Hyperbolic geometry on the unit ball of $B(H)^n$ and dilation theory

Simple item page
dc.creatorPopescu, Gelu
dc.date2008-10-03
dc.date.accessioned2026-07-07T10:07:29Z
dc.date.available2026-07-07T10:07:29Z
dc.descriptionIn 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 $L_{B,A}$ of the minimal isometric dilations of $A, B\in [B(H)^n]_1^-$, 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 $L_{B,A}$ 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 $L_{B,A}$ and obtain new characterizations for the Harnack domination (resp. equivalence) in $[B(H)^n]_1^-$. We express $\|L_{B,A}\|$ in terms of the reconstruction operators $R_A$ and $R_B$, and obtain a Schwartz-Pick lemma for contractive free holomorphic functions on $[B(H)^n]_1$ with respect to the intertwining operator $L_{B,A}$. As a consequence, we deduce a Schwartz-Pick lemma for operator-valued multipliers of the Drury-Arveson space, with respect to the hyperbolic metric.
dc.description26 pages, to appear in Indiana Univ. Math. J
dc.identifierhttps://arxiv.org/abs/0810.0656
dc.identifierhttp://arxiv.org/abs/0810.0656
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/170656
dc.subjectFunctional Analysis
dc.subjectOperator Algebras
dc.subject46L52; 32F45; 47A20; 47A56; 32Q45
dc.titleHyperbolic geometry on the unit ball of $B(H)^n$ and dilation theory
dc.typetext

Files

Collections

arXiv