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 −1 B invariant subspaces for the operator T B of multiplication by B in a scale of Dirichlet-type spaces.
We give a new characterization for the boundedness of composition operator followed by differentiation operator acting on Bloch-type spaces and calculate its essential norm in terms of the n-th power of the induced analytic self-map on the unit disk. From which some sufficient and necessary conditions of compactness of the operator follow immediately.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.