Invariant subspaces of compressions of the Hardy shift on some parametric spaces
Abstract
We study the class of operators Sα,β obtained by compressing the Hardy shift on the parametric spaces H2α, β corresponding to the pair \α,β\ satisfying |α|2+|β|2=1. We show, for nonzero α,β, each Sα,β is indeed a shift Mz on some analytic reproducing kernel Hilbert space and present a complete classification of their invariant subspaces. While all such invariant subspaces are cyclic, we show, unlike other classical shifts, they may not be generated by their corresponding wandering subspaces ( Sα,β). We provide a necessary and sufficient condition along this line and show, for a certain class of α, β, there exist Sα,β-invariant subspaces such that ≠ [ Sα,β]Sα,β.
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