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 d 2 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 = ( T 1 ,…, T d ) of commuting operators on a Hilbert space such that || T 1 x 1 + ⋯ + T d x d || 2 ≦ || x 1 || 2 + ⋯ + || x d || 2 for all x 1 , … , x d ∈ . If is a separable Hilbert space then we write H d 2 ( ) ≅ H d 2 ⊗ for the space of -valued H d 2 functions and we use M z = (,…, ) to denote the tuple of multiplication by the coordinate functions. We consider M z -invariant subspaces ℳ ⊆ H d 2 ( ). 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) d m for every λ ∈ \ σ e ( M z |ℳ ). As a corollary we prove that if T = ( T 1 ,…, 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 .