On the Weyl Spectrum: Spectral Mapping Theorem and Weyl’s TheoremU

It is shown that if T is a dominant operator or an analytic quasi-hyponormal operator on a complex Hilbert space and if f is a function analytic on a neighborhood of s T., then sw f T..sf sw T.., where s T. and sw T. stand respectively for the spectrum and the Weyl spectrum of T; moreover, Weyl’s theorem holds for f T.qF if ‘‘dominant’’ is replaced by ‘‘M-hyponormal,’’ where F is any finite rank operator commuting with T. These generalize earlier results for hyponormal operators. It is also shown that there exist an operator T and a finite rank operator F commuting with T such that Weyl’s theorem holds for T but not for TqF. This answers negatively a problem raised by K. K. Oberai Illinois J. Math. 21, 1977, 84]90.. However, if T is required to be isoloid, then the statement that Weyl’s theorem holds for T will imply it holds for TqF.