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arXiv · 2201.09226

On analytic structure of weighted shifts on generalized directed semi-trees

Gargi Ghosh·Somnath Hazra
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Abstract

Inspired by natural classes of examples, we define generalized directed semi-tree and construct weighted shifts on the generalized directed semi-trees. Given an $n$-tuple of directed directed semi-trees with certain properties, we associate an $n$-tuple of multiplication operators on a Hilbert space $\mathscr{H}^2(\beta)$ of formal power series. Under certain conditions, $\mathscr{H}^2(\beta)$ turns out to be a reproducing kernel Hilbert space consisting of holomorphic functions on some domain in $\mathbb C^n$ and the $n$-tuple of multiplication operators on $\mathscr{H}^2(\beta)$ is unitarily equivalent to an $n$-tuple of weighted shifts on the generalized directed semi-trees. Finally, we exhibit two classes of examples of $n$-tuple of operators which can be intrinsically identified as weighted shifts on generalized directed semi-trees.

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Gargi Ghosh, Somnath Hazra. 2022-01-23. On analytic structure of weighted shifts on generalized directed semi-trees. https://arxiv.org/abs/2201.09226

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