Invariant subspaces of compressions of the Hardy shift on some parametric spaces
Volume 287 / 2026
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 $, that 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 $\mathcal {M}$ are cyclic, we show that, unlike other classical shifts, they may not be generated by their corresponding wandering subspaces $\mathcal {M}\ominus S_{\alpha ,\beta }\mathcal {M}$. We provide a necessary and sufficient condition along this line and show that, for a certain class of $\alpha , \beta $, there exist $S_{\alpha ,\beta }$-invariant subspaces $\mathcal {M}$ such that $\mathcal {M}\neq [\mathcal {M}\ominus S_{\alpha ,\beta }\mathcal {M}]_{S_{\alpha ,\beta }}$.
