝒩 (νI−A) )−1 , Kν z = f

means that f ∈ 𝒟(A), (νI − A)f = 0 and ΦX f = z.

Lemma 3.2 ([2, Lemma 2.4]). For every (ν, μ) ∈ (ρ(A1 ))2 , under the conditions (ℋ1 )– (ℋ3 ), we obtain: (a) Kν − Kμ = (μ − ν)(νI − A1 )−1 Kμ ; (b) If for some ν ∈ ρ(A1 ) the operator Kν is closable, then the same is true for all ν, with the closure satisfying K ν − K μ = (μ − ν)(νI − A1 )−1 K μ . Assumptions about the component C of the matrix 𝒜 are formulated in the hypothesis (ℋ4 ) as follows: (ℋ4 ) The domains of the operators A and C are 𝒟(A) ⊂ 𝒟(C) ⊂ XA ,

with a bounded operator C(νI − A1 )−1 ∈ ℒ(XA , Y), for ν ∈ ρ(A1 ). Remark 3.3. (a) The last hypothesis, together with the closed graph theorem, asserts that, for ν ∈ ρ(A1 ), the operator Cν := C(νI − A1 )−1 from X into Y is bounded. (b) Since the assumptions (ℋ1 )–(ℋ3 ) are required, for ν ∈ ρ(A1 ), we obtain, for x ∈ 𝒟(A), that (νI − A)x = (νI − A1 )(I − Kν ΦX )x. Moreover, we will assume that: (ℋ5 ) The operator Kν is bounded from ΦX (𝒟(A)) into X, where K ν denotes its extension by continuity to ΦX (𝒟(A)), for some (hence for all) ν ∈ ρ(A1 ). (ℋ6 ) 𝒟(B) ∩ 𝒟(D) ⊂ 𝒟(ΨY ). The dense sets Yi , i = 1, 2, in Y are defined as Y1 := {y ∈ 𝒟(B) ∩ 𝒟(D) : ΨY y ∈ ΦX (𝒟(A))} and Y2 := {y ∈ 𝒟(B) ∩ 𝒟(ΨY ) : ΨY y ∈ ΦX (𝒟(A)) = Z1 }. We denote by

Left–right essential spectra of one-sided operator matrix and application

| 167

∘

– ΨY , the continuous extension of ΨY |Y1 on the all space Y; – ΨY |Y2 , a bounded operator from Y2 into Z. (ℋ7 ) The operator D is assumed to be a densely defined linear operator with ρ(D) ≠ 0 and for which the set Y1 is a core of D. (ℋ8 ) For the operators B and C, we suppose that – 𝒟(B) = Y; – The operators (νI − A1 )−1 B and C[−Kν ΨY + (νI − A1 )−1 B] are bounded on 𝒟(B) and on Y2 , for some (hence for all) ν ∈ ρ(A1 ), respectively. (ℋ9 ) ρ(M ν ) ≠ 0, where Mν := D + C[−Kν ΨY + (νI − A1 )−1 B] is defined on the set Y1 . All the assumptions cited above are used to provide a fine description of the closure of such an operator matrix 𝒜 originated from the work of S. Charfi et al. in [3] and they are needed to formulate our result. Theorem 3.4 ([3]). Suppose that the hypotheses (ℋ1 )–(ℋ9 ) are fulfilled. Then, for some (hence for all) ν ∈ ρ(A1 ), 𝒜 is a closable operator on X × Y with closure 𝒜 described as I Cν

𝒜 := νI − (

−(

0 0

0 νI − A1 )( I 0

0 I )( 0 νI − D

Gν ) I

0 ), Rν

(3.1)

where ∘

Gν := −[−K ν ΨY + (νI − A1 )−1 B] and Rν := −C[−Kν ΨY + (νI − A1 )−1 B]. Remark 3.5. Under the assumptions (ℋi ), 1 ≤ i ≤ 9, and for ν ∈ ρ(A1 ) ∩ ρ(M ν ) ∩ ρ(D), we have: (a) The resolvent expression of the Schur-complement M ν is given by (νI − M ν )−1 := (νI − D)−1 − (νI − M ν )−1 Rν (νI − D)−1 . (b) The expression of the resolvent of 𝒜 is expressed as (νI − A1 )−1

(νI − 𝒜)−1 = (

+Gν (νI − M ν )−1 Cν −(νI − M ν )−1 Cν

−Gν (νI − M ν )−1 ). (νI − M ν )−1

(3.2)

168 | S. Bouzidi and I. Walha Now, we are in a position to use our approach and obtain the description of spectral properties of such a model of operator matrix 𝒜 involving the theory of Fredholm perturbation. Our advances are aimed at formulating some conditions which ensure a fine characterization of some essential spectra of the operator matrix 𝒜. For this, we will associate to the operator 𝒜 a diagonal operator matrix 𝒬 as follows: 𝒬 := (

A1 0

0 ). D

Theorem 3.6. Suppose that the assumptions (ℋi ), 1 ≤ i ≤ 9 are fulfilled, for ν ∈ ρ(A1 ). Then, if for some (hence for all) ν ∈ ρ(A1 ) ∩ ρ(D) ∩ ρ(Mν ), the operators (a) Rν , (νI − D)−1 Cν , and Gν (νI − D)−1 are left-Fredholm perturbations, and we have σel (𝒜) := σel (A1 ) ∪ σel (D); (b) Rν , (νI − D)−1 Cν , and Gν (νI − D)−1 are right-Fredholm perturbations, and we have σer (𝒜) := σer (A1 ) ∪ σer (D); (c) Rν , (νI − D)−1 Cν , and Gν (νI − D)−1 are Fredholm perturbations, and we have σew (𝒜) := σew (A1 ) ∪ σew (D). Proof. The proof of this result relies on Theorem 2.5. To apply it, we first need to verify its assumptions, that is, to prove the property of Fredholm perturbation for the difference of the resolvent expressions between the operators νI − 𝒜 and νI − 𝒬. Indeed, let ν ∈ ρ(A1 ) ∩ ρ(M ν ) ∩ ρ(D). Then we have ν ∈ ρ(𝒜) ∩ ρ(𝒬) and (νI − 𝒜)−1 − (νI − 𝒬)−1 Gν (νI − D)−1 Cν

:= ( (

−Gν (νI − M ν )−1 Rν (νI − D)−1 Cν −(νI − D)−1 Cν

+(νI − M ν )−1 Rν (νI − D)−1 Cν

−Gν (νI − D)−1

+Gν (νI − M ν )−1 Rν (νI − D)−1 −(νI − M ν )−1 Rν (νI − D)−1

). )

(a) From the assumptions, the operators Gν (νI − D)−1 Cν , −Gν (νI − M ν )−1 Rν (νI − D)−1 Cν , +Gν (νI − M ν )−1 Rν (νI − D)−1 , (νI − M ν )−1 Rν (νI − D)−1 Cν , and −(νI − M ν )−1 Rν (νI − D)−1 are left-Fredholm perturbations since the class of left Fredholm perturbation is a two-sided ideal of the set of all bounded operators. According to Theorem 2.4, one has (νI − 𝒜)−1 − (νI − 𝒬)−1 ∈ ℱlb (X × Y). Consequently, σel (𝒜) := σel (A1 ) ∪ σel (D).

Left–right essential spectra of one-sided operator matrix and application

| 169

(b) In this case, we will adopt the same reasoning as for the above part. It is sufficient to replace ℱlb (X × Y) by ℱrb (X × Y). (c) An immediate consequence of parts (a) and (b) reveals that σew (𝒜) := σew (A1 ) ∪ σew (D),

(3.3)

i(νI − 𝒜) := i(νI − A1 ) + i(νI − D).

(3.4)

with

Remark 3.7. (a) For other types of essential spectrum, the result of Theorem 3.6 is still valid. Unfortunately, for Schechter and Browder essential spectra, some sufficient conditions will be added to make it true. (b) If ΦX ≡ ΨY ≡ 0, we return to the case of an unbounded operator matrix with maximal domain. So, the results obtained by A. Y. Konstantinov et al. in [12] are provided in this case. Throughout, our advances for a one-sided operator matrix model are new and more general in comparison with those developed in [2, 3, 7, 9, 12]. An easy technique for the computation of some essential spectra of an example of integro-differential operator in a nuclear space will be derived in the next section to illustrate the importance of the theoretical results.

4 Application to a physical example of transport operators Our advances on spectral theory of an operator matrix consist in demonstrating how the model of transport equations, namely 𝒜ℋ (

f f f ) = 𝒯ℋ ( )+K( ) g g g

with specific boundary condition (

fi 0 )=( 0 gi = ℋ(

H fo )( o ) H g fo ), go

170 | S. Bouzidi and I. Walha where H is an abstract bounded linear operator defined on suitable boundary spaces, 𝒯ℋ (

𝜕f f −v 𝜕x (x, v) − σ1 (v)f (x, v) ) := ( g 0

:= (

T 0

0 ) (x, v) − σ2 (v)g(x, v) −v 𝜕g 𝜕x

0 f )( ) TH g

0 K

and K := ( K2 K31 ), where Kj , j = {1, 2, 3} are bounded linear operators defined on X := L1 ((−a, a) × (−1, 1), dxdv), a > 0, by {

Kj : X 󳨀→ X,

1

f 󳨃󳨀→ Kj f : (x, v) 󳨃󳨀→ ∫−1 κj (x, v, v󸀠 )f (x, v󸀠 ) dv󸀠 ,

can be treated within our framework. Here the functions are as follows: – σj (⋅) is called the collision frequency; – κj (⋅, ⋅, ⋅) is called the scattering kernel; – ( gf ) represents the angular density of particles such as gas molecules, photons, neutrons, etc., in a homogeneous slab of thickness 2a. Our assumptions are in general formulated as follows: σj (⋅) ∈ L∞ (−1, 1), { { κ : (−a, a) × (−1, 1) × (−1, 1) 󳨀→ ℝ is a measurable function, { { j H ∈ ℒ(X o , X i ), { where the boundary spaces X o and X i are identified as X o := L1 (−a × (−1, 0), |v|dv) × L1 (a × (0, 1), |v|dv) and X i := L1 (−a × (0, 1), |v|dv) × L1 (a × (−1, 0), |v|dv). To make our model of transport operators have the form of a one-sided operator matrix as Section 3, we will first define the partial Sobolev space 𝒲 by 𝒲 := {f ∈ X : v

𝜕f ∈ X}. 𝜕x

In the theory of transport operators, we know that any function φ ∈ 𝒲 has traces on {−a}×(−1, 0) and {a}×(0, 1) which belong to the spaces X o and X i , respectively (see, for instance, [5]). These trace functions will be denoted by φo and φi , corresponding to the

Left–right essential spectra of one-sided operator matrix and application | 171

outgoing and the incoming fluxes (“o” for outgoing and “i” for incoming), respectively, and expressed as: { { { { { { { { {

φi : v ∈ (0, 1) 󳨀→ φ(−a, v), φi : v ∈ (−1, 0) 󳨀→ φ(a, v), φo : v ∈ (−1, 0) 󳨀→ φ(−a, v), φo : v ∈ (0, 1) 󳨀→ φ(a, v).

An explanation of the component entries of the transport operator matrix model is provided as follows: – The closed linear operator T is defined on its maximal domain as T : 𝒟(T) ⊆ X 󳨀→ X, { { { 𝜕f f 󳨃󳨀→ Tf : (x, v) 󳨃󳨀→ −v 𝜕x (x, v) − σ1 (v)f (x, v), { { { { 𝒟(T) := 𝒲 ; –

The streaming operator TH is defined as TH : 𝒟(TH ) ⊆ X 󳨀→ X, { { { (x, v) − σ2 (v)g(x, v), g 󳨃󳨀→ TH g : (x, v) 󳨃󳨀→ −v 𝜕g { 𝜕x { { i o { 𝒟(TH ) := {g ∈ 𝒲 : g = Hg }.

The corresponding transport problem is supplemented with a boundary condition i o modeled by the relation ( gf ) = ℋ( gf ) and used to clarify the boundary operators ΦX and ΨY of the previous section having the following form: {

ΦX : 𝒲 ⊂ X 󳨀→ X i , f 󳨃󳨀→ ΦX (f ) = f i ,

and

{

ΨX : 𝒲 ⊂ X 󳨀→ X i ,

g 󳨃󳨀→ ΨX (g) = Hg o .

Consequently, the above physical equation may be translated into the one-sided operator matrix having the form of Section 3 and formulated as 𝒜ℋ := (

T K2

K1 ) TH + K3

on 𝒟(𝒜ℋ ) := {(

f ) ∈ 𝒟(T) × 𝒟(TH ) : f i = Hg o } , g

with X = Y, Z = Z1 = X i , A := T, B := K1 , C := K2 , D := TH + K3 , ΦX := ΦX , and ΨY := ΨX . Physically, the operator A1 of the theoretical part of this paper corresponds to the transport operator with vacuum boundary condition (that is, H ≡ 0). So, we identify such an operator as follows: {

A1 := T|𝒲∩𝒩 (ΦX ) ,

𝒟(A1 ) := {f ∈ 𝒟(T) : f i = 0}.

