Sequence of Multivalued Linear Operators Converging in the Generalized Sense
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Sequence of Multivalued Linear Operators Converging in the Generalized Sense Aymen Ammar1
· Aref Jeribi1 · Nawrez Lazrag1
Received: 17 November 2019 / Accepted: 12 January 2020 © Iranian Mathematical Society 2020
Abstract To study the Weyl essential spectrum of a sequence of closed multivalued linear operators converging in the generalized sense, we seek a counterpart of this convergence for closed linear relations as defined by Kato (Perturbation theory for linear operators, 2nd edn. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer, Berlin, 1976). Furthermore, we apply the obtained results to determine the Weyl essential spectrum of a 2 × 2 matrix of closed multivalued linear operators. Keywords Sequence of linear relations · Essential spectrum · Gap · Convergence in the generalized sense Mathematics Subject Classification 47A06 · 39B42
1 Introduction The theory of multivalued linear operators appeared in functional analysis motivated by the need to consider adjoints of non-densely defined linear differential operators (see Von Neumann in [22] to first appearance) and also by the need to study the inverses of certain operators used in the study of some Cauchy problems associated with parabolic type equations in Banach spaces (see, example [16]). During the past
Communicated by Mohammad S. Moslehian.
B
Aymen Ammar [email protected] Aref Jeribi [email protected] Nawrez Lazrag [email protected]
1
Department of Mathematics, University of Sfax, Faculty of Sciences of Sfax, Soukra Road Km 3.5, B. P. 1171, 3000 Sfax, Tunisia
123
Bulletin of the Iranian Mathematical Society
last years, a number of papers have appeared on the theory of multivalued linear operator (see [3,4,11–13]). One of the issues which have been studied concerning the subject of linear relations is the spectral theory, especially the theory of the essential spectra. In recent years, the concept of the essential spectrum of a multivalued linear operator was introduced and studied by several mathematicians. In [14], Cross introduced this concept on a complex normed space in terms of the nullity and the deficiency of its complete closure. Afterwards, in [1], Álvarez introduced several essential spectra of a linear relation on a normed space, and investigated the closedness and the emptiness of such essential spectra. Subsequently, in [23], Wilcox gave a definition of the five distinct essential spectra of linear relations on Banach spaces in terms of semi-Fredholm properties and the index. She applied the theory of Fredholm relations to show that essential spectra of linear operators can be extended naturally to linear relations. Among these approaches, we are interested in the Weyl essential spectrum defined by σw (T ) :=
σ (T + K ),
K ∈KT (X )
where KT (X ) := {K ∈ K R(X ) : D(K ) ⊃ D(T ), K (0) ⊂ T (0)}, when T is a closed linear relation on a Banach space X , and K R(X ) is the set of compact linear relations on X . The notion of a gap between linear subspaces and linear operators was introduced by Kre
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