Module tensor product of subnormal modules need not be subnormal

摘要

Let k : D x D -> be a diagonal positive definite kernel and let H-k denote the associated reproducing kernel Hilbert space of holomorphic functions on the open unit disc D. Assume that zf is an element of H whenever f is an element of H. Then H is a Hilbert module over the polynomial ring C[z] with module action p . f bar right arrow pf. We say that H-k is a subnormal Hilbert module if the operator M-z of multiplication by the coordinate function z on H-k is subnormal. N. Salinas (1988) [11] asked whether the module tensor product H-k1 circle times c[z] H-k2 of subnormal Hilbert modules H-k1 and H-k2 is again subnormal. In this regard, we describe all subnormal module tensor products L-a(2)(D, w(s1)) circle times c[z] L-a(2)(D, w(s2)),where L-a(2)(D, w(s)) denotes the weighted Bergman Hilbert module with radial weight