# Multiplicities, invariant subspaces and an additive formula

## TL;DR

This paper establishes an additive formula for the multiplicity of certain commuting operator tuples on Hilbert spaces, specifically for invariant subspaces of Dirichlet, Hardy, and Bergman spaces, revealing that the multiplicity equals the sum of individual multiplicities.

## Contribution

The paper introduces a new additive formula for the multiplicity of joint invariant subspaces of commuting operator tuples on classical function spaces, extending understanding of their structure.

## Key findings

- The multiplicity of the joint invariant subspace equals the sum of individual multiplicities.
- The formula applies to subspaces of Dirichlet, Hardy, and Bergman spaces over the unit polydisc.
- The result holds for zero-based shift invariant subspaces, with the sum equal to the space dimension n.

## 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 $M_{\boldsymbol{z}} = (M_{z_1}, \ldots, M_{z_n})$-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 \[ \mbox{mult}_{M_{\boldsymbol z}|_{ (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^\perp}} (\mathcal{Q}_1 \otimes \cdots \otimes \mathcal{Q}_n)^\perp = \sum_{i=1}^n (\mbox{mult}_{M_z|_{\mathcal{Q}_i^\perp}} (\mathcal{Q}_i^{\bot})) = n. \] A similar result holds for the Bergman space over the unit polydisc.