172 | S. Bouzidi and I. Walha Moreover, its essential spectra is localized in the half-plane as: σek (A1 ) := {ν ∈ ℂ : Rel ν ≤ −ν1∗ },

ek = {e1, e2, ew, ess},

where νj∗ := lim inf σj (v), |v|󳨀→0

j = 1, 2.

The following lemma is crucial to verify the assumption (ℋ5 ). Lemma 4.1 ([9]). Let ν ∈ ρ(A1 ). The operator Kν is bounded by (Rel ν + ν1∗ )−1 and is defined on X i by { { { { { { { { { { { { { { { { { { { { {

Kν : X i 󳨀→ X f 󳨃󳨀→ Kν f = χ(0,1) (v)Kν+ f + χ(−1,0) (v)Kν− f , with Kν− f (x, v) := f (a, v)e

−(σ1 (v)+ν)|a−x| |v|

Kν+ f (x, v) := f (−a, v)e

−1 < v < 0,

,

−(σ1 (v)+ν)|a+x| |v|

,

0 < v < 1.

For such a physical model of transport operator matrix 𝒜ℋ , its corresponding Schur-complement, Mν , is expressed as follows: Mν := TH + K3 − K2 [−Kν ΨX + (νI − A1 )−1 K1 ],

for ν ∈ ρ(A1 ).

Remark 4.2. Clearly, while Kj ∈ ℒ(X), for j = {1, 2, 3}, A1 and TH are two closed, densely defined linear operators which have nonempty resolvent sets. The set Y1 := {g ∈ 𝒟(TH ) : ΨX g ∈ X i } := 𝒟(Mν ) may be regarded as a core of the operator TH + K3 . So, the assumptions ℋj , j = {1, . . . , 4, 6, . . . , 9} are still verified. Let us introduce the following definition for the collision operators Kj , for j = {1, 2, 3}, due to B. Lods [13]. Definition 4.3. The collision operator Kj , defined above, is said to be regular if the subset {k(x, ⋅, v󸀠 ), (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} is relatively weakly compact in L1 ((−1, 1), dv). The following lemma may be essential to derive our advances. Lemma 4.4 ([4]). Let the scalar ν ∈ ρ(A1 ) be such that rσ ((νI − A1 )−1 K3 ) < 1 (rσ (⋅) the spectral radius). 󸀠 ) (a) If the subset { k2 (x,⋅,v , (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} is relatively weakly compact in |v󸀠 | L1 ((−1, 1), dv), then K2 (νI − A1 )−1 ∈ 𝒲 (X), for Re ν > −ν1∗ .

Left–right essential spectra of one-sided operator matrix and application | 173

) (b) If { k1 (x,⋅,v , (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} is a relatively weakly compact subset of |v󸀠 | L1 ((−1, 1), dv), then 󸀠

K1 (νI − TH − K3 )−1 ∈ 𝒲 (X). Remark 4.5. Using Lemma 4.4 with the weak compactness assumption of H, we conclude, for ν ∈ ρ(A1 ) ∩ ρ(TH + K3 ) such that rσ ((νI − A1 )−1 K3 ) < 1, and the subsets

) ) , (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} and { k1 (x,⋅,v , (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} are relatively { k2 (x,⋅,v |v󸀠 | |v󸀠 | weakly compact in L1 ((−1, 1), dv), that 󸀠

󸀠

Rν := −K2 [−Kν ΨX + (νI − A1 )−1 K1 ] ∈ 𝒲 (X), (νI − D)−1 Cν := −(νI − TH − K3 )−1 K2 (νI − A1 )−1 ∈ 𝒲 (X), and Gν (νI − D)−1 := −[−Kν ΨX + (νI − A1 )−1 K1 ](νI − TH − K3 )−1 ∈ 𝒲 (X). By the above arguments, we state our result for the transport operator matrix model by using the notions and the theory of Fredholm perturbations. Theorem 4.6. Assume that H ∈ 𝒲 (X), rσ ((νI −A1 )−1 K3 ) < 1, for ν ∈ ρ(A1 ), K3 is a regular

) ) operator, while { k2 (x,⋅,v , (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} and { k1 (x,⋅,v , (x, v󸀠 ) ∈ (−a, a) × (−1, 1)} |v󸀠 | |v󸀠 | 󸀠

󸀠

are relatively weakly compact subsets of L1 ((−1, 1), dv). Then σek (𝒜ℋ ) := {ν ∈ ℂ : Re ν ≤ − min(ν1∗ , ν2∗ )},

ek = {el, er, ew}.

Proof. We denote 𝒬ℋ := (

A1 0

0 ). TH + K3

Let ν ∈ ρ(A1 ) ∩ ρ(Mν ) ∩ ρ(TH + K3 ), then ν ∈ ρ(𝒜ℋ ) ∩ ρ(𝒬ℋ ) and (νI − 𝒜ℋ )−1 − (νI − 𝒬ℋ )−1 may be written as (νI − 𝒜ℋ )−1 − (νI − 𝒬ℋ )−1 Gν (νI − TH − K3 )−1 Cν −Gν (νI − Mν )−1 Rν (νI − TH − K3 )−1 Cν := (

−(νI − TH − K3 )−1 Cν +(νI − Mν )−1 Rν (νI − TH − K3 )−1 Cν

−Gν (νI − TH − K3 )−1 +Gν (νI − Mν )−1 Rν (νI − TH − K3 )−1 ). −(νI − Mν ) Rν (νI − TH − K3 ) −1

−1

Taking into account Lemma 4.4 and Remark 4.5, we infer that the component entries of (νI − 𝒜ℋ )−1 − (νI − 𝒬ℋ )−1 are weakly compact on X.

174 | S. Bouzidi and I. Walha Hence, as a result of Theorem 3.6, we get σek (𝒜ℋ ) := σek (A1 ) ∪ σek (TH + K3 ),

ek ∈ {el, er, ew}.

Consequently, according to Remark 4.3 in [4] and using Eq. (2.1), we deduce: σek (A1 ) := {ν ∈ ℂ : Rel ν ≤ −ν1∗ },

ek ∈ {el, er, ew}.

Therefore, K3 is a regular operator and H ∈ 𝒲 (X), it follows that σek (TH + K3 ) := {ν ∈ ℂ : Rel ν ≤ −ν2∗ },

ek ∈ {el, er, ew}.

5 Conclusion The aim of this work was to provide significant advances in the theory of one-sided operator matrices involving the notions of Fredholm perturbation theory which are used to analyze the stability problem of such an operator matrix. Our approach was to provide a fine computation of some essential spectra of a transport operator matrix model with a specific boundary condition in a nuclear space.

Bibliography [1]

F. V. Atkinson, H. Langer, R. Mennicken, A. A. Shkalikov, The essential spectrum of some matrix operators. Math. Nachr. 5–20 (1994). [2] A. Batkai, P. Binding, A. Dijksma, R. Hrynivo, H. Langer, Spectral problems for operator matrices. Math. Nachr. 1408–1429 (2005). [3] S. Charfi, I. Walha, On relative essential spectra of block operator matrices and application. Bull. Korean Math. Soc. 681–698 (2016). [4] M. Dammak, A. Jeribi, N. Moalla, Essential spectra of some matrix operators and application to two-group transport operators with general boundary condition. J. Math. Anal. Appl. 1071–1090 (2006). [5] R. Dautray, J. L. Lions, Analyse mathématique et calcul numérique. Masson 9 (1988). [6] I. C. Gohberg, A. S. Markus, I. A. Feldman, Normally solvable operators and ideals associated with them. Am. Math. Soc. Transl. 63–84 (1967). [7] A. Jeribi, Spectral Theory and Applications of Linear Operators and Block Operator Matrices (Springer-Verlag, New York, 2015). [8] A. Jeribi, Linear Operators and Their Essential Pseudospectra (CRC Press, Boca Raton, 2018). [9] A. Jeribi, I. Walha, Gustafson, Weidmann, Kato, Wolf, Schechter and Browder essential spectra of some matrix operator and application to a two-group transport equation. Math. Nachr. 67–86 (2011). [10] A. Jeribi, N. Moalla, S. Yengui, Some results on perturbation theory of matrix operators, M-essential spectra and application to an example of transport operators. http://arxiv.org/ licenses/nonexclusive-distrib/1.0/.

Left–right essential spectra of one-sided operator matrix and application | 175

[11] T. Kato, Perturbation Theory of Linear Operators (Springer, New York, 1996). [12] A. Y. Konstantinov, Spectral theory of some matrix differential operators of mixed order. Ukr. Math. J. 1064–1072 (1998). [13] B. Lods, On linear kinetic equations involving unbounded cross-sections. Math. Methods Appl. Sci. 1049–1075 (2004). [14] R. Nagel, Well-posedness and positivity for systems of linear evolution equations. Conf. Semin. Mat. Univ. Bari 29 (1985). [15] R. Nagel, Towards a matrix theory for unbounded operator matrices. Math. Z. 57–68 (1989). [16] R. Nagel, Characteristic equations for the spectrum of generators. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 703–717 (1997). [17] J. Qi, S. Chen, Essential spectra of singular matrix differential operators of mixed order. J. Differ. Equ. 4219–4235 (2011). [18] A. A. Shkalikov, On the essential spectrum of matrix operators. Math. Notes 945–949 (1995).

Amor Fahem, Aref Jeribi, and Najib Kaddachi

Existence of Solutions for an integral equation of Chandrasekhar type in Banach algebras with respect to the weak topology Abstract: In this work, we study the existence of a solution for a coupled system of fractional integral equations of Chandrasekhar type in a suitable Banach algebra. This result is an application of a fixed point theorem for a 2 × 2 block operator matrix where the imputes are weakly sequentially continuous operators. Keywords: Banach algebra, Chandrasekhar equation, weakly sequentially continuous MSC 2010: 32A65, 47H08, 47H30

1 Introduction In this paper, we are concerned with the existence results for a system of two quadratic integral equations of Chandrasekhar type given as t

t { x(t) = f1 (t, x(t)) + g(t, y(t))(∫0 t+s u1 (s, y(s)) ds) ⋅ v, { t t { y(t) = f2 (t, y(t)) + (∫0 t+s u2 (s, x(s)) ds) ⋅ u,

v ∈ X\{0}, t ∈ [0, b], u ∈ X\{0}, t ∈ [0, b],

(1.1)

where X is a Banach algebra satisfying the following condition: (𝒫 ) {

For any sequences {xn } and {yn } of X such that xn ⇀ x and yn ⇀ y, one has xn ⋅ yn ⇀ x ⋅ y, where ⇀ denotes weak convergence.

Note that the system (1.1) may be remodeled as a fixed-point problem of the 2 × 2 block operator matrix (

A C

B ⋅ B󸀠 ). D

(1.2)

In this direction, H. H. G. Hashem in [5] used some results of [7] to prove the existence of a solution for a system of two quadratic integral equations of Chandrasekhar type by applying a fixed point theorem for the block operator matrix (1.2) defined on nonempty, bounded, closed and convex subsets of Banach algebras. Amor Fahem, Aref Jeribi, Department of Mathematics, Faculty of Sciences of Sfax, University of Sfax, Soukra Road Km 3.5 B.P. 1171, 3000 Sfax, Tunisia, e-mails: [email protected], [email protected] Najib Kaddachi, University of Kairouan, Faculty of Science and Technology of Sidi Bouzid, Agricultural University City Campus, 9100 Sidi Bouzid, Tunisia, e-mail: [email protected] https://doi.org/10.1515/9783110598193-013

178 | A. Fahem et al. The remainder of this paper is structured as follows. In Section 2, we give some preliminary results. In addition, we give a reformulation of Theorem 3.4 in [6]. Finally, we apply Theorem 2.5 to prove the existence result for the system (1.1).

