Maximal invariant subspaces for a class of operators

Abstract

In this note, we characterize maximal invariant subspaces for a class of operators. Let T be a Fredholm operator and $1-TT^{*}\in\mathcal{S}_{p}$ for some p≥1. It is shown that if M is an invariant subspace for T such that dim M⊖TM<∞, then every maximal invariant subspace of M is of codimension 1 in M. As an immediate consequence, we obtain that if M is a shift invariant subspace of the Bergman space and dim M⊖zM<∞, then every maximal invariant subspace of M is of codimension 1 in M. We also apply the result to translation operators and their invariant subspaces.