Nearly invariant subspaces for operators in Hilbert spaces

Abstract

For a shift operator $T$ with finite multiplicity acting on a separable infinite dimensional Hilbert space we represent its nearly $T^{-1}$ invariant subspaces in terms of invariant subspaces under the backward shift. Going further, given any finite Blaschke product $B$, we give a description of the nearly $T_{B}^{-1}$ invariant subspaces for the operator $T_B$ of multiplication by $B$ in a scale of Dirichlet-type spaces.