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

Abstract

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 $|\alpha {|}^2+|\beta {|}^2=1$. 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 $\clm$ are cyclic, we show, unlike other classical shifts, they may not be generated by their corresponding wandering subspaces $(\clm\ominus S_{α,β}\clm)$. 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 $\clm$ such that $\clm\neq [\clm\ominus S_{α,β}\clm]_{S_{α,β}}$.