2 Preliminaries and mains results Let X be a Banach algebra with the norm ‖ ⋅ ‖ and the zero element θ. Denote by ℬ(X) the group of all nonempty bounded subsets of X and by 𝒲 (X) the subfamily of ℬ(X) containing all weakly compact subsets of X. We remind that the notion of the measure of weak noncompactness β on ℬ(X) was introduced by De Blasi [3] as follows: β(S) = inf{r > 0 : there exists K ∈ 𝒲 (X) such that S ⊆ K + Br }, where Br is the closed ball in X centered at 0 with radius r. Definition 2.1. An operator T : X 󳨀→ X is said to be weakly sequentially continuous on X if, for every sequence {xn } ⊂ X with xn ⇀ x, we have Txn ⇀ Tx. Definition 2.2. An operator T : X 󳨀→ X is called 𝒟-Lipschitz if there exists a continuous and nondecreasing function ϕT : ℝ+ 󳨀→ ℝ+ satisfying ‖Tx − Ty‖ ≤ ϕT (‖x − y‖) for all x, y ∈ X with ϕT (0) = 0. Sometimes we call the function ϕT a 𝒟-function of T on X. Let S ⊂ X and T : S 󳨀→ X. If T is bounded and β(T(S1 )) < β(S1 ) for any S1 ∈ ℬ(S) with β(S1 ) > 0, then T is called β-condensing. In the following, we will use several lemmas which were established in [1]. Lemme 2.3. Let S be a nonempty, closed, convex subset of a Banach space X. Assume that F : S 󳨀→ S is a weakly sequentially continuous and condensing mapping with respect to β. In addition, if F(S) is bounded, then F has at least one fixed point in S. Lemme 2.4. If V ∈ ℬ(X) and K ∈ 𝒲 (X), then β(V ⋅ K) ≤ ‖K‖β(V). In [6], A. Jeribi, B. Krichen, and N. Kaddachi proved the next result in case of Lipschitz mappings. Now, we give the proof for the case of 𝒟-Lipschitz maps. Theorem 2.5. Let X be a Banach algebra satisfying Condition (𝒫 ) and let S be a nonempty, convex, and closed subset of X. Suppose that A, C : S 󳨀→ X and B, B󸀠 , D : X 󳨀→ X are weakly sequentially continuous operators satisfying the following conditions: (i) A, B, and C are 𝒟-Lipschitz with the 𝒟-functions ϕA , ϕB , and ϕC , respectively; (ii) B󸀠 (S) is weakly relatively compact;

Existence results for a system of Chandrasekhar’s equation

| 179

(iii) A(S) and B(S) are bounded; (iv) D is a contraction with constant k and C(S) ⊆ (I − D)(S); (v) Ax + B(I − D)−1 Cx ⋅ B󸀠 (I − D)−1 Cx ∈ S for all x ∈ S. Then the operator matrix (1.2) has at least one fixed point in S × S as soon as ϕA (r) + 1 MϕB ∘ ( 1−k ϕC )(r) < r, where M = ‖B󸀠 (I − D)−1 C(S)‖.

3 Existence theorem In the section we prove the existence of solutions for the coupled system (1.1) in the space C([0, 1], X) of all continuous functions on [0, 1] endowed with the norm ‖ ⋅ ‖∞ , where X is a Banach algebra satisfying Condition (𝒫 ). Clearly, C(J, X) becomes a Banach algebra satisfying Condition (𝒫 ) (see [1]). Let us now introduce the following assumptions needed in the sequel: (ℋ0 ) The function fi : J × X → X, i = 1, 2 is such that: (a) fi is a contraction with constant ki ; (b) fi is weakly sequentially continuous with respect to the second variable; (b) Mi = sup(t,x)∈J×X |fi (t, x)|. (ℋ1 ) The function ui : J × X → X, i = 1, 2 is such that: (a) t 󳨃󳨀→ ui (t, x) is measurable for each x ∈ X; (b) x 󳨃󳨀→ ui (t, x) is weakly sequentially continuous for almost t ∈ J; (c) ui is a contraction with constant li with respect to the second variable; (d) ‖ui (t, x)‖ ≤ mi (t) ∈ L1 (J) for all (t, x) ∈ J × X, and b

1 mi (s) ds. (e) λi = sup ∫0 t+s (ℋ2 ) The function g : J × X → X is such that: (a) g is weakly sequentially continuous with respect to the second variable; (b) There exist constant L > 0, K > 0 such that

0 < g(t, x(t)) − g(t, y(t)) ≤

L(x − y) , K + (x − y)

for all t ∈ J and x, y ∈ X with x ≥ y. Moreover, L ≤ K, and (c) ‖g‖ = N. Theorem 3.1. Assume that the assumptions (ℋ0 )–(ℋ2 ) hold. Furthermore, if {

L‖u‖‖m1 ‖ ≤ K, k1 +

then the system (1.1) has a solution.

1 bl ‖v‖ 1−k2 2

≤ 1,

180 | A. Fahem et al. Proof. Define a subset S in 𝒞 ([0, 1], X) by S = {x ∈ 𝒞 ([0, 1], X), ‖x‖ ≤ inf{λ1 N‖v‖ + M1 , λ2 ‖v‖ + M2 }}. Note that the problem (1.1) can be written in the following form: x(t) A )=( y(t) C

(

B ⋅ B󸀠 x(t) )( ), D y(t)

where A, B, C, D, and B󸀠 on S are defined by { { { { { { { { { { { { { { { { { { { { {

(Ax)(t) = f1 (t, x(t)), (By)(t) = g(t, y(t)), t

t u (s, x(s)) ds) t+s 2

(Cx)(t) = (∫0

⋅ v,

t ∈ [0, 1] and v ∈ X\{0},

(Dy)(t) = f2 (t, y(t)), t

(B󸀠 y)(t) = (∫0

t u (s, y(s)) ds) t+s 1

⋅ u,

t ∈ [0, 1] and u ∈ X\{0}.

In order to apply Theorem 2.5, we have to justify the following claims: Claim 1. A, B, and C are 𝒟-Lipschitz under our assumptions. Claim 2. We show that C(S) ⊆ (I − D)(S). Indeed, let x ∈ S be a fixed point. Define a mapping {

ϕx : 𝒞 ([0, 1], X) 󳨀→ 𝒞 ([0, 1], X), y 󳨃󳨀→ Cx + Dy.

From assumption (ℋ0 ), we infer that the operator ϕx is a k2 -contraction, then an application of Banach fixed point theorem yields the existence of a unique point y ∈ 𝒞 ([0, 1], X) such that y = Cx + Dy, and consequently C(S) ⊆ (I − D)(𝒞 ([0, 1], X)). Since y ∈ 𝒞 ([0, 1], X), there is t ∗ ∈ [0, 1] such that ‖y‖∞

󵄨󵄨 󵄨󵄨 t ∗ ∗ 󵄨󵄨 t 󵄨 󵄨 󵄨󵄨 p2 (s, x(s)) ds)v󵄨󵄨󵄨 + |f2 (t ∗ , x(t ∗ )| ≤ λ2 ‖v‖ + M2 . = 󵄨󵄨󵄨y(t ∗ )󵄨󵄨󵄨 ≤ 󵄨󵄨󵄨(∫ ∗ 󵄨󵄨 󵄨󵄨 t + s 󵄨 󵄨 0

Claim 3. By using assumption (ℋ1 ), we have MψB ∘ (

1 ψ )(r) + ψA (r) ≤ r. 1 − k2 c

Next, let us fix an arbitrary y ∈ 𝒞 ([0, 1], X) and x ∈ S such that y = Ax + B(I − D)−1 Cx ⋅ B󸀠 (I − D)−1 Cx. Then 󵄩󵄩 󵄩 󵄩󵄩y(t)󵄩󵄩󵄩 ≤ M1 + N‖v‖λ1 . Then by Theorem 2.5, we conclude that the system (1.1) has a fixed point in S × S.

Existence results for a system of Chandrasekhar’s equation

| 181

We have the following particular cases that constitute the versions of quadratic integral equations of Chandrasekhar type: (a) In the special case when x = y, f1 (t, x(t)) = 1, g(t, x(t)) = x(t), u1 (t, x(t)) = φ(t)x(t), f2 (t, y(t)) = y(t), and u2 (t, x(t)) = 0, QIE (1.1) is reduced to QIE t

x(t) = 1 + x(t) ∫ 0

t φ(s)x(s) ds. t+s

(3.1)

The Chandrasekhar’s integral equation (3.1) has been discussed in [2] which investigated different aspects of the solutions under suitable conditions. t a (b) If we take f1 (t, x(t)) = a(t) ∫0 u(t, s, x(s)) ds, g(t, x(t)) = ∫0 v(t, s, x(s)) ds, f2 (t, y(t)) = t+s y(t), and u1 (t, s, x(s)) = t u(t, s, x(s)), we obtain the following integral equation: a

t

a

x(t) = a(t) ∫ u(t, s, x(s)) ds + (∫ v(t, s, x(s)) ds) ⋅ (∫ u(t, s, x(s)) ds). 0

0

(3.2)

0

Equation (3.2) was examined in the paper [4] and some special cases of this equation were considered in [8, 9].

Bibliography [1] A. Ben Amar, S. Chouayekh, A. Jeribi, Fixed point theory in a new class of Banach algebras and application. Afr. Math. 24, 705–724 (2013). [2] S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960). [3] F. S. De Blasi, On a property of the unit sphere in a Banach space. Bull. Math. Soc. Sci. Math. Roum. 21(69), 259–262 (1977). [4] B. C. Dhage, On α-condensing mappings in Banach algebras. Math. Stud. 63, 146–152 (1994). [5] H. H. G. Hashem, Solvability of a 2 × 2 block operator matrix of Chandrasekhar type on a Banach algebras. Filomat 16, 5169–5175 (2017). [6] A. Jeribi, N. Kaddachi, B. Krichen, Existence results for a system of nonlinear integral equations in Banach algebras under weak topology. Fixed Point Theory 18, 247–267 (2017). [7] N. Kaddachi, A. Jeribi, B. Krichen, Fixed point theorems of block operator matrices on Banach algebras and an application to functional integral equations. Math. Methods Appl. Sci. 36, 659–673 (2013). [8] M. A. Krasnoselskii, Topological Methods in the Theory of Nonlinear Integral Equations (Pergamon, New York, 1964). [9] B. Rzepecki, Measure of noncompactness and Krasnoselskii’s fixed point theorem. Bull. Acad. Polon. Scz. St. Sci. Math. Astron. Phys. 24, 861–865 (1976).

A. Aberqi, J. Bennouna, and M. Elmassoudi

Strong evolution problems in Musielak spaces Abstract: Our motivation in the current paper is to study the solvability of nonlinear evolution problems in inhomogeneous Musielak spaces staying away from coercivity limitations on the subsequent lower request term. Keywords: Musielak space, evolution equations, entropy solutions MSC 2010: Primary 35K86, Secondary 35K55, 35A01

1 Introduction In this article, we study the following unilateral nonlinear parabolic problem: 𝜕t g(x, v) − div(σ(x, t, v, Dv) + K(x, t, v)) = f

v≥ζ

a. e. in OT ,

u|𝜕O×(0,T) = 0,

in OT ,

g(x, v(x, 0)) = g(x, v0 ) in O,

(1.1) (1.2) (1.3)

where O ⊂ ℝd is a bounded Lipschitz domain (d ≥ 2) and the cylinder OT = O × (0, T) (T > 0). We will assume that φ is a generalized N-function (see Preliminaries); g(x, ⋅)

is a strictly increasing 𝒞 1 (ℝ)-function,

(1.4)

g ∈ L∞ (O × ℝ), and g(x, 0) = 0. There exist λ > 0 and (g 1 , g 2 ) ∈ L∞ (O) ∩ Lφ (O) such that λ ≤ 𝜕s g(x, s) ≤ g 1 (x),

󵄨󵄨 󵄨 2 󵄨󵄨Dx (𝜕s g(x, s))󵄨󵄨󵄨 ≤ g (x)

a. e. x ∈ O, ∀s ∈ ℝ,

(1.5)

σ : OT × ℝ × ℝd → ℝd is a Carathéodory function such that for a. e. x ∈ O and for all s ∈ ℝ, ξ , ξ ∗ ∈ ℝd , ξ ≠ ξ ∗ , 󵄨󵄨 󵄨 −1 󵄨󵄨σ(x, t, s, ξ )󵄨󵄨󵄨 ≤ ν(a0 (x, t) + φx P(x, |s|)) with a0 ∈ Eφ (OT ), (σ(x, t, s, ξ ) − σ(x, t, s, ξ ∗ ))(ξ − ξ ∗ ) > 0,

(1.6)

σ(x, t, s, ξ ).ξ ≥ αφ(x, |ξ |).

(1.8)

(1.7)

Furthermore, K : OT × ℝ → ℝ is a Carathéodory function such that α 󵄨󵄨 󵄨 −1 󵄨󵄨K(x, t, s)󵄨󵄨󵄨 ≤ a(x, t) + r(x, t)φx φ(x, 0 |s|), δ A. Aberqi, Sidi Mohamed Ben Abdellah University, ENSA, Fez, Morocco J. Bennouna, M. Elmassoudi, Sidi Mohamed Ben Abdellah University, Faculté des Sciences, Fez, Morocco https://doi.org/10.1515/9783110598193-014

(1.9)

184 | A. Aberqi et al. where 0 < α0 < 1, ‖r(x, t)‖L∞ (OT ) < min( α α+1 ; (2α function which belongs to Eφ (OT );

0

λα

0 +1)‖g

1‖

1

2

f ∈ L1 (OT ),

L∞ +α0

), q : OT → ℝ+ is positive (1.10)

(v0 , g(⋅, v0 )) ∈ (L (O)) .

