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

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 $L_{B,A}$ 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 $\delta$ 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.