In this paper we discuss some topological properties of the set F(S0, T, z) := {x ∈ X : S0x ∈ Tx+ z} , where T is a nonlinear multi-valued mappings and S0 is a single-valued mappings acting on a Banach space X. This study is based on a new concept, the so called weakly relative demicompactness for nonlinear operators.

/pdf/on-the-class-of-relatively-weakly-demicompact-nonlinear-3vytspqgoy.pdf

On the class of relatively weakly demicompact nonlinear operators

In this paper, we investigate the concept of generalized weakly demicompact operators with respect to weakly closed densely defined linear operators. We give their relationship with Fredholm and upper semi-Fredholm operators. In particular a characterization by means of upper semi-Browder spectrum is given. Moreover, we provide some sufficient conditions on the inputs of a closable block operator matrix to ensure the generalized weak demicompactness of its closure. Our results generalize many known ones in the literature.

Spectral Properties Involving Generalized Weakly Demicompact Operators

The concept of the line graph of a given graph is so natural that it has been independently discovered by many authors. Of course, each author gave it a different name: It was called the interchange graph by Ore [272], derivative by H. Sachs [297], derived graph by L. W. Beineke [52], edge-to-vertex dual by M. Reed [291], coverning graph by G. Kirchhoff [189], and adjoint by V. Menon [247].

Spectral Graph Theory

On some classes of demicompact linear relation and some results of essential pseudospectra

In this paper, we study the essential and the structured essential
 pseudospectra of closed densely defined linear operators acting on a Banach
 space X. We start by giving a refinement and investigating the stability of
 these essential pseudospectra by means of the class of demicompact linear
 operators. Moreover, we introduce the notion of pseudo demicompactness and
 we study its relationship with pseudo upper semi-Fredholm operators. Some
 stability results for the Gustafson essential pseudospectrum involving
 pseudo demicompact operators is given.

/pdf/essential-pseudospectra-involving-demicompact-and-bpzns7m4xq.pdf

Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation results

Relative essential spectra involving relative demicompact unbounded linear operators

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Relative essential spectra involving relative demicompact unbounded linear operators