Study of nearly invariant subspaces with finite defect in Hilbert spaces.

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 H as a generalization of nearly T - 1 invariant subspaces introduced by Liang and Partington in Complex Anal. Oper. Theory15(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. Theory92 (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 D α for α ∈ [ - 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. Theory15(1) (2021) 17 pp). [ABSTRACT FROM AUTHOR]