(1.11)

Let ζ be a measurable function with values in ℝ such that ζ ∈ W01 Eφ (OT ) ∩ L∞ (OT ),

𝜕ζ ∈ L1 (OT ) 𝜕t

and

v0 ≥ ζ ,

(1.12)

and let Kζ = {v ∈ W01,x Lφ (OT ) : v ≥ ζ a. e. in OT }. The solvability of (1.1)–(1.3) is well known in the classical framework (see [5, 13, 12, 2, 1, 6]) and the references therein. In the framework of modular spaces, we refer to [9] and [10] for the linear evolution case and the case where K = 0. Our motivation in this paper is to sum up [9, 10] and extend the theory either to include the lower nonlinear term K or without the △2 -condition on φ and φ, which presents some intricacy in understanding the double matching. The lack of regularity on the obstacle function and the energy space which does not have sufficient topological properties motivate the study of these equations. The structure of this article is as follows: In Section 2, we define a generalized N-function and the inhomogeneous Musielak spaces. In Section 3, the main result is formulated in Theorem 3.1. First of all, we show that if we impose certain conditions on a sequence of solution of the approximate problems of (1.1)–(1.3), then the limit of this sequence becomes the solution of our problem, and then we go back to prove the validity of these conditions.

2 Preliminaries Let φ : O × ℝ+ → ℝ be such that (i) t → φ(⋅, t) is an N-function; (ii) x → φ(x, ⋅) is a measurable function. Then such a function φ is called a generalized N-function. For more details, see [8]. Proposition 2.1 ([8]). If P ≪ φ and for all t > 0, supx∈O P(x, t) < ∞, then for all ϵ > 0, there exists Cϵ > 0 such that P(x, t) ≤ φ(x, ϵt) + Cϵ

for a. e. x ∈ O, for all t > 0.

The Musielak space Lφ (O) is defined as v Lφ (O) = {v : O → ℝ measurable: ϱM,O ( ) < ∞, for some λ > 0}, λ where ϱφ,O (v) = ∫O φ(x, |v(x)|) dx.

(2.1)

Strong evolution problems in Musielak spaces | 185

Furthermore, φ(x, s) = supt≥0 (st − φ(x, t)) is the conjugate generalized N-function of φ, while Eφ (O) = {v : O → ℝ, measurable functions: ∫ φ(x, O

|v(x)| ) dx < ∞ ∀λ > 0}. λ

The Musielak–Sobolev space is defined as follows: W 1 Lφ (O) = {v ∈ Lφ (O) : Dα v ∈ Lφ (O),

∀|α| ≤ 1},

endowed with the norm ‖v‖1φ,O = inf{λ > 0 : ∑ ϱM,O |α|≤1

(Dα v ) ≤ 1}. λ

All throughout this paper, the two complementary generalized N-functions φ and φ satisfy the conditions of Theorem 2.1 in [4]. Lemma 2.1 ([3]). Assuming that φ(x, ⋅) decreases with respect to one of the coordinates of x, there exists a constant δ > 0 which depends only on O such that ∫ φ(x, |v|) dx dt ≤ ∫ φ(x, δ|Dv|) dx dt. OT

(2.2)

OT

Now we define energy space as follows: W 1,x Lφ (OT ) = {v ∈ Lφ (OT ) : Dαx v ∈ Lφ (OT ), ∀α ∈ ℕd , |α| ≤ 1}, where α ∈ ℕd ⋅ Lemma 2.2 ([8]). If O ⊂ ℝd is a set with segment property, and α < g ∈ ℝ, then if ∈ (W01,x Lφ (O × (α, g)) ∩ L1 (O × (α, g)))∗ , u ∈ W01,x Lφ (O × (α, g)) ∩ L1 (O × (α, g)) and 𝜕v 𝜕t we have v ∈ 𝒞 ([α, g], L1 (O)).

3 Main results Let φ and P be two generalized N-functions such that φ is decreasing in x and P ≪ φ. Theorem 3.1. Under the assumptions (1.4)–(1.12), the equation (1.1)–(1.3) admits at least one solution v satisfying the following conditions: g(x, v) ∈ L∞ (0, T; L1 (O)), v≥ζ

a. e. in OT ,

g(x, v(x, 0)) = g(x, v0 ) in O,

Tk (v) ∈ W01,x Lφ (OT ),

186 | A. Aberqi et al. T

v

0

0

v0

∫⟨𝜕s h; ∫ 𝜕s g(x, z)Tk󸀠 (z

− h) dz⟩ ds + ∫ ∫ 𝜕s g(x, s)Tk (s − h(0)) ds dx O 0

T

T

+ ∫ ∫ σ(x, s, v, Dv)DTk (u − h) dx ds + ∫ ∫ K(x, s, v)DTk (v − h) dx ds T

0O

0O

≤ ∫ ∫ fTk (v − h) dx ds,

for all k > 0,

0O

∀h ∈ Kζ ∩ L∞ (OT ) with h(T) = 0, and

𝜕h ∈ Lφ (0, T; W −1 Lφ (O)). 𝜕t

(3.1)

Proof. For any n ∈ ℕ∗ , we define the regularization gn (x, s) = g(x, Tn (s)),

σn (x, t, s, ξ ) = σ(x, t, Tn (s), ξ ),

Kn (x, t, s) = K(x, t, Tn (s)),

a. e. (x, t) ∈ OT , for all (s, ξ ) ∈ ℝ × ℝd . Consider fn ∈ L1 (OT ) : fn → f

strongly in L1 (OT ),

(3.2)

and v0n ∈ 𝒞0∞ (O) : gn (x, v0n ) → g(x, v0 )

strongly in L1 (O).

(3.3)

Consider the penalized equations: 𝜕t gn (x, vn ) − div(σn (x, t, vn , Dvn ) + Kn (x, t, vn )) = nTn (vn − ζ )− + fn { { { vn (x, t) = 0 { { { { gn (x, vn (x, 0)) = gn (x, v0n )

in OT , on ∑,

(3.4)

in O.

Let vn ∈ W01,x Lφ (OT ). Then, using (1.9), we get a(x, t) 󵄨󵄨 󵄨 ) + ϵφ(x, |Dvn |) 󵄨󵄨Kn (x, t, vn )Dvn 󵄨󵄨󵄨 ≤ φ(x, ϵ 󵄩 󵄩 + 󵄩󵄩󵄩r(⋅, ⋅)󵄩󵄩󵄩L

∞ (OT )

α 󵄨 1 󵄨 φ(x, φ−1 φ(x, 0 󵄨󵄨󵄨Tn (vn )󵄨󵄨󵄨) + ϵφ(x, |Dvn |)), ϵ x δ

thus 󵄨󵄨 󵄨 󵄩 󵄩 󵄨󵄨Kn (x, t, vn )Dvn 󵄨󵄨󵄨 ≤ dn,ϵ (x, t) + ϵ(1 + 󵄩󵄩󵄩r(⋅, ⋅)󵄩󵄩󵄩L∞ (OT ) )φ(x, |Dvn |), and by (1.8) 󵄩 󵄩 [σn (x, t, vn , Dvn ) + Kn (x, t, vn )]Dvn ≥ [α − ϵ(1 + 󵄩󵄩󵄩r(⋅, ⋅)󵄩󵄩󵄩L

∞ (OT )

)]φ(x, |Dvn |) − dn,ϵ (x, t).

187

Strong evolution problems in Musielak spaces |

Choosing ϵ such that ϵ
p. The pointwise convergence of vn combined with (3.1) gives σn (vn , Dvn )DTk (vn − γ) ⇀ σ(v, Dv)DTk (v − γ) weakly in L1 (OT ). Since Kn (vn )DTk (vn −γ) = K(Tp (vn ))DTk (Tp (vn )−γ) a. e. in OT , with p = k+‖γ‖∞ , the weak convergence of Tp (vn ) to Tp (v) as n → +∞ allows us to have Kn (vn )DTk (vn − γ) ⇀ K(v)DTk (v − γ) weakly in L1 (OT ). Using (3.2), we can easily see that fn Tk (vn −γ) → fTk (v−γ) in L1 (OT ). Since −nTn (vn − ζ )− Tk (vn − γ) ≥ 0 and γ ∈ Kζ , by letting n → +∞ in (3.6), we achieve that v satisfies (3.1). 𝜕G (x,v ) Let us show that g(⋅, v(⋅, 0)) = g(⋅, v0 ). By (3.6), we show that p𝜕t n is bounded in 1 −1,x L (OT ) + W Lφ (OT ). As a consequence, we get that Gp (x, vn ) ∈ C 0 ([0, T]; L1 (O)) (see Lemma 2.2). Whereas the smoothness of Gp implies that Gp (x, vn (x, 0)) → Gp (x, v(x, 0)) strongly in L1 (O), thus Gp (x, vn (x, 0)) = Gp (x, v0n ) converges to Gp (x, u(x, 0)) strongly in L1 (O), Gp (x, v(x, 0)) = Gp (x, v0 ) a. e. in O, and we get g(x, v(x, 0)) = g(x, v0 ) a. e. in O, when p tends to +∞. In the next two steps we will check the conditions of Proposition 3.5.

Step 1. We prove that Tk (vn ) → Tk (v)

weakly in W 1,x Lφ (OT )

(3.7)

and v ≥ ζ.

a. e. in OT .

(3.8)

Strong evolution problems in Musielak spaces |

189

For this, fix k > 0 and let a function ψn = gn (x, vn ) − g(x, ζ ) be in L1 (OT ). Then 𝜕gn (x, vn ) 𝜕g(x, ζ ) Dvn + Dζ ]χ{|ψn |≤k} 𝜕s 𝜕s

DTk (ψn ) = [

+ (Dx gn (x, vn ) − Dx g(x, ζ ))χ{|ψn |≤k}

(3.9)

and |vn | ≤

1 |ψ | + ‖ζ ‖L∞ (OT ) . λ n

(3.10)

Letting τ ∈ (0, T) and multiplying (3.4) by Tk (ψn )χ(0,τ) , we obtain τ

∫ Bk (ψn )(τ) dx + ∫ ∫ σn (x, t, vn , Dvn )DTk (ψn ) dx dt 0O

O τ

τ

+ ∫ ∫ Kn (vn )DTk (ψn ) dx dt − n ∫ ∫ Tn (vn − ζ )− Tk (ψn ) dx dt 0O

0O

󵄩 󵄩 ≤ k(‖fn ‖L1 (OT ) + 󵄩󵄩󵄩g(x, v0n ) − g(x, ζ (0))󵄩󵄩󵄩L1 (O) ),

(3.11)

r

where Gk (r) = ∫0 𝜕t g(x, s)Tk (s) ds.

From the definition of the truncation function, we easily show that ∫ Bk (ψn ) dx ≥ 0,

∀k > 0.

(3.12)

O

For the second right-hand side of (3.11), we have τ

τ

∫ ∫ σn (vn , Dvn )DTk (ψn ) dx dt = ∫ ∫ σn (vn , Dvn ) 0O

0O

𝜕gn (x, vn ) Dvn χ{|ψn |≤k} dx dt 𝜕s

τ

+ ∫ ∫ σn (vn , Dvn ) 0O

𝜕gn (x, vn ) Dζχ{|ψn |≤k} dx dt 𝜕s

τ

+ ∫ ∫ σn (vn , Dvn )(Dx gn (x, vn ) − Dx g(x, ζ ))χ{|ψn |≤k} dx dt. 0O

Using (1.8), we get τ

τ

𝜕g (x, vn ) Dvn χ{|ψn |≤k} dx dt ≥ λα ∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt. ∫ ∫ σn (vn , Dvn ) n 𝜕s 0O

0O

190 | A. Aberqi et al. Now using (1.6), (2.1), Young inequality, and Lemma 2.1, for any ϵ > 0, τ

∫ ∫ σn (vn , Dvn ) 0O

𝜕gn (x, vn ) Dζχ{|ψn |≤k} dx dt 𝜕s τ

τ

󵄩 󵄩 ≤ υ󵄩󵄩󵄩g 1 󵄩󵄩󵄩L

∞

(O) [∫ ∫ φ(x, a0 (x, t)) dx dt + ∫ ∫ φ(x, |Dζ |) dx dt] 0O

0O

󵄩 󵄩 + ϵυ󵄩󵄩󵄩g 1 󵄩󵄩󵄩L

τ ∞ (O)

∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt + Cϵ . 0O

Using again (1.6), (2.1), and Young inequality, we get τ

∫ ∫ σn (vn , Dvn )(Dx gn (x, vn ) − Dg(x, ζ ))χ{|ψn |≤k} dx dt 0O

T

T

0O

0O

󵄨 󵄨 ≤ ν[∫ ∫ φ(x, a0 (x, t)) dx dt + ∫ ∫ φ(x, 󵄨󵄨󵄨Dx gn (x, vn ) − Dx g(x, ζ )󵄨󵄨󵄨) dx dt] T

t

󵄨 󵄨 + νϵ ∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt + ν ∫ ∫ φ(x, 󵄨󵄨󵄨Dx gn (x, vn ) − Dx g(x, ζ )󵄨󵄨󵄨) dx dt. 0O

