Module tensor product of subnormal modules need not be subnormal

Let κ:D×D→C be a diagonal positive definite kernel and let Hκ denote the associated reproducing kernel Hilbert space of holomorphic functions on the open unit disc D. Assume that zf∈H whenever f∈H. Then H is a Hilbert module over the polynomial ring C[z] with module action p⋅f↦pf. We say that Hκ is a subnormal Hilbert module if the operator Mz of multiplication by the coordinate function z on Hκ is subnormal. N. Salinas (1988) [11] asked whether the module tensor product Hκ1⊗C[z]Hκ2 of subnormal Hilbert modules Hκ1 and Hκ2 is again subnormal. In this regard, we describe all subnormal module tensor products La2(D,ws1)⊗C[z]La2(D,ws2), where La2(D,ws) denotes the weighted Bergman Hilbert module with radial weightws(z)=1sπ|z|2(1−s)s(z∈D,s>0). In particular, the module tensor product La2(D,ws)⊗C[z]La2(D,ws) is never subnormal for any s≥6. Thus the answer to this question is no.