A note on Weyl’s theorem (original) (raw)
Kato type operators and Weyl's theorem
Journal of Mathematical Analysis and Applications, 2005
A Banach space operator T satisfies Weyl's theorem if and only if T or T * has SVEP at all complex numbers λ in the complement of the Weyl spectrum of T and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity. If T * (respectively, T) has SVEP and T is Kato type at all λ which are isolated eigenvalues of T of finite algebraic multiplicity (respectively, T is Kato type at all λ ∈ iso σ (T)), then T satisfies a-Weyl's theorem (respectively, T * satisfies a-Weyl's theorem).
Generalized Weyl's theorem for some classes of operators
Kyungpook Mathematical Journal, 2006
Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set σBw(A) of all λ ∈ C such that A−λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem σ Bw (A) = σ(A) \ E(A), and the B-Weyl spectrum σ Bw (A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in , if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalized Weyl's theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.
On Property (Saw) and others spectral properties type Weyl-Browder theorems
An operator T acting on a Banach space X satisfies the property (aw) if σ(T) \ σW (T) = E 0 a (T), where σW (T) is the Weyl spectrum of T and E 0 a (T) is the set of all eigenvalues of T of finite multiplicity that are isolated in the approximate point spectrum of T. In this paper we introduce and study two new spectral properties, namely (Saw) and (Sab), in connection with Weyl-Browder type theorems. Among other results, we prove that T satisfies property (Saw) if and only if T satisfies property (aw) and σ SBF − + (T) = σW (T), where σ SBF − + (T) is the upper semi B-Weyl spectrum of T .
Generalized Browder's and Weyl's theorems for Banach space operators
Journal of Mathematical Analysis and Applications, 2007
We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T . We also prove that the spectral mapping theorem holds for the B-Browder 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 )).
WEYLS THEOREM, aaa -WEYLS THEOREM, AND LOCAL SPECTRAL THEORY
Journal of the London Mathematical Society, 2003
We give necessary and sufficient conditions for a Banach space operator with the single valued extension property (SVEP) to satisfy Weyl's theorem and a-Weyl's theorem. We show that if T or T * has SVEP and T is transaloid, then Weyl's theorem holds for f (T ) for every f ∈ H(σ(T )). When T * has SVEP, T is transaloid and T is a-isoloid, then a-Weyl's theorem holds for f (T ) for every f ∈ H(σ(T )). We also prove that if T or T * has SVEP, then the spectral mapping theorem holds for the Weyl spectrum and for the essential approximate point spectrum. 1991 Mathematics Subject Classification. Primary 47A10, 47A53, 47A11; Secondary 47A15, 47B20. Key words and phrases. Weyl's theorem, Browder's theorem, a-Weyl's theorem, a-Browder's theorem, single valued extension property. The research of the first named author was partially supported by NSF grants DMS-9800931 and DMS-0099357.
Journal of Mathematical Analysis and Applications, 2006
In this note we study the property (w), a variant of Weyl's theorem introduced by Rakočević, 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). 567 be the class of all upper semi-Fredholm operators, and let Φ − (X) := T ∈ L(X): β(T ) < ∞ be the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by
A note on the a-Browder’s and a-Weyl’s theorems
Let T be a Banach space operator. In this paper, we characterize a-Browder’s theorem for T by the localized single valued extension property. Also, we characterize a-Weyl’s theorem under the condition E a (T)=π a (T), where E a (T) is the set of all eigenvalues of T which are isolated in the approximate point spectrum and π a (T) is the set of all left poles of T. Some applications are also given.
Operators obeying aaa-Weyl's theorem
Publicationes Mathematicae Debrecen
This article treatises several problems relevant to a-Weyl's theorem for bounded operators on Banach spaces. There are presented sufficient conditions for an operator T , such that a-Weyl's theorem holds for T. If a-Weyl's theorem holds for an a-isoloid operator T , and F is a finite rank operator commuting with T , then a-Weyl's theorem holds for T + F. The algebraic view point for a-Weyl's theorem is considered in the sense of the spectral mapping theorem for a special part of the spectrum. If T * is a quasihyponormal operator on a Hilbert space, f is a regular function in a neighbourhood of the spectrum of T and f is not constant on the connected components of its domain, we prove that a-Weyl's theorem holds for f (T). The article also contains some related results.
Weyl spectra and Weyl's theorem
Journal of Mathematical Analysis and Applications, 2003
Two variants of the Weyl spectrum are discussed. We find, for example, that if one of them coincides with the Browder spectrum then Weyl's theorem holds, and conversely for isoloid operators.
