Invariant Subspaces and Nevanlinna–Pick Kernels

A theorem of Beurling–Lax–Halmos represents a subspace M of H2C(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 ΦH2F, where F is a subspace of E and Φ 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 Cd, the BLH Theorem survives with F an auxiliary Hilbert space and Φ a L(F, E) valued function which is inner in the sense that the operator MΦ of multiplication by Φ is a partial isometry. Under mild additional hypotheses, when E=C, Mz, the operator of multiplication by z, is cellularly indecomposable and has the codimension one property; however, if M is invariant for Mz, M⊖MzM need not be a cyclic subspace for Mz restricted to M.