ON THE WEYL-SPECTRUM - SPECTRAL MAPPING-THEOREM AND WEYLS-THEOREM

It is shown that if T is a dominant operator or an analytic quasi-hypo normal operator on a complex Hilbert space and if f is a function anal ytic on a neighborhood of sigma(T), then sigma(W)(f(T)) =f(sigma(W)(T) ), where sigma(T) and sigma(W)(T) stand respectively for the spectrum and the Weyl spectrum of T; moreover, Weyl's theorem holds for f(T) F if ''dominant'' is replaced by ''M-hyponormal,'' where F is any fini te rank operator commuting with T. These generalize earlier results fo r 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 theor em holds for T but not for T + F. This answers negatively a problem ra ised 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 ho lds for T will imply it holds for T + F. (C) 1998 Academic Press.