Let M be a Hilbert module of holomorphic functions defined on a bounded domain 0 C m . Let M 0 be the submodule of functions vanishing to order k on a hypersurface Z in 0. In this paper, we describe the quotient module M q .

Choose an arbitrary but fixed set of n × n matrices A1, . . . , Am and let ΩA ⊂ C m be the unit ball with respect to the norm · A, where (z1, . . . , zm) A = z1A1 + · · · + zmAm op . It is known that if m ≥ 3 and B is any ball in C m with respect to some norm, say · B , then there exists a contractive linear map L : (C m , · * B ) → M k which is not completely contractive. The characterization of those balls in C 2 for which contractive linear maps are always completely contractive thus remains open. We answer this question for balls of the form ΩA in C 2 .

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