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 $B$.
Categories: math.FA