On the Weyl Spectrum: Spectral Mapping Theorem and Weyl's Theorem

Abstract

It is shown that ifTis a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and iffis a function analytic on a neighborhood of σ(T), then σw(f(T))=f(σw(T)), where σ(T) and σw(T) stand respectively for the spectrum and the Weyl spectrum ofT; moreover, Weyl's theorem holds forf(T)+Fif “dominant” is replaced by “M-hyponormal,” whereFis any finite rank operator commuting withT. These generalize earlier results for hyponormal operators. It is also shown that there exist an operatorTand a finite rank operatorFcommuting withTsuch that Weyl's theorem holds forTbut not forT+F. This answers negatively a problem raised by K. K. Oberai (Illinois J. Math.21, 1977, 84–90). However, ifTis required to be isoloid, then the statement that Weyl's theorem holds forTwill imply it holds forT+F.