Module tensor product of subnormal modules need not be subnormal

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.