Multiplicities, invariant subspaces and an additive formula

Abstract

Let $T=(T_1,\dots ,T_n)$ 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 $n\ge 2$, and let ${\mathcal{Q}}_i$, $i=1,\dots ,n$, be a proper closed shift co-invariant subspaces of the Dirichlet space or the Hardy space over the unit disc in $\mathbb{C}$. If ${\mathcal{Q}}_i^{\perp }$, $i=1,\dots ,n$, is a zero-based shift invariant subspace, then the multiplicity of the joint $M_{\mathbf{z}}=(M_{z_1},\dots ,M_{z_n})$-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.

arXiv commentModified and corrected version