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 Hilbert space 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.
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