Certain invariant subspaces for operators with rich eigenvalues

Abstract

For a connected open subset Ω of the plane and n a positive integer, let Bn(Ω) be the space introduced by Cowen and Douglas. In this article we study the spectrum of restrictions of T in order to obtain more information about the invariant subspaces of T. When n=1 and T ϵ B1(Ω) such that σ(T)=Ω¯ is a spectral set for T we use the functional calculus we have developed for such operators to give some infinite dimensional cyclic invariant subspaces for T.