Invariant subspaces for bounded operators with large localizable spectrum

Abstract:

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.