Invariant subspaces for bounded operators with large localizable spectrum

Suppose H is a complex Hilbert space and T is an element of L(H) is a bound ed operator. For each closed set F subset of C let H-T (F) denote the corre sponding spectral manifold. Let sigma (loc)(T) denote the set of all points lambda is an element of sigma (T) with the property that H-T ((V) over bar ) not equal = 0 for any open neighborhood V of lambda. In this paper we sho w 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 opera tor which is a quasiaffine transform of a subdecomposable operator with lar ge spectrum has a nontrivial invariant subspace.