Polaroid operators satisfying Weyl’s theorem

A Banach space operator T is polaroid and satisfies Weyl’s theorem if and only if T is Kato type at points λ ∈ iso σ(T) and has SVEP at points λ not in the Weyl spectrum of T. For such operators T, f(T) satisfies Weyl’s theorem for every non-constant function f analytic on a neighborhood of σ(T) if and only if f(T^∗) satisfies Weyl’s theorem.     

Generalized Browder’s Theorem and SVEP

A bounded operator a Banach space, is said to verify generalized Browder’s theorem if the set of all spectral points that do not belong to the B-Weyl’s spectrum coincides with the set of all poles of the resolvent of T, while T is said to verify generalized Weyl’s theorem if the set of all spectral points that do not belong to the B-Weyl spectrum coincides with the set of all isolated points of the spectrum which are eigenvalues. In this article we characterize the bounded linear operators T satisfying generalized Browder’s theorem, or generalized Weyl’s theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part H ₀(λI − T) as λ belongs to certain subsets of . In the last part we give a general framework for which generalized Weyl’s theorem follows for several classes of operators.     

A Note on *-paranormal Operators

In this note we show that if either T or T^* is totally *-paranormal then Weyls theorem holds for f(T) for every f , and also a-Weyls theorem holds for f(T) if T is totally *-paranormal. We prove that if either T or T^* is *-paranormal then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum.     

Weyl’s Theorem and Perturbations

In the present paper we examine the stability of Weyl’s theorem under perturbations. We show that if T is an isoloid operator on a Banach space, that satisfies Weyl’s theorem, and F is a bounded operator that commutes with T and for which there exists a positive integer n such that Fⁿ is finite rank, then T + F obeys Weyl’s theorem. Further, we establish that if T is finite-isoloid, then Weyl’s theorem is transmitted from T to T + R, for every Riesz operator R commuting with T. Also, we consider an important class of operators that satisfy Weyl’s theorem, and we give a more general perturbation results for this class.     

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having property (H) and study Weyl’s theorems, and related results for operators which satisfy this property. We show that a- Weyl’s theorem holds for every decomposable operator having property (H). We also show that a-Weyl’s theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator T_μ of a group algebra L¹(G), G a locally compact abelian group, satisfies a-Weyl’s theorem. Similar results are given for multipliers of other important commutative Banach algebras.     

Browder’s Theorems through Localized SVEP

A bounded linear operator T ∈ L(X) on aBanach space X is said to satisfy “Browder’s theorem” if the Browder spectrum coincides with the Weyl spectrum. T ∈ L(X) is said to satisfy “a-Browder’s theorem” if the upper semi-Browder spectrum coincides with the approximate point Weyl spectrum. In this note we give several characterizations of operators satisfying these theorems. Most of these characterizations are obtained by using a localized version of the single-valued extension property of T. In the last part we shall give some characterizations of operators for which “Weyl’s theorem” holds.     

Tensor products, multiplications and Weyl’s theorem

Tensor productsZ=T ₁⊗T ₂ and multiplicationsZ=L _{T 1} R _{T 2} do not inherit Weyl’s theorem from Weyl’s theorem forT ₁ andT ₂. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifT _i,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσ _w(T_i) ofT _i, and if the operatorsT _i are Kato type at the isolated points ofσ(T_i), thenZ andZ* satisfy Weyl’s theorem.     

A note on generalized Weyl's theorem

We prove that if either T or T^∗ has the single-valued extension property, then the spectral mapping theorem holds for B-Weyl spectrum. If, moreover T is isoloid, and generalized Weyl's theorem holds for T, then generalized Weyl's theorem holds for f(T) for every f∈H(σ(T)). An application is given for algebraically paranormal operators.     

Weyl’s Theorem for Algebraically Paranormal Operators

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyls theorem holds for f(T) for every f $\in$ H((T)); (ii) a-Browders theorem holds for f(S) for every S $\prec$ T and f $\in$ H((S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Riesz Idempotent and Algebraically M-hyponormal Operators

Let T be an M-hyponormal operator acting on infinite dimensional separable Hilbert space and let be the Riesz idempotent for λ₀, where D is a closed disk of center λ₀ which contains no other points of σ (T). In this note we show that E is self-adjoint and As an application, if T is an algebraically M-hyponormal operator then we prove : (i) Weyl’s theorem holds for f(T) for every (ii) a-Browder’s theorem holds for f(S) for every and f ∈ H(σ(S)); (iii) the the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.     

Polaroid operators,SVEP and perturbed Browder,Weyl theorems

A Banach space operatorT ɛB(X) is polaroid,T ɛP, if the isolated points of the spectrum ofT are poles of the resolvent ofT. LetPS denote the class of operators inP which have have SVEP, the single-valued extension property. It is proved that ifT is polynomiallyPS andA ɛB(X) is an algebraic operator which commutes withT, thenf(T+A) satisfies Weyl’s theorem andf(T ^*+A ^*) satisfiesa-Weyl’s theorem for everyf which is holomorphic on a neighbourhood of σ(T+A).     

