Topological Uniform Descent Research Articles

Topological Uniform Descent and Judgement of A-Weyl's Theorem

In this paper, a-Browder's theorem and a-Weyl's theorem for bounded linear operators are studied by means of the property of the topological uniform descent. The sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space holding a-Browder's theorem and a-Weyl's theorem are established. As a consequence of the main result, the new judgements of a-Browder's theorem and a-Weyl's theorem for operator function are discussed.

Judgement of two Weyl type theorems for bounded linear operators

Let H be an infinite dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ? B(H) is said to satisfy property (UW?) if ?a(T)\?ea(T) = ?(T), where ?a(T) and ?ea(T) denote the approximate point spectrum and the essential approximate point spectrum of T respectively, ?(T) denotes the set of all poles of T. T ? B(H) satisfiesa-Weyl?s theorem if ?a(T)\?ea(T) = ?a 00(T), where ?a 00(T) = {? ? iso?a(T) : 0 < n(T ??I) < ?}. In this paper, we give necessary and sufficient conditions for a bounded linear operator and its function calculus to satisfy both property (UW?) and a-Weyl?s theorem by topological uniform descent. In addition, the property (UW?) and a-Weyl?s theorem under perturbations are also discussed.

Property $(W_{E})$ and topological uniform descent

We characterize the property $(W_{E})$ in terms of topological uniform descent, and give the transfer of property $(W_{E})$ from bounded linear operators to functions of operators. Moreover, the perturbation of property $(W_{E})$ under power finite rank operators is investigated. As an application, the property for hypercyclic operators and supercyclic operators is investigated.

New Decompositions for Classes of Operators with Topological Uniform Descent

The article describes a new decomposition property for operators with topological uniform descent, like Kato type operators, as well as new results on the stability of this class of operators under perturbations by operators with finite-range power based on topological descent notion, from which we can generalize many perturbation results for a large classes of operators by extending to Banach spaces known techniques on Hilbert spaces. As application of our resuts we obtain that is a lower semi B-Weyl operator if and only if , where is a lower semi B-Browder operator and , for some . Our methods generalize to Banach spaces some results obtained by Aiena for operators acting on Hilbert spaces.

Topological Uniform Descent, Quasi-Fredholmness and Operators Originated from Semi-B-Fredholm Theory

In this paper we study operators originated from semi-B-Fredholm theory and as a consequence we get some results regarding boundaries and connected hulls of the corresponding spectra. In particular, we prove that a bounded linear operator T acting on a Banach space, having topological uniform descent, is a BR operator if and only if 0 is not an accumulation point of the associated spectrum $$\sigma _\mathbf{R}(T)=\{\lambda \in \mathbb {C}:T-\lambda I\notin \mathbf{R}\}$$ , where $$\mathbf{R}$$ denote any of the following classes: upper semi-Weyl operators, Weyl operators, upper semi-Fredholm operators, Fredholm operators, operators with finite (essential) descent and $$\mathbf{BR}$$ the B-regularity associated to $$\mathbf{R}$$ as in Berkani (Studia Mathematica 140(2):163–174, 2000). Under the stronger hypothesis of quasi-Fredholmness of T, we obtain a similar characterisation for T being a $$\mathbf{BR}$$ operator for much larger families of sets $$\mathbf{R}.$$

Topological uniform descent and compact perturbations

Let $$\mathcal {H}$$ be an infinite dimensional separable complex Hilbert space and $$\mathcal {B(H)}$$ the algebra of all bounded linear operators on $$\mathcal {H}$$ . In this paper, we characterize some features of the topological uniform descent resolvent set $$\rho _\tau (T)$$ for an operator $$T\in \mathcal {B(H)}$$ , and give a classification of the components of $$\rho _\tau (T)$$ . Then using topological uniform descent spectrum and $$\rho _\tau (T)$$ , we discuss the SVEP and further characterize those operators for which the SVEP is preserved under compact perturbations.

Property (ω) and topological uniform descent

We give the necessary and sufficient condition for a bounded linear operator with property (ω) by means of the induced spectrum of topological uniform descent, and investigate the permanence of property (ω) under some commuting perturbations by power finite rank operators. In addition, the theory is exemplified in the case of algebraically paranormal operators.

ON THE EQUIVALENCE FOR PROPERTY (

Property(ω)is a variant of Weyl's theorem.By the property of consistence in Fredholm index,the judgement of property(ω)is studied.Using the relation between a new spectrum defined in view of the property of consistence in Fredholm index and the topological uniform descent,the necessary and sufficient condition for an operator and its conjugate to satisfy property(ω)are given.In addition,the property(ω)of operator matrices is studied.

