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arXiv · 2405.09670

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

Susmita Das
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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 $\clm\neq [\clm\ominus S_{\alpha,\beta}\clm]_{S_{\alpha,\beta}}$.

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Susmita Das. 2024-05-15. Invariant subspaces of compressions of the Hardy shift on some parametric spaces. https://arxiv.org/abs/2405.09670

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