The Structure of Inner Multipliers on Spaces with Complete Nevanlinna Pick Kernels

Let k be the reporducing kernel for a Hilbert space H(k) of nanlytic functions on Bd, the open unit ball in C, d ≥ 1. k is called a complete NP kernel, if k0 ≡ 1 and if 1 − 1/kλ(z) is positive definite on Bd × Bd. Let D be a separable Hilbert space, and consider H(k) ⊗ D ∼= H(k,D), and think of it as a space of D-valued H(k)-functions. A theorem of McCullough and Trent, [10], partially extends the Beurling-Lax-Halmos theorem for the invariant subspaces of the Hardy spaceH2(D). They show that if k is a complete NP kernel and if D is a separable Hilbert space, then for any scalar multiplier invariant subspace M of H(k,D) there exists an auxiliary Hilbert space E and as multiplication operator Φ : H(k, E) −→ H(k,D) such that Φ is a partial isometry and M = ΦH(k, E). Such multiplication operators are called inner multiplication operators and they satisfy ΦΦ∗ = the projection onto M. In this paper we shall show that for many interesting complete NP kernels the analogy with the Beurling-Lax-Halmos theorem can be strengthened. We show that for almost every z ∈ Bd the nontangential limit φ(z) of the multiplier φ : Bd −→ B(E ,D) associated with an inner multiplication operator Φ is a partial isometry and that rankφ(z) is equal to a constant almost everywhere. The result applies to certain weighted Dirichlet spaces and to the symmetric Fock spaceH d . In particular, our result implies that the curvature invariant of W. Arveson ([5]) of a pure contractive Hilbert module of finite rank is an integer. The answers a question of W. Arveson, [5].