Weyl's theorem through local spectral theory (original) (raw)
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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.
Weyl type theorems for operators satisfying the single-valued extension property
Journal of Mathematical Analysis and Applications, 2007
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, a-Weyl’s theorem and single-valued extension property
Let T be a bounded linear operator on an infinite-dimensional complex Banach space X. T is said to satisfy Weyl’s theorem if σ(T)∖σ W (T)-π 00 (T), where σ(T) and σ W (T)={λ∈C:λI-T is Fredholm with ind(λI-T)=0} denote the spectrum and the Weyl spectrum of T, respectively, and π 00 (T) consists of isolated eigenvalues of T with finite multiplicities. A classical result of H. Weyl shows that Hermitian operators satisfy Weyl’s theorem. T is said to have the single-valued extension property (SVEP) if for every open subset U of the complex plane, the only analytic function f:U→X which satisfies (λI-T)f(λ)=0 for all λ in U is the function f≡0. The present paper starts by proving that if T or T * has the SVEP, then Weyl’s theorem holds for T if and only if one of ten equivalent spectral conditions is true. Such conditions involve various fine parts of σ(T), and the quasinilpotent part, analytic core, hyperrange, co-rank, descent and reduced minimum modulus of λI-T. This generalizes the pre...
Generalized a-Weyl's Theorem and the Single-Valued Extension Property
2006
Let T be a bounded linear operator acting on a Banach space X such that T or T* has the SVEP. We prove that the spectral mapping theorem holds for the semi-essential approximate point spectrum ?SBF-+ (T); and we show that generalized a-Browder's theorem holds for f(T) for every analytic function f defined on an open neighbourhood U of [sigma](T): Moreover, we give a necessary and sufficient condition for such T to obey generalized a-Weyl's theorem. An application is given for an important class of Banach space operators.
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.
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.
Proceedings of the American Mathematical Society
The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder’s theorem to hold.
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 Remark on Weyl Type Theorems
The study of operators satisfying Weyl's theorem, gen-eralized Weyl's theorem, Browder's theorem , the single valued extension property (SVEP), and Bishop's property are of signifi-cant interest, and are currently being done by a number of math-ematicians around the world. In one of there papers R.Curto and Y.M.Han have shown that an algebraically paranormal operator has the single valued extension property in separable Hilbert space. They also showed Weyl's theorem holds for algebraically paranor-mal operator in a separable Hilbert space. In this paper we prove that the single valued extension property holds without the sep-arability condition and we show that an algebraically paranormal operator obey's Weyl's theorem in a general Hilbert space without the separability condition. We also show that all results about gen-eralized Weyl's theorem for several classes of operators containing normal operators obtained by many mathematicians are not true in ...