Weyl spectra and Weyl's theorem

Young Min Han¹ ; Woo Young Lee²

¹ Department of Mathematics Sungkyunkwan University Suwon 440-746, South Korea
² Department of Mathematics Sungkyunkwan University Suwon 440-746 South Korea

Studia Mathematica, Tome 148 (2001) no. 3, pp. 193-206

Résumé

“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 $T$.

Détail

DOI : 10.4064/sm148-3-1

Keywords: weyls theorem operator hilbert space statement complement spectrum weyl spectrum coincides isolated eigenvalues finite multiplicity paper consider weyls theorem survives polynomials operators under quasinilpotent compact perturbations first reduced each its finite dimensional eigenspaces weyl spectrum obeys spectral mapping theorem further reduction isoloid every polynomial weyls theorem holds results perturbations follows finite isoloid operator commutes either compact quasinilpotent weyls theorem transmitted noncommutative perturbation theorem spectrum has holes finitely many isolated points compact operator weyls theorem holds holds