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

TL;DR

This paper classifies invariant subspaces of certain compressed Hardy shift operators on parametric spaces, revealing unique properties about their cyclicity and generation by wandering subspaces.

Contribution

It provides a complete classification of invariant subspaces for these operators and identifies conditions where they are not generated by wandering subspaces.

Findings

All invariant subspaces are cyclic.

Invariant subspaces may not be generated by wandering subspaces.

Existence of invariant subspaces not generated by their wandering subspaces.

Abstract

We study the class of operators $S_{\alpha,\beta}$ obtained by compressing the Hardy shift on the parametric spaces $H^2_{\alpha, \beta}$ 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 $(\clm\ominus S_{\alpha,\beta}\clm)$. 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 $\clm\neq [\clm\ominus S_{\alpha,\beta}\clm]_{S_{\alpha,\beta}}$.