Multiplicities, invariant subspaces and an additive formula

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.