Hyperbolic geometry on the unit ball of B(H)n and dilation theory
Abstract
In this paper we continue our investigation concerning the hyperbolic geometry on the noncommutative ball [B(H)n]1-, where B(H) is the algebra of all bounded linear operators on a Hilbert space H, and its implications to noncommutative function theory. The central object is an intertwining operator LB,A of the minimal isometric dilations of A, B∈ [B(H)n]1-, which establishes a strong connection between noncommutative hyperbolic geometry on [B(H)n]1- and multivariable dilation theory. The goal of this paper is to study the operator LB,A and its connections to the hyperbolic metric δ on the Harnack parts of [B(H)n]1-. We study the geometric structure of the operator LB,A and obtain new characterizations for the Harnack domination (resp. equivalence) in [B(H)n]1-. We express \|LB,A\| in terms of the reconstruction operators RA and RB, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on [B(H)n]1 with respect to the intertwining operator LB,A. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.