0O

Combining (1.9) and (2.1) with Lemma 2.1, we have τ

∫ ∫ Kn (vn )DTk (ψn ) dx dt 0O τ

T

a(x, t) 󵄩 󵄩 ) dx dt + ϵ ∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt] ≤ 󵄩󵄩󵄩g 1 󵄩󵄩󵄩L [∫ ∫ φ(x, ∞ ϵ 0O

0O

T

T

0O

0O

󵄩 󵄩 + 󵄩󵄩󵄩g 1 󵄩󵄩󵄩L [∫ ∫ φ(x, a(x, t)) dx dt + ∫ ∫ φ(x, |Dζ |) dx dt] ∞

T

T

0O

0O

󵄨 󵄨 + ∫ ∫ φ(x, a(x, t)) dx dt + ∫ ∫ φ(x, 󵄨󵄨󵄨Dx gn (x, vn ) − Dx g(x, ζ )󵄨󵄨󵄨) dx dt τ

󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩g 1 󵄩󵄩󵄩L 󵄩󵄩󵄩r(x, t)󵄩󵄩󵄩L (O ) (α0 + 1) ∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt ∞ ∞ T 0O

τ

󵄩 󵄩 󵄩 󵄩 + 󵄩󵄩󵄩g 1 󵄩󵄩󵄩L 󵄩󵄩󵄩r(x, t)󵄩󵄩󵄩L (O ) [α0 ∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt + ∫ φ(x, |Dζ |) dx dt] ∞ ∞ T 0O

OT

191

Strong evolution problems in Musielak spaces | τ

󵄩 󵄩 + 󵄩󵄩󵄩r(x, t)󵄩󵄩󵄩L

∞ (OT

) [α0 ∫ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt 0O

󵄨 󵄨 + ∫ φ(x, 󵄨󵄨󵄨Dx gn (x, vn ) − Dx g(x, ζ )󵄨󵄨󵄨) dx dt]. OT

Finally, using (1.8), we get τ

τ

1 ∫ ∫ φ(x, DTk (vn ))χ{|ψn |≤K} dx dt − n ∫ ∫ Tn (vn − ζ )− Tk (ψn ) dx dt ≤ kC1 + Cϵ , C󸀠 0O

0O

where 1 󵄩 󵄩 󵄩 󵄩 󵄩 󵄩 = λα − ϵ[(ν + 1)󵄩󵄩󵄩g 1 󵄩󵄩󵄩L + ν] − 󵄩󵄩󵄩r(x, t)󵄩󵄩󵄩L (O ) [(2α0 + 1)󵄩󵄩󵄩g 1 󵄩󵄩󵄩L + α0 ]. ∞ ∞ T ∞ C2 Choosing ϵ such that ϵ
0 and k = ‖g 1 ‖L∞ (η + ‖ζ ‖L∞ ), using (3.13), we have 󵄨 󵄨 ∫ φ(x, 󵄨󵄨󵄨DTη (vn )󵄨󵄨󵄨) dx dt ≤ ∫ φ(x, |Dvn |)χ{|ψn |≤k} dx dt ≤ ηC3 + C4 ,

Qτ

where C3 = kC2 C1 and C4 = C2 Cϵ .

Qτ

192 | A. Aberqi et al. Using (2.2), we have η inf φ(x, )meas{|vn | > η} ≤ x∈O δ

φ(x,

∫

|Tη (vn )| δ

{|vn |>η}

) dx dt

τ

󵄨 󵄨 ≤ ∫ ∫ φ(x, 󵄨󵄨󵄨DTη (vn )󵄨󵄨󵄨) dx dt ≤ ηC3 + C4 , 0O

then meas{|vn | > η} ≤

ηC3 + C4 η , infx∈O φ(x, δ )

for all n and η.

Furthermore, if t.h(t) ≤ ess infx∈O φ(x, t), where h : ℝ → ℝ+ is such that limt→+∞ h(t) = +∞, then we get limη→∞ meas{|vn | > η} = 0. Proposition 3.2. If vn satisfies (16), then we have vn → v

a. e. in OT , where v is a measurable function on OT ; 1

gn (x, vn ) → g(x, v)

a. e. in OT , g(x, v) ∈ L∞ (0, T, L (O));

σn (Tk (vn ), DTk (vn )) ⇀ κk

N

in (Lφ (OT )) , for σ(ΠLφ , ΠEφ ),

(3.15) (3.16) (3.17)

for some κk ∈ (Lφ (OT ))N . Proof. We first prove (3.15) and (3.16). Let zl be an increasing C 2 (ℝ)-function such that {

zl (r) = r zl (r) = l

if |r| ≤ 2l , if |r| ≥ l.

Multiplying (3.4) by zl󸀠 (vn ), we get l 𝜕Gn,θ (x, vn )

𝜕t

= div(zl󸀠 (vn )σn (vn , Dvn )) − zl󸀠󸀠 (vn )σn (vn , Dvn )Dvn + div(zl󸀠 (vn )Kn (vn )) − zl󸀠󸀠 (vn )Kn (vn )Dvn + fn zl󸀠 (vn ) θ 𝜕gn (x,s) 󸀠 zl (s)ds. 𝜕s

l where Gn,θ (x, θ) = ∫0

󸀠 { zl (vn ) l { 𝜕Gn,z (x,vn ) { 𝜕t

in 𝒟(OT ),

(3.18)

Then from (3.13), we obtain

is bounded in W01,x Lφ (OT ), is bounded in W −1,x Lφ (OT ) + L1 (OT ),

and by (1.5) we have that zl (vn ) is compact in L1 (OT ). This gives the a. e. convergence of Tl (vn ), as well as the a. e. convergence of un .

Strong evolution problems in Musielak spaces |

193

Adopting the same proof used in [2], combined with (3.18) and (3.13), we get l (x, vn )dx ≤ lC4 + C5 , for almost all t in (0, T). Using the pointwise conver∫O Gn,θ

l gence of vn and Gn,θ (x, vn ) and letting l → +∞ we reach the conclusion that g(x, v) ∈ 1 L∞ (0, T, L (O)). Now we prove (3.17). To prove the boundedness of {σn (Tk (vn ), DTk (vn ))}n in (Lφ (O))N for all k > 0, it is enough to use (1.6) and (3.13).

Step 2. We need the following result. Lemma 3.1 ([7]). If vn satisfies (3.4), then lim

lim

m→+∞ n→+∞

∫

σ(vn , Dvn )Dvn dx dt = 0.

(3.19)

{m≤|vn |≤m+1}

Now, we will focus on the accompanying last two conditions of Proposition 3.1, namely that σ(Tk (vn ), DTk (vn ))DTk (vn ) ⇀ σ(Tk (v), DTk (v))DTk (v)

weakly in L1 (OT )

(3.20)

and Dvn → Dv

a. e. in OT .

(3.21)

Let us define the function ψ(v, ζ ) = g(x, v) − g(x, ζ ). For μ > 0, we consider Tk (ψ(v, ζ ))μ (x, t) to be the approximation in time of Tk (ψ(v, ζ )). For the properties of this function, we refer to Landes [11]. We remark that (Tk (ψ(v, ζ )))μ → Tk (ψ(v, ζ )) a. e. in OT , weakly-* in L∞ (OT ) and in W01,x Lφ (OT ), and we have ‖(Tk (ψ(v, ζ )))μ ‖L∞ (OT ) ≤ k. Let Sl (r) = r { { { supp(S󸀠l ) ⊂ [−2l, 2l], { { { 3 󸀠󸀠 { ‖Sl ‖L∞ (ℝ) ≤ l

if |r| ≤ l, for any l ≥ 1.

Denote by ϵ(l, μ, n) the value ϵ(n, μ, l) → 0 as l, μ, n → +∞. Lemma 3.2 ([14, 13]). Let S be a C ∞ -function such that S(r) = r for |r| ≤ k, and supp(S󸀠 ) is compact. We have, for k > 0, T

lim

lim ∫⟨𝜕t g(x, vn ), S󸀠 (g(x, vn ) − g(x, ζ ))Tk (g(x, vn ) − g(x, ζ ))

μ→+∞ n→+∞

0

− (Tk (ψ(v, ζ )))μ ⟩(L1 +W −1 L

φ ;L∞ ∩W

1L

φ)

dt ≥ 0.

194 | A. Aberqi et al. Lemma 3.3. For k ≥ 0, the sequence vn satisfies lim ∫ σ(vn , DTk (vn ))DTk (vn ) dx dt ≤ ∫ κk DTk (vn ) dx dt.

n→+∞

OT

OT

Proof. Let plug-in the test function S󸀠l (ψn )(Tk (ψn ) − (Tk (ψn ))μ ) for l > 0 and μ > 0, where ψn = g(x, vn ) − g(x, ζ ). μ For fixed l ≥ 0, letting Θn = Tk (ψn ) − (Tk (ψ))μ , we obtain T

∫⟨𝜕t gn (x, vn ), S󸀠l (ψn )Θμn ⟩dt + ∫ σn (vn , Dvn )S󸀠l (ψn )DΘμn dx dt 0

OT μ 󸀠 μ + ∫ σn (vn , Dvn )S󸀠󸀠 l (ψn )D(ψn )Θn dx dt + ∫ Kn (vn )Sl (ψn )DΘn dx dt OT

+ ∫

OT μ Kn (vn )S󸀠󸀠 l (ψn )D(ψn )Θn dx dt

OT

= ∫

− ∫ nTn (vn − ζ )− S󸀠l (ψn )Θμn dx dt OT

fn S󸀠l (ψn )Θμn dx dt.

(3.22)

OT

For a fixed k > 0, we prove the following limits: T

lim

lim ∫⟨𝜕t gn (x, vn ); S󸀠l (ψn )Θμn ⟩dt ≥ 0

for any l ≥ k,

(3.23)

lim ∫ Kn (vn ))S󸀠l (ψn )DΘμn dx dt = 0

for any l ≥ 1,

(3.24)

μ→+∞ n→+∞

lim

0

μ→+∞ n→+∞

OT

lim

μ lim ∫ Kn (vn )S󸀠󸀠 l (ψn )D(ψn )Θn dx dt = 0,

μ→+∞ n→+∞

(3.25)

OT

lim

μ lim lim ∫ σn (vn , Dvn ))S󸀠󸀠 l (ψn )D(ψn )Θn dx dt = 0,

l→+∞ μ→+∞n→+∞

lim

(3.26)

OT

lim ∫ nTn (vn − ζ )− S󸀠l (ψn )Θμn dx dt = 0,

μ→+∞ n→+∞

(3.27)

OT

lim

lim ∫ fn S󸀠l (ψn )Θμn dx dt = 0.

μ→+∞ n→+∞

(3.28)

OT

Proof of (3.23). A direct consequence of Lemma 3.2, taking S = Sl , with l ≥ k. μ

Proof of (3.24). For fixed μ > 0, Θn ⇀ Θμ = Tk (ψ(v, ζ )) − (Tk (ψ(v, ζ )))μ weakly in W01,x Lφ (OT ). Now we remark that 󵄩󵄩 μ 󵄩󵄩 󵄩󵄩Θn 󵄩󵄩L∞ (OT ) ≤ 2k

for any n, μ > 0,

(3.29)

Strong evolution problems in Musielak spaces |

195

and then we deduce that Θμn → Θμ

a. e. in OT and L∞ (OT ),

(3.30)

weakly-* when n → +∞. One has supp S󸀠l ⊂ [−2l, −l] ∩ [l, 2l] for any fixed l ≥ 1 and μ μ n > l + 1, then Kn (vn )S󸀠l (ψn )DΘn = Kn (vn )S󸀠l (T2l (ψn ))DΘn a. e. in OT since supp S󸀠l ⊂ [−2l, 2l]. On the other hand, Kn (vn )S󸀠l (T2l (ψn )) → K(v)S󸀠l (T2l (ψ(u, ζ ))) a. e. in OT and |Kn (vn )S󸀠l (T2l (ψn ))| ≤ 2la(x, t) + 2lr(x, t)φ−1 x φ(x, W01,x Lφ (OT )

μ

As Θ ⇀ 0 weakly in

α0 l) λ

for l ≥ 1.

when μ → +∞, we obtain (3.24).

Proof of (3.25). For any fixed l ≥ 1 and n > 2l, μ 󸀠󸀠 μ Kn (vn )S󸀠󸀠 l (ψn )D(ψn )Θn = Kn (vn )Sl (T2l (ψn ))DT2l (ψn )Θn ,

a. e. in OT .

By (3.29)–(3.30), letting n → +∞, μ 󸀠󸀠 μ Kn (vn )S󸀠󸀠 l (T2l (ψn ))Θn → K(v)Sl (T2l (ψ(v, ζ )))Θ ,

a. e. in OT .

