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

We study the class of operators $S_{\alpha ,\beta }$ obtained by compressing the Hardy shift on the parametric spaces $H_{\alpha ,\beta }^2$ corresponding to the pair $\{\alpha ,\beta \}$ satisfying $\mathrm{\mid }\alpha {\mathrm{\mid }}^2+\mathrm{\mid }\beta {\mathrm{\mid }}^2=1$. We show, for nonzero $\alpha ,\beta$, each $S_{\alpha ,\beta }$ is indeed a shift $M_z$ on some analytic reproducing kernel Hilbert space and present a complete classification of their invariant subspaces. While all such invariant subspaces $\text{clm}$ are cyclic, we show, unlike other classical shifts, they may not be generated by their corresponding wandering subspaces $(\text{clm}\ominus S_{\alpha ,\beta }\text{clm})$. We provide a necessary and sufficient condition along this line and show, for a certain class of $\alpha ,\beta$, there exist $S_{\alpha ,\beta }$-invariant subspaces $\text{clm}$ such that $\text{clm}\mathrm{\ne }[\text{clm}\ominus S_{\alpha ,\beta }\text{clm}{]}_{S_{\alpha ,\beta }}$.