Module tensor product of subnormal modules need not be subnormal

TL;DR

This paper investigates whether the tensor product of subnormal Hilbert modules remains subnormal, providing specific examples with weighted Bergman modules and showing that in many cases, the tensor product is not subnormal.

Contribution

It characterizes subnormal module tensor products of weighted Bergman Hilbert modules and demonstrates that such tensor products are not necessarily subnormal, answering a question posed by Salinas.

Findings

Tensor product of certain weighted Bergman modules is not subnormal for s ≥ 6.

Provides explicit characterization of subnormal tensor products.

Answers negatively to Salinas's question on subnormality preservation.

Abstract

Let $\kappa : \mathbb D \times \mathbb D \to \mathbb C$ be a diagonal positive definite kernel and let $\mathscr H_{\kappa}$ denote the associated reproducing kernel Hilbert space of holomorphic functions on the open unit disc $\mathbb D$. Assume that $zf \in \mathscr H$ whenever $f \in \mathscr H.$ Then $\mathscr H$ is a Hilbert module over the polynomial ring $\mathbb C[z]$ with module action $p \cdot f \mapsto pf$. We say that $\mathscr H_{\kappa}$ is a subnormal Hilbert module if the operator $\mathscr M_{z}$ of multiplication by the coordinate function $z$ on $\mathscr H_{\kappa}$ is subnormal. %If $\kappa_1$ and $\kappa_2$ are two diagonal positive definite kernels then so is their pointwise (tensor) product $\kappa:=\kappa_1\kappa_2$. In [Oper. Theory Adv. Appl, 32: 219-241, 1988], N. Salinas asked whether the module tensor product $\mathscr H_{\kappa_1} \otimes_{\mathbb C[z]} \mathscr H_{\kappa_2}$ of subnormal Hilbert modules $\mathscr H_{\kappa_1}$ and $\mathscr H_{\kappa_2}$ is again subnormal. In this regard, we describe all subnormal module tensor products $L^2_a(\mathbb D, w_{s_1}) \otimes_{\mathbb C[z]} L^2_a(\mathbb D, w_{s_2})$, where $L^2_a(\mathbb D, w_s)$ denotes the weighted Bergman Hilbert module with radial weight $$w_s(z)=\frac{1}{s \pi}|z|^{\frac{2(1-s)}{s}}~(z \in \mathbb D, ~s > 0).$$ In particular, the module tensor product $L^2_a(\mathbb D, w_{s}) \otimes_{\mathbb C[z]} L^2_a(\mathbb D, w_{s})$ is never subnormal for any $s \geq 6$. Thus the answer to this question is no.