Invariant subspaces for operators whose spectra are Caratheodory regions

In this paper it is shown that if an operator T satisfies parallel to p(T)parallel to <= parallel to p parallel to(sigma(T)) for every polynomial p and the polynomially convex hull of sigma(T) is a Caratheodory region whose accessible boundary points lie in rectifiable Jordan arcs on its boundary, then T has a nontrivial invariant subspace. As a corollary, it is also shown that if T is a hyponormal operator and the outer boundary of sigma(T) has at most finitely many prime ends corresponding to singular points on partial derivative D and has a tangent at almost every point on each Jordan arc, then T has a nontrivial invariant subspace. (C) 2010 Elsevier Inc. All rights reserved.