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A theorem of Beurling Lax Halmos represents a subspace M of H 2 C (D) the Hardy space of analytic functions with values in the Hilbert space E and square summable power series invariant for multiplication by z as 8H 2 F , where F is a subspace of E and 8 is an inner function with values in L(F, E). When the Hardy space is replaced by the Hilbert space H(k) determined by a Nevanlinna Pick kernel k, such as the Dirichlet kernel or the row contraction kernel on the ball in C d , the BLH Theorem survives with F an auxiliary Hilbert space and 8 a L(F, E) valued function which is inner in the sense that the operator M 8 of multiplication by 8 is a partial isometry. Under mild additional hypotheses, when E=C, M z , the operator of multiplication by z, is cellularly indecomposable and has the codimension one property; however, if M is invariant for M z , M M z M need not be a cyclic subspace for M z restricted to M. Academic Press

Sarason has remarked that several interpolation theorems for analytic functions defined in the unit disc can be unified in an operatorial frame in Hilbert space; this line of research has lead to the commutant lifting theorem of Sz-Nagy and Foias, which is a basic result in the theory of contractions on a Hilbert space. In the present paper we obtain a generalized interpolation theorem for analytic (operator-valued) functions defined in the polydisc, analogous to Sarason's one variable result. The main tool is a representation theorem due to Agler.

Abstract. The main purpose of this paper is to extend and refine some work of Agler-McCarthy and Amar concerning the Corona problem for the polydisk and the unit ball in C n .