Weakly demicompact linear operators and axiomatic measures of weak noncompactness

Abstract

In this paper, we study the relationship between the class of weakly demicompact linear operators, introduced in [KRICHEN, B.—O’REGAN, D.: On the class of relatively weakly demicompact nonlinear operators, Fixed Point Theory 19 (2018), 625–630], and measures of weak noncompactness of linear operators with respect to an axiomatic one. Moreover, some Fredholm and perturbation results involving the class of weakly demicompact linear operators are investigated. Our results are then used to investigate the relationship between the relative essential spectrum of the sum of two linear operators and the relative essential spectrum of each of these operators.

The Class of Demi-Strongly Order Bounded Operators

“…The demi notation was used firstly in [6] by Petryshyn. The class of weakly demicompact operators was studied in [7]. After that, the class of demi Dunford-Pettis operators was introduced in [8], and the class of order weakly demicompact operators was studied in [9].…”

ε-Pseudo Weak-Demicompactness for 2 × 2 Block Operator Matrices

“…The family of demicompact operators on X is denoted by DC(X). For more details on this subject, the reader can see [4,5,13,14,15,16,17]. In Fredholm theory, the first two papers were developed by Petryshyn W.V.…”

The Class of Demi-Order Norm Continuous Operators

“…The demi notation was used firstly in the article named "Construction of fixed points of demicompact mappings in Hilbert space" by Petryshyn in 1966 [1]. Krichen B. and O'Regan D. studied some results of the class of weakly demicompact linear operators in 2019 [2]. After, in [3], Benkhaled H., Hajji M., and Jeribi A. introduced the class of demi Dunford-Pettis operators which are a generalization of Dunford Pettis operators.…”