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Multiplicities, invariant subspaces and an additive formula

Published online by Cambridge University Press:  30 April 2021

Arup Chattopadhyay
Affiliation:
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati781039, India (arupchatt@iitg.ac.in, 2003arupchattopadhyay@gmail.com)
Jaydeb Sarkar
Affiliation:
Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore560059, India (jay@isibang.ac.in, jaydeb@gmail.com)
Srijan Sarkar
Affiliation:
Department of Mathematics, Indian Institute of Science, Bangalore560012, India (srijans@iisc.ac.in, srijansarkar@gmail.com)

Abstract

Let $T = (T_1, \ldots , T_n)$ be a commuting tuple of bounded linear operators on a Hilbert space $\mathcal{H}$. The multiplicity of $T$ is the cardinality of a minimal generating set with respect to $T$. In this paper, we establish an additive formula for multiplicities of a class of commuting tuples of operators. A special case of the main result states the following: Let $n \geq 2$, and let $\mathcal{Q}_i$, $i = 1, \ldots , n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb {C}$. If $\mathcal{Q}_i^{\bot }$, $i = 1, \ldots , n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\textbf {z}} = (M_{z_1}, \ldots , M_{z_n})$-invariant subspace $(\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{\perp }$ of the Dirichlet space or the Hardy space over the unit polydisc in $\mathbb {C}^{n}$ is given by

\[ \mbox{mult}_{M_{\textbf{{z}}}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}}}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^{{\perp}} = \sum_{i=1}^{n} (\mbox{mult}_{M_z|_{\mathcal{Q}_i^{{\perp}}}} (\mathcal{Q}_i^{\bot})) = n. \]
A similar result holds for the Bergman space over the unit polydisc.

Type
Research Article
Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society

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Footnotes

Dedicated to Professor Kalyan Bidhan Sinha on the occasion of his 75th birthday.

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