ALGEBRAIC EXTENSION OF ＊-A OPERATOR

In this paper, we study various properties of algebraic extension of *-A operator.Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid.And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.     

拟绝对-*-κ-仿正规算子的谱性质

引入了拟绝对-*-k-仿正规算子,获得了拟绝对-*-k-仿正规算子的一个充要条件.并证明了拟绝对-*-k-仿正规算子在0≤k≤1上是有限上升的,作为此性质的应用,证明了若T是拟绝对-*-k-仿正规算子,其中0≤k≤1,则Weyl谱和本质近似点谱的谱映射定理成立.最后证明了若T是拟绝对-*-k-仿正规算子,其中0≤k≤1,则σ_(ja)(T)\{0}=σ_a(T)\{0}.     

Variations on Weyl's theorem

In this note we study the property (w), a variant of Weyl's theorem introduced by Rako?evi?, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property (w) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property (w) for T (respectively T^*) coincide whenever T^* (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property (w).     

Property (w) and perturbations III

The 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 core K(λ₀I−T)={0} for some λ₀∈C. The 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.     

On Drazin invertibility

The left Drazin spectrum and the Drazin spectrum coincide with the upper semi--Browder spectrum and the -Browder spectrum, respectively. We also prove that some spectra coincide whenever or satisfies the single-valued extension property.

     

Property (w) and perturbations II

This note is a continuation of a previous article [P. Aiena, M.T. Biondi, Property (w) and perturbations, J. Math. Anal. Appl. 336 (2007) 683-692] 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.     

Algebraic extension of *-A operator

In this paper, we study various properties of algebraic extension of *-A operator. Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid. And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.     

B-Browder spectra and localized SVEP

In this paper we study the relationships between the B-Browder spectra and some other spectra originating from Fredholm theory and B-Fredholm theory. This study is done by using the localized single valued extension property. In particular, we shall see that many spectra coincide in the case that a bounded operator T, or its dual T*, or both, admits the single valued extension property.      

SVEP for multipliers on a faithful commutative Banach algebra

We give some sufficient conditions that each multiplier on a faithful commutative Banach algebra has SVEP. On the other hand, we show that there exist a faithful commutative Banach algebra and a multiplier on it without SVEP. Such examples of multipliers can actually be found within the class of multiplication operators on unital commutative Banach algebras. This answers in negative a question that is stated as Open problem 6.2.1 by Laursen and Neumann, 2000.