Sie sind hier: Home :: Bereich NIMMT :: Mathematik

Jim Gleason, Stefan Richter, Carl Sundberg

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

Let be the open unit ball in ?^d, d ? 1, and H_d² be the space of analytic functions on determined by the reproducing kernel (1 ? ? z, ? ?)^?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 ? ?H_d²( ). The fiber dimension of ?? is defined to be . We show that if ?? has finite positive fiber dimension m, then the essential Taylor spectrum of M_z |? , ?_e(M_z |? ), equals ? plus possibly a subset of the zero set of a nonzero bounded analytic function on and ind(M_z ? ?) |? = (?1)^dm  for every ? ?  ?_e(M_z |? ). As a corollary we prove that if T = (T₁,?,T_d ) is a pure d-contraction of finite rank, then ?_e(T ) ?  is contained in the zero set of a nonzero bounded analytic function and (?1)^d ind(T ? ?) = ? (T ) for all ? ?  ?_e(T ). Here ?(T ) denotes Arveson’s curvature invariant. We will also show that for d > 1 there are such d-contractions with ?_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 ?^d.

Journal fur die reine und angewandte Mathematik (Crelles Journal), Walter de Gruyter

Print ISSN: 0075-4102
Volume: 2005, 10/2005
Seiten: 49 - 76

Zum Artikel (extern)

Alle verfügbaren Artikel dieser Zeitschrift anzeigen

 

Bereich NIMMT