Weyl spectra and Weyl's theorem
“Weyl's theorem” for an operator on a Hilbert space is the statement that the complement in the spectrum of the Weyl spectrum coincides with the isolated eigenvalues of finite multiplicity. In this paper we consider how Weyl's theorem survives for polynomials of operators and under quasinilpotent or compact perturbations. First, we show that if $T$ is reduced by each of its finite-dimensional eigenspaces then the Weyl spectrum obeys the spectral mapping theorem, and further if $T$ is reduction-isoloid then for every polynomial $p$, Weyl's theorem holds for $p(T)$. The results on perturbations are as follows. If $T$ is a “finite-isoloid” operator and if $K$ commutes with $T$ and is either compact or quasinilpotent then Weyl's theorem is transmitted from $T$ to $T+K$. As a noncommutative perturbation theorem, we also show that if the spectrum of $T$ has no holes and at most finitely many isolated points, and if $K$ is a compact operator then Weyl's theorem holds for $T+K$ when it holds for $T$.