A closed subspace of a vector-valued Hardy space is almost invariant under the backward shift exactly when it is almost invariant under the forward shift, with explicit Toeplitz-Hankel range forms.
O’Loughlin, Nearly invariant subspaces with applications to tru ncated Toeplitz operators, Com- plex Analysis and Operator Theory 14 (2020), no
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Almost invariant subspaces of shift operators and products of Toeplitz and Hankel operators
A closed subspace of a vector-valued Hardy space is almost invariant under the backward shift exactly when it is almost invariant under the forward shift, with explicit Toeplitz-Hankel range forms.