Abstract
In this paper, we introduce and study the essential pseudospectra of closed, densely defined linear operators in the Banach space. We start by giving the definition and we investigate the characterization, the stability and some properties of these essential pseudospectra.
Highlights
We denote by L(X, Y ) resp., C(X, Y ) the set of all bounded linear operators from X into Y and we denote by K(X, Y ) the subspace of compact operators from X into Y
Jeribi defined the notion of pseudo-Browder essential spectra of densely closed, linear operators in the Banach space by: 1 σb,ε(T ) = σb(T ) ∪ λ ∈ C such that Rb(λ, T ) > ε, where Rb(λ, T ) = (λ − T )|Kλ −1 I − Pλ + Pλ and by convention we write Rb(λ, T ) = ∞ if Rb(λ, T ) is unbounded or nonexistent, i.e., if λ is in the spectrum σb(T )
When dealing with Weyl pseudospectra of closed, densely defined linear operators on Banach spaces, one of the main problems consists of studying the invariance of the Weyl pseudospectrum of these operators subjected to various kinds of perturbation
Summary
Let T be a closed linear operator on a Banach space X . There are several and in general non-equivalent definitions of the essential spectrum of a linear operator on a Banach space. Jeribi defined the notion of pseudo-Browder essential spectra of densely closed, linear operators in the Banach space by: 1 σb,ε(T ) = σb(T ) ∪ λ ∈ C such that Rb(λ, T ) > ε , where Rb(λ, T ) = (λ − T )|Kλ −1 I − Pλ + Pλ and by convention we write Rb(λ, T ) = ∞ if Rb(λ, T ) is unbounded or nonexistent, i.e., if λ is in the spectrum σb(T ).
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