Invariant subspaces of compressions of the Hardy shift on some parametric spaces

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.