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

We study the class of operators S, obtained by compressing the Hardy shift on the parametric spaces H², corresponding to the pair \, \ satisfying ||²+||²=1. We show, for nonzero, , each S, is indeed a shift Mᵦ 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 (S, ). We provide a necessary and sufficient condition along this line and show, for a certain class of, , there exist S, -invariant subspaces such that S, ₒ_,.