Nearly invariant subspaces for operators in Hilbert spaces

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 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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