On the index of invariant subspaces in spaces of analytic functions of several complex variables

Abstract

Let be the open unit ball in ℂ^d, d ≧ 1, and H_d² be the space of analytic functions on determined by the reproducing kernel (1 − 〈 z, λ 〉)⁻¹. This reproducing kernel Hilbert space serves a universal role in the model theory for d -contractions, i.e. tuples T = (T₁,…,T_d ) of commuting operators on a Hilbert space such that ||T₁x₁ + ⋯ + T_d x_d ||² ≦ ||x₁||² + ⋯ + ||x_d ||² for all x₁, … ,x_d ∈ . If  is a separable Hilbert space then we write H_d²( ) ≅ H_d² ⊗  for the space of -valued H_d² functions and we use M_z = (,…, ) to denote the tuple of multiplication by the coordinate functions. We consider M_z-invariant subspaces ℳ ⊆H_d²( ). The fiber dimension of ℳ  is defined to be . We show that if ℳ  has finite positive fiber dimension m, then the essential Taylor spectrum of M_z |ℳ , σ_e(M_z |ℳ ), equals ∂ plus possibly a subset of the zero set of a nonzero bounded analytic function on and ind(M_z − λ) |ℳ = (−1)^dm  for every λ ∈  \σ_e(M_z |ℳ ). As a corollary we prove that if T = (T₁,…,T_d ) is a pure d-contraction of finite rank, then σ_e(T ) ∩  is contained in the zero set of a nonzero bounded analytic function and (−1)^d ind(T − λ) = κ (T ) for all λ ∈  \σ_e(T ). Here κ(T ) denotes Arveson’s curvature invariant. We will also show that for d > 1 there are such d-contractions with σ_e(T ) ∩  ≠ ∅. These results answer a question of Arveson, [William Arveson, The Dirac operator of a commuting d-tuple, J. Funct. Anal. 189(1) (2002), 53–79]. We also prove related results for the Hardy and Bergman spaces of the unit ball and unit polydisc of ℂ^d.