0000000000530578
AUTHOR
Jesús R. Guillén
Property (R) for Bounded Linear Operators
We introduce the spectral property (R), for bounded linear operators defined on a Banach space, which is related to Weyl type theorems. This property is also studied in the framework of polaroid, or left polaroid, operators.
Property (R) under perturbations
Property (R) holds for a bounded linear operator $${T \in L(X)}$$ , defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.
Property (w) for perturbations of polaroid operators
Abstract 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 approximate-point 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's theorem for perturbations of paranormal operators
A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Weyl's theorem" if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl's theorem for every algebraic operator K which commutes with T.
A Unifying Approach to Weyl Type Theorems for Banach Space Operators
Weyl type theorems have been proved for a considerably large number of classes of operators. In this paper, by introducing the class of quasi totally hereditarily normaloid operators, we obtain a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. This framework also entails Weyl type theorems for perturbations f(T + K), where K is algebraic and commutes with T, and f is an analytic function, defined on an open neighborhood of the spectrum of T + K, such that f is non constant on each of the components of its domain.