Property (w) and perturbations II

Abstract

This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) [683][684][685][686][687][688][689][690][691][692] concerning the stability of property (w), a variant of Weyl's theorem, for a bounded operator T acting on a Banach space, under finite-dimensional perturbations K commuting with T . A counterexample shows that property (w) in general is not preserved under finite-dimensional perturbations commuting with T , also under the assumption that T is a-isoloid. Definitions and basic resultsProperty (w) is a variant of Weyl's theorem that has been introduced by V. Rakočević in [14] and studied in more recent papers [3,4]. To introduce all these concepts, let X denote an infinite-dimensional complex Banach space and let L(X) be the algebra of all bounded linear operators on X. For an operator T ∈ L(X) we shall denote by α(T ) the dimension of the kernel ker T , and, by β(T ) the codimension of the range T (X). We recall that T ∈ L(X) is said to be upper semi-Fredholm if α(T ) < ∞ and T (X) is closed, while T ∈ L(X) is said to be lower semi-Fredholm if β(T ) < ∞. Let Φ + (X) and Φ − (X) denote the class of all upper semi-Fredholm operators and the class of all lower semi-Fredholm operators, respectively.The upper semi-Weyl operators are defined as W + (X) := T ∈ Φ + (X): indT 0 , the lower semi-Weyl operators are defined as W − (X) := T ∈ Φ − (X): indT 0 . The set of Weyl operators is defined by W (X) := W + (X) ∩ W − (X) = T ∈ Φ(X): indT = 0 . ✩ This research was supported by Fondi ex-60, 2005, Universitá di Palermo.

Property (w) and perturbations III

“…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.…”

“…Furthermore, from definition of SVEP we have σ a (T ) does not cluster at λ ⇒ T has SVEP at λ. (4) In particular, if the point spectrum σ p (T ) (= the set of all eigenvalues of T ) is empty then T satisfies SVEP. An important subspace in local spectral theory is the quasi-nilpotent part of T defined by…”

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.…”

“…Property (w) and its perturbation properties has been studied in very recent papers ( [11], [6], [10], [3]). The following diagram resume the relationships between Weyl's theorems, a-Browder's theorem and property (w).…”

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.…”