An additive formula for multiplicities on reproducing kernel Hilbert spaces

Arup Chattopadhyay, Jaydeb Sarkar, Srijan Sarkar

Published 2018-12-13Version 1

In this paper, we compute the exact rank of a non-trivial co-doubly commuting submodule of analytic reproducing kernel Hilbert modules over $\mathbb{C}[z_1,\ldots,z_n]$. More precisely, let $\mathcal{H} = \mathcal{H}_{1} \otimes \ldots \otimes \mathcal{H}_{n}$ be an analytic reproducing kernel Hilbert module over $\mathbb{C}[z_1,\ldots,z_n]$. Let $\mathcal{S}$ be a co-doubly commuting submodule of $\mathcal{H}$, that is , \[ \mathcal{S} = (\mathcal{Q}_1 \otimes \ldots \otimes \mathcal{Q}_n)^{\bot}, \] where $\mathcal{Q}_i$ are quotient modules of $\mathcal{H}_{i}$. Then our result states that \[ \mbox{rank}~\Big(\big(\mathcal{Q}_1\otimes \cdots \otimes \mathcal{Q}_n\big)^{\perp}\Big) = \sum_{i=1}^n \mbox{rank}~\big(\mathcal{Q}_i^{\bot}). \] This immediately answers an open question in the special case where $\mathcal{H}_{i} = H^2(\mathbb{D})$ for all $i \in \lbrace 1,\ldots,n \rbrace$ posed in [3]. As an immediate consequence we have the following observation: Let $\mathcal{S}$ be a co-doubly commuting submodule in $H^2(\mathbb{D}^n)$. Then for $m \leq n$, rank ($\mathcal{S})=m$, implies $\mathcal{S}=\Theta H^2(\mathbb{D}^n)$ for some $n-m$ variables inner function $\Theta\in H^{\infty}(\mathbb{D}^{n-m})$. In particular, for $n=2$ and $m=1$ it positively meets the observations made by R. G. Douglas and R. Yang in [7].