On the essential spectra of some matrix of linear relations

Abstract

In this paper, we investigate a detailed treatment of some subsets of essential spectra of a closed multivalued linear operator. On the following, we will establish some results on perturbation theory of 2 × 2 matrix of multivalued linear operators. Copyright © 2013 John Wiley & Sons, Ltd.

A Characterization of the Essential Spectra of $$2\times 2$$ Block Matrices of Linear Relations

“…In [1], the authors exhibited a characterization of the Weyl essential spectrum of T by means of Fredholm linear relations by…”

“…In [10], Jeribi determined the essential spectra of the closure of the operator L 0 under certain conditions. Recently, in [1], Álvarez et al tackled the essential spectra of closed linear relations in Banach spaces. They investigated some properties of the spectral theory of the matrix of multivalued linear operators.…”

“…Relying on the Closed Graph Theorem for linear relations in[7, Theorem III.4.2], we deduce that R −1 is bounded single valued. The use of[1, Corollary 3.2] allows us to conclude that (R −1 ) −1 ST = RST is closable and RST = RST = RST . We define the linear relation L 0 acting on the product of two Banach spaces X × Y given by the block matrix of linear relation as…”

Stability of the essential spectrum of the diagonally and off-diagonally dominant block matrix linear relations

“…Several papers also focused on semi-Fredholm linear relations and other classes related to them (see for example [2,6,7]). However, many authors have recently shed light on 2 × 2 operators matrices and block matrices of linear relations (see for example [1,2,5,11,15,19]). Precisely, in [19], Tretter considered X and Y as two Banach spaces, and the linear operator A as given by the block operator matrix…”

Congruent spectrum for compact linear relations

“…Fundamental notions known for operators have been extended to the frame of linear relations. Some related important results have been carried out in various contexts: Spectral analysis (see [2], [4], [9]), classification (see [5], [13]), perturbation problem (see [3], [6], [7], [11]), decomposition and functional calculus (see [1], [14], [15]). A standard reference work in the field is the Cross's book [10].…”