Study of nearly invariant subspaces with finite defect in Hilbert spaces

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 Complex Anal. Oper. Theory 15(1) (2021) 17 pp. 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 Int. Equations Oper. Theory 92 (2020) 1–15. 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, as was done recently by Liang and Partington for defect zero case (see Section 3 of Complex Anal. Oper. Theory 15(1) (2021) 17 pp).