A characterization of the essential approximation pseudospectrum on a Banach space

Abstract

One impetus for writing this paper is the issue of approximation pseudospectrum introduced by M. P. H.Wolff in the journal of approximation theory (2001). The latter study motivates us to investigate the essential approximation pseudospectrum of closed, densely defined linear operators on a Banach space. We begin by defining it and then we focus on the characterization, the stability and some properties of these pseudospectra.

Essential pseudospectra involving demicompact and pseudodemicompact operators and some perturbation results

“…Note that if ε tends to 0, we recover the well-known definitions of essential spectra of T. In [18], M. P. H. Wolff has given a motivation to study the essential approximate pseudospectrum. In [3], the notion of the essential approximate pseudospectrum was extended by devoting the studies to the essential approximate spectrum. For ε > 0 and T ∈ C(X),…”

“…♦[1] Let X be a Banach space, T ∈ C(X) and ε > 0. Then, λ σ e5,ε (T) if, and only if, for all D ∈ L(X) such that D < ε, we haveλ − T − D ∈ Φ(X) and i(λ − T − D) = 0.♦[3] Let T ∈ C(X) and ε > 0. The following properties are equivalent.…”

Generalized trace pseudo-spectrum of matrix pencils

“…In this paper, we interest by a generalization of eigenvalues called generalized trace pseudo-spectrum for an element in the matrix algebra to give more information about the matrix pencils of the form λV −U. For more information on various details on the above concepts, properties and applications of pseudo-spectrum [2,3,6,7,9], condition spectrum [1,4,5] and determinant spectrum [8]. Now, we introduce the new concept of the generalized trace pseudo-spectrum in the following definition.…”

The essential approximate pseudospectrum and related results

“…For ε > 0, it can be shown that σ ε (T) is a larger set and is never empty. The pseudospectra of T are a family of strictly nested closed sets, which grow to fill the whole complex plane as ε → ∞ (see [3,8,18,19]). From these definitions, it follows that the pseudospectra associated with various ε are nested sets.…”