Property (w) and perturbations III

Abstract

SVEP Polaroid operatorThe property (w) is a variant of Weyl's theorem, for a bounded operator T acting on a Banach space. In this note we consider the preservation of property (w) under a finite rank perturbation commuting with T , whenever T is polaroid, or T has analytical coreThe preservation of property (w) is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations. The theory is exemplified in the case of some special classes of operators.In this paper we continue the study of the class of linear bounded operators defined on Banach spaces that verify property (w), a variant of Weyl's theorem introduced by V. Rakočević in [23] and studied in a more recent paper [8]. The preservation of property (w) under certain classes of perturbations has been investigated in [3,4,7]. In this paper we give further results on the preservation of property (w) in some special cases and improve previous results. Moreover, the theory is applied to several classes of operators. We begin by given some preliminary definitions and basic results.Let X be an infinite-dimensional complex Banach space and denote by L( X) the algebra of all bounded linear operators on X . A bounded operator T ∈ L( X) is said to be an upper semi-Fredholm operators if α(T ) := dim ker T < ∞ and T (X) is closed, while T ∈ L( X) is said to be lower semi-Fredholm if β(T ) := codim T (X) < ∞. Let Φ + (X) and Φ − (X) denote the class of all upper semi-Fredholm operators. The index of a semi-Fredholm operator is defined as ind T := α(T ) − β(T ). T ∈ L( X) is said to be a Fredholm operator if T ∈ Φ + (X) ∩ Φ − (X). The upper semi-Weyl operators W + (X) are defined as the class of upper semi-Fredholm operators having ind T 0. The lower semi-Weyl operators W − (X) are defined as the class of lower semi-Fredholm operators having ind T 0. The class of Weyl operators is defined byThese classes of operators generate the following spectra: the Weyl spectrum defined by σ w (T ) := λ ∈ C: λI − T / ∈ W (X) , the upper semi-Weyl spectrum (in literature called also Weyl essential approximate point spectrum) defined by σ uw (T ) := λ ∈ C: λI − T / ∈ W + (X) ,

Property (R) for Bounded Linear Operators

“…This property, that we call property (R), means that the isolated points of the spectrum σ(T ) of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder (see later for definitions). Property (R) is strictly related to a strong variant of classical Weyl's theorem, the so-called property (w) introduced by Rakočević in [26], and more extensively studied in recent papers ( [11], [3], [6], [8], [10]). We shall characterize property (R) in several ways and we shall also describe the relationships of it with the other variants of Weyl's theorem.…”

Tensor products and property (w)

“…The implication of Proposition 1 fails for commuting quasinilpotents [3]; however, if A ⊗ B is finitely isoloid, then we have the following. …”

“…If in the above, Q 1 and Q 2 are nilpotents then (A + Q 1 ) ⊗ (B + Q 2 ) is the perturbation of A ⊗ B by a commuting nilpotent operator. The following proposition is immediate from [3,Theorem 3.8].…”

“…Operators satisfying property (w) have been studied in a number of papers in the recent past, see [1][2][3] for additional references. It is known that an operator T satisfying property (w) satisfies W t, hence Bt, but the reverse implication is generally false; property (w) neither implies nor is implied by a-W t. Property (w) does not survive perturbations by commuting Riesz, even quasinilpotent, operators [1].…”

Generalized Weyl's Theorem for Functions of Operators and Compact Perturbations

“…This paper is also inspired by [1,3,4], where the stability of property (w) under some perturbations is studied. The aim of this paper is to study the stability of generalized Weyl's theorem under analytic functional calculus and (small) compact perturbations.…”