Study of nearly invariant subspaces with finite defect in Hilbert spaces

TL;DR

This paper characterizes nearly inverse invariant subspaces with finite defect for shift operators in Hilbert spaces, linking them to backward shift invariant subspaces and providing concrete representations in Dirichlet-type spaces.

Contribution

It generalizes the concept of nearly inverse invariant subspaces with finite defect and offers a concrete representation in Dirichlet-type spaces for finite Blaschke products.

Findings

Characterization of nearly T^{-1} invariant subspaces with finite defect.

Connection to backward shift invariant subspaces in Hardy spaces.

Concrete representation in Dirichlet-type spaces for finite Blaschke products.

Abstract

In this article, we briefly describe nearly $T^{-1}$ invariant subspaces with finite defect for a shift operator $T$ having finite multiplicity acting on a separable Hilbert space $\mathcal{H}$ as a generalization of nearly $T^{-1}$ invariant subspaces introduced by Liang and Partington in \cite{YP}. In other words we characterize nearly $T^{-1}$ invariant subspaces with finite defect in terms of backward shift invariant subspaces in vector-valued Hardy spaces by using Theorem 3.5 in \cite{CDP}. Furthermore, we also provide a concrete representation of the nearly $T_B^{-1}$ invariant subspaces with finite defect in a scale of Dirichlet-type spaces $\mathcal{D}_\alpha$ for $\alpha \in [-1,1]$ corresponding to any finite Blashcke product $B$.