Demicompact linear operators, essential spectrum and some perturbation results

Abstract

Key words Demicompact linear operator, essential spectrum, Fredholm and semi-Fredholm operators MSC (2010) 47A53In this paper, we present some results on Fredholm and upper semi-Fredholm operators involving demicompact operators. Our results generalize many known ones in the literature, in particular those obtained by Petryshyn in [27] and Jeribi et al. in [1], [22]. They are used to establish a fine description of the Schechter essential spectrum of closed densely defined operators, and to investigate the essential spectrum of the sum of two bounded linear operators defined on a Banach space by means of the essential spectrum of each of the two operators.

Spectral theory for polynomially demicompact operators

“…More precisely, it was shown in [1] the invariance of the Schechter essential spectrum on Banach spaces by means of polynomially compact perturbations. The same result has been proved by W. Chaker et al in [4] for the class Λ X , where…”

“…W. V. Petryshyn has proved, in [13], that I −T is a Fredholm operator and i(I −T) = 0 for every condensing operator T. The same result has been proved by W. Chaker et all in [4], for T ∈ Λ X . For more results in this direction the reader can refers to [8,9].…”

“…When dealing with essential spectra of closed, densely defined linear operators on Banach spaces, one of the main problems consists in studying the invariance of the essential spectra of these operators subjected to various kinds of perturbation. Among the works in this direction we quote, for example [1,4,7]. More precisely, it was shown in [1] the invariance of the Schechter essential spectrum on Banach spaces by means of polynomially compact perturbations.…”

“…Using Theorem 3.1 in[4] and Theorem 2.1 in[5], we deduce that if P is a complex polynomial such that P(0) = 0 and T ∈ L(X), then P(T) ∈ Φ + (X) if, and only if, T ∈ Φ + (X).♦ If T ∈ Θ X , then I − T ∈ Φ(X) and i(I − T) = 0. ♦ Proof.…”

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

“…Recently, the concept of demicompactness with respect to a closed densely defined linear operator, as a generalization of the class of demicompact operator introduced by Petryshyn in [18]. This description involved the use of notions that were originally developed for demicompactness in linear spaces by W. Chaker, A. Jeribi, and B. Krichen [11] and systematically treated in the context of the bundle operator in [16] by B. Krichen. Lately, in [9] A. Ammar, H. Daoud and A. Jeribi defined the demicompact of a linear relation by T : D(T ) ⊆ X → X is said to be demicompact if for every bounded sequence {x n } in D(T ) such that Q I−T (I − T )x n → y ∈ X/(I − T )(0), there is a convergent subsequence of {Q T x n }.…”