Since α 3 󵄨󵄨 −1 󸀠󸀠 μ󵄨 󵄨󵄨K(v)Sl (T2l (ψ(u, ζ )))Θ 󵄨󵄨󵄨 ≤ [r(x, t)φx φ(x, 0 l)] l λ

a. e. in OT

and Θμ →mod 0 in W01,x Lφ (OT ), we get (3.25). Proof of (3.26). Since supp S󸀠l ⊂ [−2l, −l] ∩ [l, 2l] for any fixed l ≥ 1, we get 󵄨󵄨 󵄨󵄨 󵄨󵄨 󵄨 󸀠󸀠 μ 󵄨󵄨 ∫ σn (vn , Dvn )Sl (ψn )D(ψn )Θn dx dt 󵄨󵄨󵄨 󵄨󵄨 󵄨󵄨 OT

󵄩 󵄩󵄩 ≤ 󵄩󵄩󵄩S󸀠󸀠 l 󵄩 󵄩L

∞ (ℝ)

󵄩 󵄩 ≤ 3󵄩󵄩󵄩Θμn 󵄩󵄩󵄩L

󵄩󵄩 μ 󵄩󵄩 󵄩󵄩Θn 󵄩󵄩L∞ (OT )

1 ∞ (OT ) l

∫

∫

σn (vn , Dvn ))D(ψn ) dx dt

l≤|(ψn |≤2l

σn (vn , Dvn ))D(ψn ) dx dt

|(ψn |≤2l

1 󵄩 󵄩 ≤ 3󵄩󵄩󵄩Θμn 󵄩󵄩󵄩L (O ) ∫ σn (vn , Dvn )Dvn dx dt, ∞ T l |vn |≤s

with s = ‖ζ ‖L∞ (OT ) + 2lλ for any l ≥ 1, any n > 2l and any μ > 0. By Lemma 3.3, it is possible to establish (3.26). μ

Proof of (3.28). By (3.2), the pointwise convergence of vn and Θn , and its boundedness, for all μ > 0, l ≥ 1, by letting n tend to +∞, lim ∫ fn Sl󸀠 (ψn )Θμn dx dt = ∫ f S󸀠l (ψ(u, ζ ))Θμ dx dt.

n→+∞

OT

OT

196 | A. Aberqi et al. μ

Fix l ≥ 1, knowing that ‖(Tk (v))μ ‖L∞ (OT ) ≤ max(‖Tk (v)‖L∞ (OT ) , ‖Θn ‖L∞ (O) ) ≤ 2k, ∀μ, k > 0, it is easy to deduce (3.28). Proof of (3.27). Similar to (3.28). Finally, we adopt the same techniques used in [14] to obtain the claim of Lemma 3.4. Lemma 3.4. For any k ≥ 0, the sequence vn satisfies T

∫ ∫[σ(Tk (vn ), DTk (vn )) − σ(Tk (vn ), DTk (v))][DTk (vn ) − DTk (v)] dx dt → 0,

(3.31)

0O

κk = σ(Tk (v), DTk (v)) a. e. in OT , as n → +∞, σ(Tk (vn ), DTk (vn ))DTk (vn ) ⇀ σ(Tk (u), DTk (v))DTk (v),

(3.32) 1

weakly in L (OT ).

(3.33)

Remark 3.1. Note the (3.31) allows us to deduce the almost everywhere convergence of the gradient {Dvn } in OT . This completes the proof of Proposition 3.1 and Theorem 3.1.

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[12] M. M. Porzio, Existence of solutions for some noncoercive parabolic equations. Discrete Contin. Dyn. Syst. 5(3), 553–568 (1999). [13] H. Redwane, Existence results for a class of nonlinear parabolic equations in Orlicz spaces. Adv. Syst. Appl. 2, 241–264 (2007). [14] H. Redwane, Existence of a solution for a class of nonlinear parabolic systems. Electron. J. Qual. Theory Differ. Equ. 24 (2007).

Ibtissem Djerrar, Leïla Alem, and Lahcène Chorfi

Numerical solution of an axisymmetric inverse heat conduction problem Abstract: In this paper we are considering a boundary value axisymmetric inverse heat conduction problem inside the cylinder 0 ≤ r ≤ b. Our aim is to reconstruct the temperature f (t) = u(b, t) on the boundary r = b from measured temperature g δ (t) in the interior point 0 < r1 < b. Using Laplace transform, for the direct problem the solution will be represented as a Fourier Bessel series. Then the inverse problem is reduced to an integral equation of type Af = g δ . Such an equation is ill-posed, hence we use the Tikhonov regularization method. Finally, we conclude with numerical examples. Keywords: Ill-posed problem, Tikhonov regularization, radially symmetric heat equation, Laplace transform MSC 2010: 35K05, 65N06, 35R30

1 Introduction The heat conduction problem arises from many physical and engineering problems, but it is often impossible to directly measure the desired physical quantity. One way to measure this quantity is to solve an IHCP. The problem in Cartesian coordinates was studied by many authors [12, 11, 15]. The mollification method and its application to ill-posed problems is introduced in [13]. In [3, 4, 5, 2, 14] the authors have treated an axisymmetric IHCP by different methods. Our aim is to study an axisymmetric IHCP using the Laplace transform method. More precisely, we consider an axisymmetric inverse problem for the heat equation inside the cylinder 0 ≤ r ≤ b. We present the mathematical model of the problem by an axisymmetric heat conduction problem: Find the temperature u(r, t) inside the disk D(0, b) = {(x1 , x2 ) ∈ ℝ2 , r 2 = x12 + x22 ≤ b2 } such that 2

𝜕u = 𝜕𝜕ru2 + 1r 𝜕u , { { 𝜕t 𝜕r { { { { { u(r, t) is bounded for r → 0, { { { u(r1 , t) = g(t), { { { { { u(r, 0) = 0,

r ∈ ]0, b[, t > 0, t > 0, 0 < r1 < b, t ≥ 0,

(1.1)

r ∈ [0, b].

Ibtissem Djerrar, Leïla Alem, Lahcène Chorfi, Lab. LMA, Badji Mokhtar University, Annaba, Algeria https://doi.org/10.1515/9783110598193-015

200 | I. Djerrar et al. Our goal is to find the temperature f (t) = u(b, t), δ

t ≥ 0,

(1.2)

δ

from g (t) with ‖g−g ‖ ≤ δ, δ is the level noise. It is known that this problem is severely ill-posed [7, 4].

2 Direct problem We suppose that u(b, t) = f (t) is known and consider the direct problem as follows: 2

𝜕u = 𝜕𝜕ru2 + 1r 𝜕u , { { 𝜕t 𝜕r { { { { { u(r, t) is bounded for r → 0, { { { u(b, t) = f (t), { { { { { u(r, 0) = 0,

r ∈ ]0, b[, t ≥ 0, t ≥ 0,

(2.1)

r ∈ [0, b], t ≥ 0.

2.1 Uniqueness of the solution Theorem 2.1. There exists at most one solution for the problem (2.1) in the space 0

2

1

2

ℋ = C ([0, +∞[, H (0, b; rdr)) ∩ C (]0, +∞[, L (0, b; rdr)).

Proof. The theorem was proved in [7].

2.2 Representation of the solution The resolution of the problem (2.1) is based on the Laplace transform method with respect to the variable t. We define the Laplace transform F(s) = ℒf (s) of f (t) by +∞

F(s) = ∫ e−st f (t) dt,

ℜ(s) > σ.

0

We assume that f is piecewise-continuous with exponential growth, i. e., |f (t)| ≤ Ceσt for t ≥ 0, σ ≥ 0 (see [6]). Let U(r, s) = ℒu(r, ⋅) and F(s) = ℒf (t). We formulate the problem (2.1) as follows: U 󸀠󸀠 + 1r U 󸀠 − sU = 0, { { { U(b, s) = F(s), { { { { U(r, s) is bounded for r → 0, where U 󸀠 =

𝜕U . 𝜕r

r ∈ (0, b), (2.2) t ≥ 0,

Numerical solution of an axisymmetric inverse heat conduction problem

| 201

The equation in (2.2) is the modified Bessel differential equation. The general solution of this equation is given by U(r, s) = C0 I0 (√sr) + C1 K0 (√sr),

(2.3)

where I0 (z), K0 (z) are modified Bessel functions of the first and second kind, respectively [1]. The solution of the problem (2.2) is given by U(r, s) = F(s)

I0 (r√s) . I0 (b√s)

(2.4)

Using the properties of the Bessel functions, the solution is written as U(r, s) = F(s)

J0 (ir√s) . J0 (ib√s)

(2.5)

The function J0 (ib√s) possesses a sequence of simple roots sn such that sn = −(

2

αn ), b

where αn ≃ (n + 43 )π, n → +∞. Applying the inverse Laplace transform and the convolution theorem [6], we obtain t

r α J0 (αn b ) sn t u(r, t) = 2 ∑ n2 e ∫ f (τ)e−sn τ dτ n=1 b J1 (αn ) ∞

0

2

(sn = −(

αn ) ). b

(2.6)

Proposition 2.1. Suppose that f (t) ∈ C 1 ([0, +∞[) with f (0) = 0, f (t) = 0 for t ≥ T. Then the series (2.6) defines a solution u in ℋ. Proof. We use arguments as in [7, Theorem 2].

2.3 Approximation by FDM In this section we want to approximate the problem (2.1) using the finite difference method [9] with an implicit scheme. In order to overcome the singularity at r = 0, we use the change of variable, r = 2√x. Hence the unknown function v(x, t) = u(2√x, t) solves the problem 2

𝜕v 𝜕v 𝜕v = x 𝜕x { 2 + 𝜕x , { 𝜕t { { { 𝜕v { { 𝜕x (0, t) = 0, { { { v(b1 , t) = f (t), { { { { { v(x, 0) = 0,

x ∈ ]0, b1 [, t ≥ 0 (b1 = b2 /4), t ≥ 0, t ≥ 0,

(2.7)

x ∈ [0, b1 ].

The homogeneous Neumann condition at x = 0 follows from an argument of symmetry.

202 | I. Djerrar et al. Discretization We use the following notations: k is the time step, h is the space step, T = Mk, b1 = Nh, tn = nk, xi = (i − 1)h, and vin = v(xi , tn ). We now propose the scheme vn −vn−1

x

n n n i i = h2i (vi+1 ), − 2vin + vi−1 ) + h1 (vin − vi−1 { { k { 0 v = 0, { { { in n n { vN+1 = fn , v1 = v0 ,

i = 1, . . . , N, n = 1, . . . , M, i = 1, . . . , N,

(2.8)

n = 1, . . . , M.

The scheme is implicit. Indeed, we have n n (β − xi R)vi−1 + (1 + 2Rxi − β)vin − xi Rvi+1 = vin−1

with R =

k h2

(2.9)

and β = hk .

Theorem 2.2 ([9]). 1. The scheme is unconditionally stable. 2. The scheme is consistent of order O(k + h). For the numerical experiments, we consider the following example. Test 1 We consider the boundary condition u(b, t) = f (t) = χ[1,2] , with b = 2 and T = 3. Here χI is the characteristic function of the interval I. We compute g(t) = u(r1 , t) (r1 < b) with two methods. We denote by gex the truncated series (2.6) up to N = 20 and by gapp the approximate solution using the implicit finite difference scheme. In Figure 1, we plot gex and gapp . The illustration confirms that the approximation is (very) good.

Figure 1: Test 1. Exact solution gex and numerical solution gapp , for r1 = 1, N = 40, and M = 70.

3 Inverse problem Now we suppose that the function u(r1 , t) = g(t) is given as an exact data. The inverse problem (1.1)–(1.2) is then reduced to the resolution of the Volterra integral equation

Numerical solution of an axisymmetric inverse heat conduction problem

| 203

of the first kind, namely t

Af (t) := ∫ k(t − τ)f (τ) dτ = g(t),

(3.1)

0

with k is the kernel given by r αn J0 (αn b1 ) sn t e . 2 J (α ) 1 n n=1 b ∞

k(t) = 2 ∑

(3.2)

To solve equation (3.1) numerically, we approximate the kernel k by the truncated series r αn J0 (αn b1 ) sn t e . 2 J (α ) 1 n n=1 b N

kN (t) = 2 ∑

(3.3)

We solve the approximate equation AN f = g by the Tikhonov regularization method (see [10]). If the data g is noisy, the regularization parameter α is selected by Morosov principle. Theorem 3.1 (See [10]). Let K : H1 → H2 be an injective compact operator (between Hilbert spaces) with a dense range. Let g = Kf and g δ ∈ H2 satisfy 󵄩󵄩 δ 󵄩 (3.4) 󵄩󵄩g − g 󵄩󵄩󵄩 ⩽ δ. Then there exists a unique α = α(δ) such that 󵄩󵄩󵄩Kf α(δ) − g δ 󵄩󵄩󵄩 = δ and 󵄩 󵄩

f α(δ) → f , δ → 0.

3.1 Numerical examples We consider the following examples. Test 1 Let f (t) = χ[1,2] (t). The exact data g is computed by using FDM and the perturbed data is g δ = g + δσ(t), with σ being a Gaussian random distribution. Test 2 The second example is the triangle function 2(t − 1), 1 < t < 1.5, { { f (t) = { 2(2 − t), 1.5 < t < 2, { otherwise. { 0, In the computation we use the parameters b = 2, r1 = 1 and r1 = 1.9, T = 5, and M = N = 80.

