A note on property (gb) and perturbations

An operator $T \in \mathcal{B}(X)$ defined on a Banach space $X$ satisfies property $(gb)$ if the complement in the approximate point spectrum $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{SBF_{+}^{-}}(T)$ coincides with the set $\Pi(T)$ of all poles of the resolvent of $T$. In this note we continue to study property $(gb)$ and the stability of it, for a bounded linear operator $T$ acting on a Banach space, under perturbations by nilpotent operators, by finite rank operators, by quasi-nilpotent operators commuting with $T$. Two counterexamples show that property $(gb)$ in general is not preserved under commuting quasi-nilpotent perturbations or commuting finite rank perturbations.Comment: 10 page

Generalized Kato decomposition, single-valued extension property and approximate point spectrum

AbstractIn this paper, we define the generalized Kato spectrum of an operator, and obtain that the generalized Kato spectrum differs from the semi-regular spectrum on at most countably many points. We study the localized version of the single-valued extension property at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ0∈C in the case that λ0I−T admits a generalized Kato decomposition. From this characterization we shall deduce several results on cluster points of some distinguished parts of the spectrum

A note on Browder spectrum of operator matrices

AbstractWhen A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the Hilbert space H⊕K of the form MC=(AC0B). In this note, it is shown that the following results in [Hai-Yan Zhang, Hong-Ke Du, Browder spectra of upper-triangular operator matrices, J. Math. Anal. Appl. 323 (2006) 700–707]W3(A,B,C)=W1(A,B,C)(in line 17 on p. 705) and⋂C∈B(K,H)σb(MC)=(⋂C∈B(K,H)σ(MC))∖[ρb(A)∩ρb(B)] are not always true, although the authors tried to fill the gap in their proofs by proposing an additional condition in [H.-Y. Zhang, H.-K Du, Corrigendum to “Browder spectra of upper-triangular operator matrices” [J. Math. Anal. Appl. 323 (2006) 700–707], J. Math. Anal. Appl. 337 (2007) 751–752]. A counterexample is given and then we show that under one of the following conditions:(i)σsu(B)=σ(B);(ii)intσp(B)=ϕ;(iii)σ(A)∩σ(B)=ϕ;(iv)σa(A)=σ(A), we have⋂C∈B(K,H)σb(MC)=σle(A)∪σre(B)∪W(A,B)∪σD(A)∪σD(B), where W(A,B)={λ∈C:N(B−λ)andH/R(A−λ)¯are not isomorphic up to a finitedimensional subspace}

LOCALIZED SVEP AND THE COMPONENTS OF QUASI-FREDHOLM RESOLVENT SET

In this paper, new characterizations of the single valued extension property are given, for a bounded linear operator T acting on a Banach space and its adjoint T*, at Λ0 C in the case that Λ0 I - T is quasi-Fredholm. With the help of a classical perturbation result concerning operators with eventual topological uniform descent, we show the constancy of certain subspace valued mappings on the components of quasi-Fredholm resolvent set. As a consequence, we obtain a classification of these components

The Intersection of Upper and Lower Semi-Browder Spectrum of Upper-Triangular Operator Matrices

When A∈B(H) and B∈B(K) are given, we denote by MC the operator acting on the infinite-dimensional separable Hilbert space H⊕K of the form MC=(AC0B). In this paper, it is proved that there exists some operator C∈B(K,H) such that MC is upper semi-Browder if and only if there exists some left invertible operator C∈B(K,H) such that MC is upper semi-Browder. Moreover, a necessary and sufficient condition for MC to be upper semi-Browder for some C∈G(K,H) is given, where G(K,H) denotes the subset of all of the invertible operators of B(K,H)