Hyperbolic geometry on noncommutative balls

In this paper, we study the hyperbolic geometry of noncommutative balls generated by the joint operator radius ${\omega }_{\rho }$, $\rho \in (0,\mathrm{\infty }]$, for $n$-tuples of bounded linear operators on a Hilbert space. In particular, ${\omega }_1$ is the operator norm, ${\omega }_2$ is the joint numerical radius, and ${\omega }_{\mathrm{\infty }}$ is the joint spectral radius. We provide mapping theorems, von Neumann inequalities, and Schwarz type lemmas for free holomorphic functions on noncommutative balls, with respect to the hyperbolic metric ${\delta }_{\rho }$, the Carath' eodory metric $d_K$, and the joint operator radius ${\omega }_{\rho }$.