204 | I. Djerrar et al. Discussion We distinguish two kinds of results. 1. Exact data (δ = 0). Figure 2 confirms the efficiency of the algorithm.

Figure 2: Exact and regularized solution, with δ = 0, N = M = 80, r1 = 1, T = 5: (a) Test 1 and (b) Test 2.

2.

Perturbed data (δ = 0.01). Figure 3 shows that the algorithm diverges in (a) or the oscillations are amplified in (b). In this case a pretreatment is useful. For this, we use the mollification method [13] which consists in filtering the data g δ by the 2 1 exp (− νt 2 ) where ν → 0 convolution gνδ = ρν ∗ g δ with the kernel ρν (t) = ν√π is the radius of mollification. Following the analysis in [13], we take ν = c√δ

with a constant c chosen by the tests. Figure 4 shows that the oscillations are damped.

Figure 3: Exact and regularized solution with a noise δ = 0.01, N = M = 80, r1 = 1, T = 5, without mollification: (a) Test 1 and (b) Test 2.

Numerical solution of an axisymmetric inverse heat conduction problem

| 205

Figure 4: Exact and regularized solution with a noise δ = 0.01, N = M = 80, with mollification (ν = 0.1): (a) Test 1 and (b) Test 2.

For r1 = 1.8 (r1 close to the ends of the interval [0, b]), the reconstruction is less good as it is shown in Figure 5. The reconstruction is also deteriorated for r = 0.4 (near r = 0). We use singular decomposition and Tikhonov-Morozov algorithms developed by Hansen [8].

Figure 5: Exact and regularized solution with mollification ν = 0.1, r1 = 1.8: (a) Test 1 and (b) Test 2.

4 Conclusion Throughout this research we were interested in an inverse heat conduction problem with a radial variable. Our approach was based on the Laplace transform method for the direct problem and the Tikhonov regularization method for the inverse problem, which was reduced to a Volterra integral equation of the first kind. The numerical tests show the efficiency of the method when the data is exact (δ = 0). However, if the data

206 | I. Djerrar et al. is perturbed (with δ = 0.01), we apply the mollification to the data g δ . Finally, we have noticed that the reconstruction is better when the time of the observation T is large enough.

Bibliography [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover Publications, Inc., New York, 1972). W. Cheng, Regularization and stability estimates for an inverse source problem of the radially symmetric parabolic equation. J. Inequal. Appl. 2015, 136 (2015). W. Cheng, C.-L. Fu, Solving the axisymmetric inverse heat conduction problem by a wavelet dual least-squares method. Bound. Value Probl. 2009, 260941 (2009). W. Cheng, C.-L. Fu, Two regularization methods for an axisymmetric inverse heat conduction problem. J. Inverse Ill-Posed Probl. 17(2), 159–172 (2009). W. Cheng, C.-L. Fu, A modified Tikhonov regularization method for an axisymmetric backward heat equation. Acta Math. Sin. Engl. Ser. 26(11), 2157–2164 (2010). V. Ditkine, A. Proudnikov, Transformation intégrales et calcul opérationnel. Traduit du russe (MIR, Moscou, 1978). I. Djerrar, L. Alem, L. Chorfi, Regularization method for the radially symmetric inverse heat conduction problem. Bound. Value Probl. 2017, 159 (2017). P. C. Hansen, Regularization tools version 4.0 for Matlab 7.3. Numer. Algorithms 46, 189–194 (2007). http://www2.imm.dtu.dk/~pcha/Regutools/. R. Herbin, Analyse numérique des équations aux dérivées partielles. Engineering school, Marseille 2011. cel.archives-ouvertes.fr/cel-00637008. A. Kirsh, An Introduction to the Mathematical Theory of Inverse Problems. Series AMS, vol. 120 (Springer, 2011). P. K. Lamm, A survey of regularization methods for first-kind Volterra equations. Preprint of Mathematics Dept, Michigan State University, USA. P. K. Lamm, L. Eldén, Numerical solution of first-kind Volterra equations by sequential Tikhonov regularization. SIAM J. Numer. Anal. 34(4), 1432–1450 (1997). D. A. Murio, The Mollification Method and the Numerical Solution of Ill-Posed Problems (John Wiley & Sons, Interscience, New York, 1993). X.-T. Xiong, On a radially symmetric inverse heat conduction problem. Appl. Math. Model. 34, 520–529 (2010). N. Yaparova, Numerical methods for solving a boundary-value inverse heat conduction problem. Inverse Probl. Sci. Eng. 22(5), 832–847 (2014).

Safia Benmansour, Atika Matallah, and Mustapha Meghnafi

Multiple solutions for nonlocal elliptic problems with concave–convex nonlinearities and weights Abstract: This work deals with the existence and multiplicity of solutions for elliptic Kirchhoff problems which are defined on a regular bounded domain in ℝ3 . Using the Nehari decomposition and Ekeland’s variational principle, we study the effect of the geometry of the nonlinearity on the multiplicity of positive solutions for this type of problems. Keywords: Variational methods, Kirchhoff equations, Nehari manifold MSC 2010: 35J20, 35IJ60, 47J30

1 Introduction In this work, we investigate the existence and multiplicity of positive solutions for the following problem: (𝒫λ ) {

−M(∫Ω |∇u|2 dx)Δu = λf (x)uq−1 + g(x)up−1 u=0

in Ω, on 𝜕Ω,

where M(t) = at + b, Ω is a regular bounded domain of ℝ3 with smooth boundary 𝜕Ω, 1 < q < 2, 4 < p < 6, a, b are positive constants, λ is a positive parameter, and f , g are continuous nonnegative functions defined on Ω. The problem (𝒫λ ) is called nonlocal because of the presence of the integral (a ∫Ω |∇u|2 dx + b), over the domain Ω which implies that the equation in (𝒫λ ) is no longer a pointwise identity. It is the elliptic version of the stationary analogue of utt − (a ∫Ω |∇u|2 dx + b) △ u = h(x, u) { { { u=0 { { { { u(x, 0) = u0 (x),

in Ω × (0, T), on 𝜕Ω × (0, T), ut (x, 0) = u1 (x),

where T is a positive constant, u0 , u1 are given functions. It was first introduced by Kirchhoff [7] as an extension of the classical D’Alembert wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length Safia Benmansour, Atika Matallah, Ecole supérieure de management de Tlemcen, BP 119, 13000 Tlemcen, Algeria Mustapha Meghnafi, University of Bechar, PO Box 417, 08000 Bechar, Algeria https://doi.org/10.1515/9783110598193-016

208 | S. Benmansour et al. of the strings produced by transverse vibrations. In such a problem, the parameters have the following meanings: u denotes the displacement, h(x, u) the external force, b is the initial tension, and a is related to the intrinsic properties of the strings (such as Young’s modulus). It is pointed out that nonlocal problems model several physical and also biological systems where u describes a process depending on the average of itself as population density. After the famous article of Lions [9] where a functional analysis approach was proposed, Kirchhoff’s problems began to call attention of researchers. The following stationary elliptic version has been intensively studied: (𝒫S ) {

−(a ∫Ω |∇u|2 dx + b)Δu = h(x, u) u=0

in Ω, on 𝜕Ω,

where Ω ⊂ ℝN and h(x, u) is a continuous function. In the literature, there are many works, and interested readers can refer to [1] where the authors obtained positive solutions for such problems by variational methods. Let us recall a brief history: 2N For a = 0, b = 1, 2 < p ≤ 2∗ (2∗ = N−2 if N ≥ 3, 2∗ = ∞ if N = 1 or N = 2) and f = g ≡ 1, Ambrosetti et al. [2] established multiple results, namely, they ensure the existence of a positive constant λ0 such that the problem (𝒫λ ) admits two positive solutions for λ ∈ (0, λ0 ), a positive solution for λ = λ0 , and no positive solution for λ > λ0 . For a = 0, b = q = 1, 2 < p ≤ 2∗ , and g ≡ 1, Tarantello [10] proved the existence of at least two solutions in a bounded domain of ℝN , N ≥ 3, under a suitable condition on f . In the case a, b > 0, p = 6 = 2∗ for N = 3 and g ≡ 1, Benmansour and Bouchekif [3] have shown the existence of two solutions under a sufficient condition on f . Recently, Lin et al. [8] considered the case 2 < p ≤ 2∗ , a = 0, and b = 1, they proved the existence of a positive constant λ0 such that the problem (𝒫λ ) admits a positive ground state solution for λ ∈ (0, λ0 ), and under suitable conditions they also studied the effect of the graph of g to establish a multiplicity result. A natural and interesting question is whether results remain valid for a > 0. Throughout this paper, we use the following notation: H = H01 (Ω), ∫ u = ∫Ω u dx, ‖u‖ = (∫Ω |∇u|2 )1/2 , and |u|2 = (∫Ω |u|2 )1/2 dx are the norms in H01 (Ω) and L2 (Ω), respectively, C denotes generic positive constants whose exact values are not important, Bra is the ball of center a and radius r, and on (1) denotes any quantity which tends to zero as n tends to infinity; Sr is the best Sobolev constant of the embedding of H01 (Ω) into Lr (Ω) for 1 < r < 6. Before giving our main results, we consider the following hypotheses: (H1) f , g ∈ C(Ω), f ≥ 0, f ≠ 0 and g > 0. (H2) There exist a1 , . . . , ak ∈ Ω such that g(ai ) = maxx∈Ω g(x), 1 ≤ i ≤ k.

Multiple solutions for nonlocal elliptic problems | 209

Let λ1 =

qSqq/2 √ab(p − 4)(p − 2) 2Sp3 √3ab(4 − q)(2 − q) (3−q)/(p−3) ] , [ √2(p − q)‖f ‖∞ (p − q)‖g‖∞

λ2 =

(p − 2)qbSqq/2 bSpp/2 (2 − q) (p−q)/(p−2) [ ] , 2(p − q)‖f ‖∞ (p − q)‖g‖∞

and λ∗ = max(λ1 , λ2 ). Our main results are: Theorem 1.1. If λ ∈ (0, λ∗ ), then there exists a positive ground state solution of problem (𝒫λ ). Theorem 1.2. Under the assumptions (H1) and (H2), the problem (𝒫λ ) admits k positive solutions for λ ∈ (0, λ∗ ). This work is organized as follows: In Section 2, we give some preliminary results which we will use later. Section 3 is devoted to the proof of the existence of the positive ground state solution. In Section 4, we construct k Palais–Smale sequences which correspond to the k maxima of g.

2 Preliminary results The energy functional associated to the problem (𝒫λ ) is given by 1 λ 1̂ p q 2 ) − ∫ g(u+ ) − ∫ f (u+ ) , Iλ (u) = M(‖u‖ 2 p q

for all u ∈ H,

̂ is the primitive of M(t) = at + b (M(0) ̂ where M(t) = 0). It is clear that Iλ is well defined, 1 of class C on H and its critical points are weak solutions of (𝒫λ ), that is, u ∈ H is said to be a weak solution of (𝒫λ ) if it satisfies p−1

(a‖u‖2 + b) ∫ ∇u∇v − ∫ g(u+ )

q−1

v − λ ∫ f (u+ )

v = 0,

for all v ∈ H.

We know that Iλ is not bounded from below on H, so we introduce the Nehari manifold given by 𝒩λ = {u ∈ H\{0} : ⟨Iλ (u), u⟩ = 0}. 󸀠

Let hu (t) = Iλ (tu) for t ≥ 0 and u ∈ H\{0}. These maps are known as fibering maps and were first introduced by Drábek and Pohozaev [6]. The set 𝒩λ is closely linked to the behavior of hu (t); for more details, see, for example, [5].

