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Invariant subspaces for bounded operators with large localizable spectrum

Author: Bebe Prunaru
Journal: Proc. Amer. Math. Soc. 129 (2001), 2365-2372
MSC (2000): Primary 47A15, 47B20; Secondary 47A11, 47L45
Published electronically: January 17, 2001
MathSciNet review: 1823920
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Suppose $H$ is a complex Hilbert space and $T\in L(H)$ is a bounded operator. For each closed set $F\subset \mathbf{C}$ let $H_{T}(F)$ denote the corresponding spectral manifold. Let $\sigma _{loc}(T)$denote the set of all points $\lambda \in \sigma (T)$ with the property that $H_{T}(\overline{V})\neq 0$ for any open neighborhood $V$ of $\lambda .$ In this paper we show that if $\sigma _{loc}(T)$ is dominating in some bounded open set, then $T$ has a nontrivial invariant subspace. As a corollary, every Hilbert space operator which is a quasiaffine transform of a subdecomposable operator with large spectrum has a nontrivial invariant subspace.

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Additional Information

Bebe Prunaru
Affiliation: Institute of Mathematics, Romanian Academy, P.O. Box 1-764, 70700 Bucharest, Romania

Keywords: Invariant subspaces, local spectral theory
Received by editor(s): December 14, 1999
Published electronically: January 17, 2001
Additional Notes: The author was partially supported by Grant 5232/1999 from ANSTI
Communicated by: Joseph A. Ball
Article copyright: © Copyright 2001 American Mathematical Society

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