Property (w) and perturbations III (original) (raw)
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Property (w) and perturbations II
Journal of Mathematical Analysis and Applications, 2008
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Property ( w) for perturbations of polaroid operators
Linear Algebra and Its Applications, 2008
A bounded linear operator T ∈ L(X) acting on a Banach space satisfies property (w), a variant of Weyl's theorem, if the complement in the approximate-point spectrum σ a (T ) of the Weyl essential approximatepoint spectrum σ wa (T ) is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property (w) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T .
Weyl type theorems for bounded linear operators
Acta Scientiarum Mathematicarum
For a bounded linear operator T acting on a Banach space let σ SBF − + (T) be the set of all λ ∈ C such that T − λI is upper semi-B-Fredholm and ind (T − λI) ≤ 0, and let E a (T) be the set of all isolated eigenvalues of T in the approximate point spectrum σ a (T) of T. We say that T satisfies generalized a-Weyl's theorem if σ SBF − + (T) = σ a (T)\E a (T). Among other things, we show in this paper that if T satisfies generalized a-Weyl's theorem, then it also satisfies generalized Weyl's theorem σ BW (T) = σ(T) \ E(T), where σ BW (T) is the B-Weyl spectrum of T and E(T) is the set of all eigenvalues of T which are isolated in the spectrum of T. 2000 AMS subject classification: 47A53, 47A55
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
Propertiesandfor Bounded Linear Operators
Journal of Mathematics, 2013
We shall consider properties which are related to Weyl type theorem for bounded linear operators , defined on a complex Banach space . These properties, that we callproperty, means that the set of all poles of the resolvent of of finite rank in the usual spectrum are exactly those points of the spectrum for which is an upper semi-Fredholm with index less than or equal to 0 and we callproperty, means that the set of all poles of the resolvent of in the usual spectrum are exactly those points of the spectrum for which is an upper semi--Fredholm with index less than or equal to 0. Properties and are related to a strong variants of classical Weyl’s theorem, the so-called property and property We shall characterize properties and in several ways and we shall also describe the relationships of it with the other variants of Weyl type theorems. Our main tool is localized version of the single valued extension property. Also, we consider the properties and in the frame of polaroid type opera...
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
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 p-hyponormal and M-hyponormal operators
Studia Mathematica, 2004
Weyl type theorems for p-hyponormal and M-hyponormal operators by Xiaohong Cao (Xi'an), Maozheng Guo (Beijing) and Bin Meng (Beijing) Abstract. "Generalized Weyl's theorem holds" for an operator when the complement in the spectrum of the B-Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues; and "generalized a-Weyl's theorem holds" for an operator when the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the isolated points of the approximate point spectrum which are eigenvalues. If T or T * is p-hyponormal or M-hyponormal then for every f ∈ H(σ(T)), generalized Weyl's theorem holds for f (T), so Weyl's theorem holds for f (T), where H(σ(T)) denotes the set of all analytic functions on an open neighborhood of σ(T). Moreover, if T * is p-hyponormal or M-hyponormal then for every f ∈ H(σ(T)), generalized a-Weyl's theorem holds for f (T) and hence a-Weyl's theorem holds for f (T). a (T) = C \ σ a (T). An operator T ∈ B(H) is called Fredholm if it has closed finite-codimensional range and finite-dimensional null space. The index of a Fredholm operator T ∈ B(H) is given by ind(T) = n(T) − d(T). An operator T ∈ B(H) is called Weyl if it is Fredholm of index zero, and Browder if it is Fredholm of finite ascent and descent, or equivalently, if T is Fredholm and T − λI is invertible for all sufficiently small λ = 0 in C. For T ∈ B(H), we write α(T) for the ascent of T and β(T) for the descent of T .