Property (ω) for operator matrices

In this note, property (ω) and property (ω1)are variants of Weyl’s theorem. By means of topological uniform descent, the sufficient and necessary conditions of a bounded linear operator defined on a Hilbert space that satisfies property (ω1) and property (ω1)is studied. Moreover, property 1 (ω1)and property (ω1)of 2 × 2 operator matrices are discussed as well.

Components of topological uniform descent resolvent set and local spectral theory

In this paper, we shall study the components of topological uniform descent resolvent set ρud(T) for a bounded linear operator T acting on a Banach space. We first show the constancy of certain subspace valued mappings on the components of ρud(T). Then using these results and, the equivalences of SVEP at a point λ for T and T* in the case that λI-T has topological uniform descent, we obtain a classification of these components. Finally, we give some applications of the classification. In particular, we give a sufficient condition on an operator T such that its topological uniform descent spectrum σud(T) coincides with some distinguished parts of its spectrum σ(T).

Weyl's type theorems and hypercyclic operators

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions: (i) If T or T* ∈▪, then generalized a-Browder's theorem holds for f(T) for every f ∈ Hol(σ(T)). (ii) If T or T* ∈▪ has topological uniform descent at all λ ∈ iso(σ(T)), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (iii) If T ∈▪ has topological uniform descent at all λ ∈E(T), then T satisfies generalized Weyl's theorem. (iv) Let T ∈▪. If T satisfies the growth condition Gd (d ≥ 1), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈▪, then, f(σSBF+−(T)) = σSBF+−(f(T)) for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈▪. If T − λI has finite ascent at every λ ∈Ea(T) and if F is of finite rank on H such that TF = FT, then T + F obeys generalized a-Weyl's theorem.

Topological uniform descent and localized SVEP

A localized version of the single-valued extension property is studied, for a bounded linear operator T acting on a Banach space and its adjoint T⁎, at the points λ0∈C such that λ0I−T has topological uniform descent (TUD for brevity). We characterize the single-valued extension property at these points for T and T⁎. We also give some applications of these results. As we give a counterexample to show that the adjoint of an operator with TUD is not necessarily with TUD, it is worth to mention that the characterizations of SVEP at these points for T⁎ cannot be obtained dually from the characterizations of SVEP at the same points for T. It is quite different from the case that λ0I−T is of Kato type or quasi-Fredholm.

SMALL ESSENTIAL SPECTRAL RADIUS PERTURBATIONS OF OPERATORS WITH TOPOLOGICAL UNIFORM DESCENT

AbstractIn this paper we consider small essential spectral radius perturbations of operators with topological uniform descent—small essential spectral radius perturbations which cover compact, quasinilpotent and Riesz perturbations.

Weyl type theorem for operator matrices

Using topological uniform descent, we give necessary and sufficient conditions for Browder's theorem and Weyl's theorem to hold for an operator $A$. The two theorems are liable to fail for $2\times 2$ operator matrices. In this paper, we explore how they

Topological uniform descent and Weyl type theorem

The generalized Weyl’s theorem holds for a Banach space operator T if and only if T or T ∗ has the single valued extension property in the complement of the Weyl spectrum (or B-Weyl spectrum) and T has topological uniform descent at all λ which are isolated eigenvalues of T. Also, we show that the generalized Weyl’s theorem holds for analytically paranormal operators.

Single valued extension property and generalized Weyl’s theorem

Let $T$ be an operator acting on a Banach space $X$, let $\sigma (T)$ and $ \sigma _{BW}(T) $ be respectively the spectrum and the B-Weyl spectrum of $T$. We say that $T$ satisfies the generalized Weyl’s theorem if $ \sigma _{BW}(T)= \sigma (T) \setminus E(T)$, where $E(T)$ is the set of all isolated eigenvalues of $T$. The first goal of this paper is to show that if $T$ is an operator of topological uniform descent and $0$ is an accumulation point of the point spectrum of $T,$ then $T$ does not have the single valued extension property at $0$, extending an earlier result of J. K. Finch and a recent result of Aiena and Monsalve. Our second goal is to give necessary and sufficient conditions under which an operator having the single valued extension property satisfies the generalized Weyl’s theorem.

On semi B-Fredholm operators

An operator T on a Banach space is called ‘semi B-Fredholm’ if for some n \in {\tf="times-b"N} the range R(T\;\!^n) of T\;\!^n is closed and the induced operator T_n on R(T\;\!^n) semi-Fredholm. Semi B-Fredholm operators are stable under finite rank perturbation, and subject to the spectral mapping theorem; on Hilbert spaces they decompose as sums of nilpotent and semi-Fredholm operators. In addition some recent generalizations of the punctured neighborhood theorem turn out to be consequences of Grabiner's theory of ‘topological uniform descent’.