210 | S. Benmansour et al. 2 4 2 q−2 Let h󸀠󸀠 ∫ f (u+ )q − (p − 1)t p−2 ∫ g(u+ )p . It is natural u (t) = 3at ‖u‖ + b‖u‖ − λ(q − 1)t to split 𝒩λ into three subsets:

𝒩λ := {u ∈ 𝒩λ : hu (1) > 0}, +

󸀠󸀠

0

󸀠󸀠

𝒩λ := {u ∈ 𝒩λ : hu (1) = 0},

and 𝒩λ := {u ∈ 𝒩λ : hu (1) < 0}, −

󸀠󸀠

which correspond to local minima, points of inflexion, and local maxima of Iλ , respectively. We give the following useful result. Lemma 2.1. Suppose that u0 is a local minimizer of Iλ in 𝒩λ and u0 ∉ 𝒩λ0 . Then Iλ󸀠 (u0 ) = 0. Proof. If u0 is a local minimizer for Iλ on 𝒩λ , then u0 is a solution of the minimization problem: min{Iλ (u); γλ (u) = 0}, where q

p

γλ (u) = a‖u‖4 + b‖u‖2 − λ ∫ f (u+ ) − ∫ g(u+ ) . Hence, by the theory of Lagrange multipliers, there exists μ ∈ ℝ such that Iλ (u0 ) = μγλ (u0 ). Thus, ⟨Iλ󸀠 (u0 ), u0 ⟩ = μ⟨γλ󸀠 (u0 ), u0 ⟩. Since u0 ∈ 𝒩λ and due to the fact that u0 ∉ 𝒩λ0 , one gets ⟨γλ󸀠 (u0 ), u0 ⟩ ≠ 0 and can conclude that μ = 0. The proof is complete. The following lemmas play a crucial role in the sequel of this work. Lemma 2.2. For all λ ∈ (0, λ∗ ) and for each u ∈ H\{0}, there exist unique t + = t + (u) ≥ u u u tmax and t − = t − (u) ≤ tmax such that t + u ∈ 𝒩λ− , t − u ∈ 𝒩λ+ , and Iλ (t + u) = maxt≥tmax Iλ (tu), − u Iλ (t u) = min0≤t≤tmax Iλ (tu). Proof. Put h󸀠u (t) = t q−1 (Hu (t) − λ ∫ f (u+ )q ), where p

Hu (t) = a‖u‖4 t 4−q + b‖u‖2 t 2−q − t p−q ∫ g(u+ ) . We know that Hu (0) = 0 and limt󳨀→+∞ Hu (t) = −∞, thus Hu (t) achieves its maxu u u imum at tmax , Hu (t) is increasing on (0, tmax ) and decreasing on (tmax , ∞). The result follows immediately. Lemma 2.3. For all λ ∈ (0, λ∗ ), we have 𝒩 0 = 0.

Multiple solutions for nonlocal elliptic problems | 211

Proof. We argue by contradiction. Suppose that there exists u ∈ 𝒩λ0 such that q

p

3a‖u‖4 + b‖u‖2 = (q − 1)λ ∫ f (u+ ) + (p − 1) ∫ g(u+ ) and u satisfies p

q

a‖u‖4 + b‖u‖2 − ∫ g(u+ ) − λ ∫ f (u+ ) = 0. From this and Sobolev inequality, we obtain 2√(4 − q)(2 − q)ab‖u‖3 ≤ a(4 − q)‖u‖4 + b(2 − q)‖u‖2 ≤ (p − q)Sp−p/2 ‖g‖∞ ‖u‖p and 2√(p − 4)(p − 2)ab‖u‖3 ≤ (p − 4)a‖u‖4 + (p − 2)b‖u‖2 ≤ λ(p − q)‖f ‖∞ Sq−q/2 ‖u‖q , consequently we get (

2Spp/2 √(4 − q)(2 − q)ab (p − q)‖g‖∞

1/(p−3)

)

≤ ‖u‖ ≤ (

1/(3−q)

λ(p − q)‖f ‖∞

2Sqq/2 √(p − 2)(p − 4)ab

)

and (

(2 − q)bSpp/2

(p − q)‖g‖∞

1/(p−2)

)

≤ ‖u‖ ≤ (

λ(p − q)‖f ‖∞ 4bSqq/2

1/(2−q)

)

,

which contradicts the fact that 0 < λ < λ∗ . Lemma 2.4. For λ ∈ (0, λ∗ ) and u ∈ 𝒩λ , there exists ε > 0 and a differentiable function t : Bε0 ⊂ H 󳨀→ ℝ+ such that t(0) = 1, t(v)(u − v) ∈ 𝒩λ for ‖v‖ < ϵ and ⟨t 󸀠 (0), v⟩ =

(4a‖u‖2 + 2b) ∫ ∇u∇v − p ∫ g(u+ )p−1 v − λq ∫ f (u+ )q−1 v h󸀠󸀠 u (1)

.

Proof. Define the map F : ℝ × H → ℝ by p

q

F(s, w) = as3 ‖u − w‖4 + bs‖u − w‖2 − sp−1 ∫ g((u − w)+ ) − λ ∫ f ((u − w)+ ) . Since F(1, 0) = 0

and

𝜕F (1, 0) = h󸀠󸀠 u (1) ≠ 0, 𝜕s

we obtain the desired result by applying the implicit function theorem at (1, 0).

212 | S. Benmansour et al. Lemma 2.5. The functional Iλ is coercive and bounded from below on 𝒩λ . Proof. For u ∈ 𝒩λ , we have a 4 b 2 1 λ p q ‖u‖ + ‖u‖ − ∫ g(u+ ) − ∫ f (u+ ) 4 2 p q 1 1 (p − q) 1 1 ‖f ‖∞ Sq−q/2 ‖u‖q . ≥ a( − )‖u‖4 + b( − )‖u‖2 − λ 4 p 2 p pq

Iλ (u) =

Then, Iλ is coercive and bounded from below in 𝒩λ . Define cλ+ = infu∈𝒩 + Iλ (u), cλ− = infv∈𝒩 − Iλ (v), and cλ = infu∈𝒩λ Iλ (u). Moreover, if λ λ uλ is a local minimum of Iλ , then we have q

p

3a‖uλ ‖4 + b‖uλ ‖2 − λ(q − 1) ∫ f (u+λ ) − (p − 1) ∫ g(u+λ ) ≥ 0, and since 𝒩λ0 = 0, we deduce that uλ ∈ 𝒩λ+ , and so cλ+ = cλ . We have the following result: Lemma 2.6. For all λ ∈ (0, λ∗ ), there exist two minimizing sequences (un ) ⊂ 𝒩λ+ and (vn ) ⊂ 𝒩λ− such that (i) Iλ (un ) < cλ+ + (ii) Iλ (vn ) < cλ− +

1 n 1 n

and Iλ (w1 ) ≥ Iλ (un ) − n1 ‖w1 − un ‖ for all w1 ∈ 𝒩λ+ ;

and Iλ (w2 ) ≥ Iλ (vn ) − n1 ‖w2 − vn ‖ for all w2 ∈ 𝒩λ− .

Proof. It is clear that Iλ is bounded in 𝒩λ . Then by using the Ekeland variational principle, we obtain two minimizing sequences (un ) ⊂ 𝒩λ+ and (vn ) ⊂ 𝒩λ− which verify (i) and (ii), respectively.

3 Proof of Theorem 1.1 To prove this theorem, we need the following result. Proposition 3.1. For all 0 < λ < λ∗ , we have cλ+ < 0. Proof. Let u ∈ 𝒩λ+ . Since q

λ(p − q) ∫ f (u+ ) > (p − 4)a‖u‖4 + (p − 2)b‖u‖2 ≥ (p − 2)b‖u‖2 , one has Iλ (u) ≤ − We now deduce that cλ+ < 0.

b(p − 2)(2 − q) 2 ‖u‖ < 0. 2pq

Multiple solutions for nonlocal elliptic problems | 213

From (i) of Lemma 2.6, we obtain a minimizing sequence (un ) ⊂ 𝒩λ+ such that Iλ (un ) < cλ+ +

1 n

and Iλ (w) ≥ I(un ) −

1 ‖w − un ‖ for all w ∈ 𝒩λ+ . n

Since (un ) is bounded in H, passing to a subsequence if necessary, we have un ⇀ uλ

weakly in H

and strongly in Lr (Ω) for 1 < r < 6.

un → uλ This implies that ∫(g(u+n )

p−1

q−1

un + λ ∫ f (u+n )

un )(un − uλ ) → 0,

and since Iλ󸀠 (un )(un − uλ ) → 0, we conclude that (aun + b) ∫ ∇un ∇(un − uλ ) → 0. Thus un → uλ

strongly in H.

4 Proof of Theorem 1.2 This section is concerned with proving the existence of k positive solutions. For this, letting ai for i ∈ {1, . . . , k} be given by the hypothesis (H2) and ρ > 0 being ρ ρ ρ such that Bai ∩ Baj = 0 for i ≠ j, 1 ≤ i, j ≤ k, and ⋃ki=1 Bai ⊂ Ω, we define βi (u) := Take r0 =

ρ 3

with ρ
0 and r0 be defined as above. If βi (u) ≤ r0 then ∫ |∇u|2 ≥ 3 ∫ |∇u|2 . ρ

Ω

Ω\Bai

The proof of this lemma is given in [4]. Proposition 4.1. Under the hypotheses (H1) and (H2), for all λ ∈ (0, λ∗ ) and j = 1, . . . , k, cj+ = infv∈𝒩 + Iλ (v) is achieved at uj ∈ 𝒩j+ which is a critical point and even a local j minimum of Iλ . Proof. As before, we have for u ∈ 𝒩i+ ⊂ 𝒩λ , Iλ (u) ≥ −λ

(p − q) ‖f ‖∞ Sq−q/2 ‖u‖q . pq

In particular, cj+ ≥ cλ ≥ −λ

(p − q) ‖f ‖∞ Sq−q/2 ‖u‖q . pq

We also know that for u ∈ 𝒩i+ ⊂ 𝒩λ+ , Iλ (u) ≤ −

b(p − 2)(2 − q) 2 ‖u‖ < 0. 2pq

Hence, we conclude that −∞ < cλ ≤ cj+ < 0. Applying the Ekeland variational principle, we obtain a minimizing sequence (uj,n )n ⊂ 𝒩j+ with the following properties: (i)

Iλ (uj,n ) < cj+ +

1 , n

(ii) Iλ (w) ≥ Iλ (uj,n ) −

1 󵄨󵄨 󵄨 󵄨∇(w − uj,n )󵄨󵄨󵄨2 , n󵄨

for all w ∈ 𝒩j+ . Taking n large enough, we get Iλ (uj,n ) < cj+ +

b(p − 2)(2 − q) 1 ≤− ‖uj,n ‖2 . n 2pq

Using the fact that uj,n ∈ 𝒩λ+ , we obtain q

(4 − p)a‖uj,n ‖4 + (2 − p)b‖uj,n ‖2 + λ(p − q) ∫ f (u+,j,n ) > 0. Since p > 4, we conclude that ∫ f (u+,j,n )q > 0, thus uj,n is not identically zero. As Iλ is coercive, (uj,n ) is bounded, so, passing to a subsequence if necessary, we have uj,n ⇀ uj weakly in H and, by the same argument as in the proof of Theorem 1.1,

Multiple solutions for nonlocal elliptic problems |

215

we conclude that uj,n → uj strongly in H. Since ‖I 󸀠 (uj,n )‖ tends to 0 as n tends to infinity, we deduce that ⟨Iλ󸀠 (uj ), w⟩ = 0,

for all w ∈ H,

i. e., uj is a solution of (𝒫λ ). In particular, uj ∈ 𝒩λ , and necessarily uj ∈ 𝒩λ+ . We have βj (uj ) = limn󳨀→∞ βj (uj,n ) ≤ r0 . To conclude that uj ∈ 𝒩j+ , we need to prove that 𝒩j+ is a closed set in H. Indeed, by Lemma 2.3, for λ ∈ (0, λ∗ ), we have 𝒩λ0 = 0 thus, we write 𝒩λ = 𝒩λ− ∪ 𝒩λ+ (𝒩λ± are closed subsets in H\{0}). For j fixed in {1, . . . , k}, we have 𝒩j+ = 𝒩λ+ ∩ βj−1 ([0, r0 ]). It suffices to prove that βj is a continuous function on 𝒩λ+ . Let (un ) ⊂ 𝒩λ+ be such that un → u in H, i. e., ∀ε > 0, ∃N0 (ε) > 0, ∀n ≥ N0 ,

󵄨󵄨 󵄨 󵄨󵄨∇(un − u)󵄨󵄨󵄨2 < ε.

We remark that u ∈ 𝒩λ+ = 𝒩λ+ and then u ≠ 0 in H and |∇un |2 = |∇u|2 + on (1). Thus 1 󵄨󵄨 󵄨2 󵄨 󵄨2 󵄨 󵄨󵄨 󵄨 ∫ ψj (x)󵄨󵄨󵄨󵄨󵄨󵄨∇un (x)󵄨󵄨󵄨 − 󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 󵄨󵄨󵄨 + 󵄨󵄨βj (un ) − βj (u)󵄨󵄨󵄨 ≤ 2 |∇un |2 󵄨 1 󵄨󵄨󵄨󵄨 󵄨 󵄨2 󵄨󵄨 1 + ∫ ψj (x)󵄨󵄨󵄨∇u(x)󵄨󵄨󵄨 󵄨󵄨󵄨 − 󵄨. 󵄨󵄨 |∇un |2 |∇u|2 󵄨󵄨󵄨 2 2 Using Hölder inequality, we obtain δε 󵄨󵄨 󵄨 . 󵄨󵄨βj (un ) − βj (u)󵄨󵄨󵄨 ≤ 4 |∇u| 2

Consequently, Iλ (uj ) = cj+ = infv∈𝒩 + Iλ (v). As Iλ (|u|) = Iλ (u), we deduce that problem j (𝒫λ ) has k positive solutions, which are distinct.

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