Weighted Frames and the Douglas-Foias problem

Abstract

Let $\mathbf{T}=(T_1,\ldots,T_n)$ be a commuting tuple of bounded linear operators on a separable Hilbert space $\mathcal{K}$, and let $\{a_α\}_{α\in\mathbb{Z}_+^n}$ be strictly positive weights. Under mild assumptions on the weights, we characterize the weighted frame vectors and weighted Parseval frame vectors, that is, those vectors $ξ$ for which $\{a_α\mathbf{T}^αξ\}_{α\in\mathbb{Z}_+^n}$ is a frame or a Parseval frame. We reduce the problem to characterizing surjective module maps from the reproducing analytic Hilbert module associated with the weights onto the module $\mathcal{K}$, equipped with the module structure defined by $\mathbf{T}$. It is shown that weighted Parseval frame vectors differ by a unimodular scalar whenever they exist. If the kernel defined by the weights has the complete Nevanlinna--Pick property, then any two associated weighted frame vectors are similar whenever they exist. We also focus on the case of the Hardy kernel on the polydisc, that is, the case $a_α=1$ for all $α\in\mathbb{Z}^n_+$. This is related to the famous Douglas--Foias problem: whether two similar quotient modules of the Hardy spaces $H^2(\mathbb{D}^n)~(n>1)$ are equal. Our result shows that the answer is negative. This leads to the conclusion that frame vectors for representations of $\mathbb{Z}_+^n$ $(n>1)$ may fail to be similar. It means that, for the case $a_α=1$, there exists a tuple $\mathbf{T}=(T_1,\dots,T_n)$ with $n>1$ on $\mathcal{K}$ and two distinct frame vectors for $\mathbf{T}$ which are not similar.