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Multiplicities, invariant subspaces and an additive formula

Published online by Cambridge University Press:  30 April 2021

Arup Chattopadhyay
Department of Mathematics, Indian Institute of Technology Guwahati, Guwahati781039, India (,
Jaydeb Sarkar
Statistics and Mathematics Unit, Indian Statistical Institute, 8th Mile, Mysore Road, Bangalore560059, India (,
Srijan Sarkar
Department of Mathematics, Indian Institute of Science, Bangalore560012, India (,


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.

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

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


Aleman, A., Richter, S. and Sundberg, C., Beurling's theorem for the Bergman space, Acta Math. 177 (1996), 275310.10.1007/BF02392623CrossRefGoogle Scholar
Apostol, C., Bercovici, H., Foias, C. and Pearcy, C., Invariant subspaces, dilation theory, and the structure of the predual of a dual algebra I, J. Funct. Anal. 63 (1985), 369404.10.1016/0022-1236(85)90093-XCrossRefGoogle Scholar
Beurling, A., On two problems concerning linear transformations in Hilbert space, Acta Math. 81 (1949), 239255.CrossRefGoogle Scholar
Chattopadhyay, A., Das, B. K. and Sarkar, J., Star-generating vectors of Rudin's quotient modules, J. Funct. Anal. 267 (2014), 43414360.10.1016/j.jfa.2014.09.024CrossRefGoogle Scholar
Chattopadhyay, A., Das, B. K. and Sarkar, J., Rank of a co-doubly commuting submodule is 2, Proc. Amer. Math. Soc. 146 (2018), 11811187.10.1090/proc/13792CrossRefGoogle Scholar
Douglas, R. and Yang, R., Operator theory in the Hardy space over the bidisk (I), Integral Equations Operator Theory 38(2) (2000), 207221.10.1007/BF01200124CrossRefGoogle Scholar
Fang, X., Additive invariants on the Hardy space over the polydisc, J. Funct. Anal. 253 (2007), 359372.10.1016/j.jfa.2007.08.011CrossRefGoogle Scholar
Hedenmalm, H., An invariant subspace of the Bergman space having the codimension two property, J. Reine Angew. Math. 443 (1993), 19.Google Scholar
Hedenmalm, H., Korenblum, B. and Zhu, K., Theory of Bergman spaces, Graduate Texts in Mathematics, Volume 199 (Springer-Verlag, New York, 2000).10.1007/978-1-4612-0497-8CrossRefGoogle Scholar
Hedenmalm, H., Richter, S. and Seip, K., Interpolating sequences and invariant subspaces of given index in the Bergman spaces, J. Reine Angew. Math. 477 (1996), 1330.Google Scholar
Izuchi, K. J., Izuchi, K. H. and Izuchi, Y., Blaschke products and the rank of backward shift invariant subspaces over the bidisk, J. Funct. Anal. 261(6) (2011), 14571468.10.1016/j.jfa.2011.05.009CrossRefGoogle Scholar
Izuchi, K. J., Izuchi, K. H. and Izuchi, Y., Ranks of invariant subspaces of the Hardy space over the bidisk, J. Reine Angew. Math. 659 (2011), 101139.Google Scholar
Izuchi, K. J., Izuchi, K. H. and Izuchi, Y., Ranks of backward shift invariant subspaces of the Hardy space over the bidisk, Math. Z. 274 (2013), 885903.10.1007/s00209-012-1100-2CrossRefGoogle Scholar
Popescu, G., Euler characteristic on noncommutative polyballs, J. Reine Angew. Math. 728 (2017), 195236.Google Scholar
Popescu, G., Invariant subspaces and operator model theory on noncommutative varieties, Math. Ann. 372 (2018), 611650.CrossRefGoogle Scholar
Richter, S., Invariant subspaces of the Dirichlet shift, J. Reine Angew. Math. 386 (1988), 205220.Google Scholar
Rudin, W., Function theory in polydiscs (New York, Benjamin, 1969).Google Scholar
Sarkar, J., Jordan blocks of $H^{2}(\mathbb {D}^{n})$, J. Operator Theory 72 (2014), 371385.10.7900/jot.2013mar16.1980CrossRefGoogle Scholar
Tomerlin, A., Products of Nevanlinna-Pick kernels and operator colligations, Integr. Equ. Oper. Theory 38(3) (2000), 350356.10.1007/BF01291719CrossRefGoogle Scholar