Abstract :
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.
