Pietro Aiena

Operators which have a closed quasi-nilpotent part

We find several conditions for the quasi-nilpotent part of a bounded operator acting on a Banach space to be closed. Most of these conditions are established for semi-Fredholm operators or, more generally, for operators which admit a generalized Kato decomposition. For these operators the property of having a closed quasi-nilpotent part is related to the so-called single valued extension property.

Single-valued extension property at the points of the approximate point spectrum

Abstract A localized version of the single-valued extension property is studied at the points which are not limit points of the approximate point spectrum, as well as of the surjectivity spectrum. In particular, we shall characterize the single-valued extension property at a point λ o ∈ C in the case that λoI−T is of Kato type. From this characterizations we shall deduce several results on cluster points of some distinguished parts of the spectrum.

On Drazin invertibility

The left Drazin spectrum and the Drazin spectrum coincide with the upper semi-B-Browder spectrum and the B-Browder spectrum, respectively. We also prove that some spectra coincide whenever T or T* satisfies the single-valued extension property.

ON THE PERTURBATION CLASSES OF CONTINUOUS SEMI-FREDHOLM OPERATORS

We prove that the perturbation class of the upper semi-Fredholm operators from X into Y is the class of the strictly singular operators, whenever X is separable and Y contains a complemented copy of C[0, 1]. We also prove that the perturbation class of the lower semi-Fredholm operators from X into Y is the class of the strictly cosingular operators, whenever X contains a complemented copy of 1 and Y is separable. We can remove the separability requirements by taking suitable spaces instead of C[0, 1] or 1. 2000 Mathematics Subject Classification. 47A53, 47A55.

Tensor products, multiplications and Weyl's theorem

Tensor productsZ=T1⊗T2 and multiplicationsZ=LT1RT2 do not inherit Weyl’s theorem from Weyl’s theorem forT1 andT2. Also, Weyl’s theorem does not transfer fromZ toZ*. We prove that ifTi,i=1, 2, has SVEP (=the single-valued extension property) at points in the complement of the Weyl spectrumσw(Ti) ofTi, and if the operatorsTi are Kato type at the isolated points ofσ(Ti), thenZ andZ* satisfy Weyl’s theorem.

SOME SPECTRAL PROPERTIES OF MULTIPLIERS ON SEMI-PRIME BANACH ALGEBRAS

Abstract We extend to arbitrary semi-prime Banach algebras some results of spectral theory and Fredholm theory obtained in [1] and [2] for multipliers defined in commutative semi-simple Banach algebras.

Property (gab) through localized SVEP

In this article we study the property (gab) for a bounded linear operator T ϵ L(X) on a Banach space X which is a stronger variant of Browder’s theorem. We shall give several characterizations of property (gab). These characterizations are obtained by using typical tools from local spectral theory. We also show that property (gab) holds for large classes of operators and prove the stability of property (gab) under some commuting perturbations.

Polaroid operators and Weyl type theorems

Weyl type theorems have been proved for a considerably large number of classes of operators. In this work, after introducing the class of polaroid operators and some notions from local spectral theory, we determine a theoretical and general framework from which Weyl type theorems may be promptly established for many of these classes of operators. The theory is exemplified by given several examples of hereditarily polaroid operators.

Some spectral mapping theorems through local spectral theory

The spectral mapping theorems for Browder spectrum and for semi-Browder spectra have been proved by several authors [14], [29] and [33], by using different methods. We shall employ a local spectral argument to establish these spectral mapping theorems, as well as, the spectral mapping theorem relative to some other classical spectra.We also prove that ifT orT* has the single-valued extension property some of the more important spectra originating from Fredholm theory coincide. This result is extended, always in the caseT orT* has the single valued extension property, tof(T), wheref is an analytic function defined on an open disc containing the spectrum ofT. In the last part we improve a recent result of Curto and Han [10] by proving that for every transaloid operatorT a-Weyl’s theorem holds forf(T) andf(T)*.

Incomparable Banach spaces and operator semigroups

Abstract. Using the notions of total incomparability and total coincomparability of Banach spaces, we define two families of operator semigroups. We show that these semigroups are minimal, in the sense that they admit a perturbative characterization. Moreover, they allow us to characterize the corresponding incomparability classes.