Weyl type theorems for bounded linear operators on Banach spaces

摘要

In 1909 H. Weyl [59] studied the spectra of all compact linear perturbations of a self-adjoint operator defined on a Hubert space and found that their intersection consisted precisely of those points of the spectrum which are not isolated eigenvalues of finite multiplicity. Later, the property established by Weyl for self-adjoint operators has been observed for several other classes of operators, for instance hyponormal operators on Hilbert spaces, Toeplitz operators [33], convolution operators on group algebras [19], and many other classes of operators defined on Banach spaces ([52]). In the literature, a bounded linear operator defined on a Banach space which satisfies this property is said to satisfy Weyl's theorem. Weaker variants of Weyl's theorem have been discussed by Harte and Lee [44], while two approximate-point spectrum versions of Weyl's theorem have been introduced by Rakocevic [54], a-Weyl's theorem ([55], [2]), and the so-called property (w) ([54], [17]). In this course we describe all Weyl type theorems, together with some their generalized versions ([25], [27]) and we show the equivalences of these theorems for classes of operators which satisfy certain "polaroid" conditions on the isolated points of the spectrum ([5], or on the isolated points of the approximate point spectrum. Our main tool is an important property, the so-called single valued extension property (SVEP), introduced by Dunford [41], [42]. The SVEP plays an important role in local spectral theory, see the monograph of Laursen and Neumann [50], and a localized version of SVEP has deep connections with Fredholm theory, see the monograph [1]. In the last part of the course we study the permanence of Weyl type theorems under a quasi-affinity, or more in general, the permanence of Weyl type theorems from an operator T ? L(X) to an operator S ? L(Y), X and Y Banach spaces, in the case that T and S are intertwined asymptotically by an operator A G L(X,Y).