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

Let k be the reproducing kernel for a Hilbert space H(k) of analytic functions on Bd, the open unit ball in Cd, 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 (J. Funct. Anal.178 (2000), 226–249) partially extends the Beurling–Lax–Halmos theorem for the invariant subspaces of the Hardy space H2(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 a 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 orthogonal 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 rank φ(z) is equal to a constant almost everywhere. The result applies to certain weighted Dirichlet spaces and to the space H2d, which is determined by the kernel kλ(z)=11−〈z,λ〉d ,λ,z∈Bd. In particular, our result implies that the curvature invariant of Arveson (Proc. Natl. Acad. Sci. USA96 (1999), 11,096–11,099) of a pure contractive Hilbert module of finite rank is an integer. This answers a question of W. Arveson (Proc. Natl Acad. Sci. USA96 (1999), 11096–11099).