Weyl's theorem through local spectral theory
Glasgow Mathematical Journal, 2002
In this paper, we show that Weyl's theorem holds for operators having the single valued extension property and quasisimilarity preserves Weyl's theorem for these operators under some assumptions for spectral subsets, respectively.
On the Weyl spectrum of a Hilbert space operator
Proceedings of the American Mathematical Society, 1972
Using the perturbation definition of the Weyl spectrum, conditions are given on a closed (possibly unbounded) linear operator T in a Hilbert space which allow the Weyl spectrum to be characterized as a subset of the spectrum of T.
On a new class of operators and Weyl type theorems
Filomat, 2013
In the present article, we introduce a new class of operators which will be called the class of k-quasi *-paranormal operators that includes '-paranormal operators. A part from other results, we show that following results hold for a k-quasi *-paranormal operator T: (i) T has the SVEP. (ii) Every non-zero isolated point in the spectrum of T is a simple pole of the resolvent of T. (iii) All Weyl type theorems hold for T. (iv) Comments and some open problems are also presented.
A Note on Weyl's Theorem for Operator Matrices
Proceedings of the American Mathematical Society, 2003
When A ∈ B(X) and B ∈ B(Y ) are given we denote by M C an operator acting on the Banach space X ⊕ Y of the form In this note we examine the relation of Weyl's theorem for A ⊕ B and M C through local spectral theory.
Weyl?s Theorem for Algebraically Paranormal Operators
Integral Equations and Operator Theory, 2003
Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyl's theorem holds for f (T ) for every f ∈ H(σ(T )); (ii) a-Browder's theorem holds for f (S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T .
Weyl and Browder theorems for operators with or without SVEP at zero
2020
The study of operators having some special spectral properties like Weyl's theorem, Browder's theorem and the SVEP has been of important interest for some time now. The SVEP is very useful in the study of the local spectral theory. In this paper, we explore the single-valued extension property (SVEP) for some operators on Hilbert spaces. We characterize operators with or without SVEP at zero and those where Weyl's and Browder's theorems hold. It is shown that if a Fredholm operator has no SVEP at zero, then zero is an accumulation point of the spectrum of the operator. It is also shown that quasi similar Fredholm operators have equal Weyl spectrum.
Spectral picture , perturbed Browder and Weyl theorems , and their variations
2017
The holes (i.e., the union of the bounded components of the complement in the complex plane), alongwith the isolated points, of the Weyl and the approximate Weyl spectrum (and their B-Fredholm avatars) play a decisive role in determining Browder and Weyl theorems type properties for Banach space operators and their perturbations. Dedicated to Marigold Maisie Duggal on her birthday
Browder's theorems and the spectral mapping theorem
Divulgaciones Matematicas
A bounded linear operator T 2 L(X) on a Banach space X is said to satisfy Browder's theorem if two important spectra, originating from Fredholm theory, the Browder spectrum and the Weyl spectrum, coin- cide. This expository article also concerns with an approximate point version of Browder's theorem. A bounded linear operator T 2 L(X) is said to satisfy a-Browder's theorem if the upper semi-Browder spec- trum 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. This paper also deals with the relationships between Browder's theorem, a-Browder's theorem and the spectral mapping theorem for certain parts of the spectrum.
Some characterizations of operators satisfying a-Browder's theorem
Journal of Mathematical Analysis and Applications, 2005
We characterize the bounded linear operators T defined on Banach spaces satisfying a-Browder's theorem, or a-Weyl's theorem, by means of the discontinuity of some maps defined on certain subsets of C. Several other characterizations are given in terms of localized SVEP, as well as by means of the quasi-nilpotent part, the hyper-kernel or the analytic core of λI − T . 531 denote the class of all upper semi-Fredholm operators, and let Φ − (X) := T ∈ L(X): β(T ) < ∞ denote the class of all lower semi-Fredholm operators. The class of all semi-Fredholm operators is defined by
Weyl’s theorem for algebraically wF(p, r, q) operators with p, r > 0 and q ≥ 1
Ukrainian Mathematical Journal, 2012
and q 1; then we establish the spectral mapping theorems for the Weyl spectrum and for the essential approximate point spectrum of T for any f 2 Hol. .T //; respectively. Finally, we examine the stability of Weyl's theorem and the a-Weyl's theorem under commutative perturbations by finite-rank operators.