On the Isolated Points of the Spectrum of Paranormal Operators

For paranormal operator T on a separable complex Hilbert space we show that (1) Weyl’s theorem holds for T, i.e., σ(T) \ w(T) = π₀₀(T) and (2) every Riesz idempotent E with respect to a non-zero isolated point λ of σ(T) is self-adjoint (i.e., it is an orthogonal projection) and satisfies that ranE = ker(T − λ) = ker(T − λ)^*.     

B-fredholm and spectral properties for multipliers in Banach algebras

The main purpose of this paper is to study spectral and B-Fredholm properties of a multiplierT acting on a semi-simple regular tauberian commutative Banach algebraA. We show thatT is a B-Fredholm operator if and only ifT is a semi B-Fredholm operator, and in this case we have the indexind(T)=0. Moreover we give some spectral properties for multipliers. Spectral mapping theorems for the Weyl’s and B-Weyl spectrum of a multiplier are also considered. Furthermore we show that Weyl’s theorem and generalized Weyl’s theorem hold for a multiplierT. Finally we give sufficient conditions for a multiplier to be a product of an invertible and an idempotent operators.     

Weyl type theorems for operators satisfying the single-valued extension property

Let T be a bounded linear operator acting on a Banach space X such that T or its adjoint T^∗ has the single-valued extension property. We prove that the spectral mapping theorem holds for the B-Weyl spectrum, and we show that generalized Browder's theorem holds for f(T) for every analytic function f defined on an open neighborhood U of σ(T). Moreover, we give necessary and sufficient conditions for such T to satisfy generalized Weyl's theorem. Some applications are also given.     

Weyl’s theorem and thin spectra

Using the notion of thin sets we prove a theorem of Weyl type for the Wolf essential spectrum ofT ∈β (H). *Further we show that Weyl’s theorem holds for a restriction convexoid operator and consequently modify some results of Berberian. Finally we show that Weyl’s theorem holds for a paranormal operator and that a polynomially compact paranormal operator is a compact perturbation of a diagnoal normal operator. A structure theorem for polynomially compact paranormal operators is also given.     

On p-quasi-hyponormal operators

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p ? 1, A ∈ p − QH, if A^∗(∣A∣^2p − ∣A^∗∣^2p)A ? 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A⁻¹(0) ⊆ A^∗-1(0), A ∈ p_∗ − QH, a necessary and sufficient condition for the adjoint of a pure p_∗ − QH operator to be supercyclic is proved. Operators in p_∗ − QH satisfy Bishop’s property (β). Each A ∈ p_∗ − QH has the finite ascent property and the quasi-nilpotent part H₀(A − λI) of A equals (A − λI)^-1(0) for all complex numbers λ; hence f(A) satisfies Weyl’s theorem, and f(A^∗) satisfies a-Weyl’s theorem, for all non-constant functions f which are analytic on a neighborhood of σ(A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p_∗ − QH.     

Generalized Browder's and Weyl's theorems for Banach space operators

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem. We also prove that the spectral mapping theorem holds for the Drazin spectrum and for analytic functions on an open neighborhood of σ(T). As applications, we show that if T is algebraically M-hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f(T), where f∈H((T)), the space of functions analytic on an open neighborhood of σ(T). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f(T), for each f∈H(σ(T)).     

ON THE SPECTRAL RADIUS OF IRREDUCIBLE AND WEAKLY IRREDUCIBLE OPERATORS IN BANACH LATPICES

Abstract

If T is an operator on a Banach lattice E we call T weakly irreducible if E contains no non-trivial T-invariant bands. We prove that if E is order complete and if the weakly irreducible operator T > 0 is in (E′_oo ? E)^⊥⊥ then T has positive spectral radéus. Prom this follows that Jentesch's theorem holds in arbitrary Banach function spaces.

If [Ttilde] denotes the restriction of T′ to E′_oo, 0 ? T an order continuous operator, then T is weakly irreducible if and only if [Ttilde]: E′_oo→E′_oo is weakly irreducible.

Finally we show that the majorizing, irreducible operator T ≥ 0, has positive spectral radius if either Tⁿ is weakly compact or E has property (P) or T is strongly majorizing.     

Weyl’s Theorem for Algebraically Quasi-class A Operators

If T or T* is an algebraically quasi-class A operator acting on an infinite dimensional separable Hilbert space then we prove that Weyl’s theorem holds for f(T) for every f ∈ H(σ(T)), where H(σ(T)) denotes the set of all analytic functions in an open neighborhood of σ(T). Moreover, if T* is algebraically quasi-class A then a-Weyl’s theorem holds for f(T). Also, if T or T* is an algebraically quasi-class A operator then we establish that the spectral mapping theorems for the Weyl spectrum and the essential approximate point spectrum of T for every f ∈ H(σ(T)), respectively. This research was supported by the Kyung Hee University Research Fund in 2007 (KHU- 20071605).