Invariant subspaces and Nevanlinna-Pick kernels

A theorem of Beurling Lax Halmos represents a subspace M of H-E(2)(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 PhiH(F)(2), where F is a subspace of E and Phi 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 Phi a L(F, E) valued function which as inner in the sense that the operator M-Phi of multiplication by Phi 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. (C) 2000 Academic Press.