Hyperbolic geometry on noncommutative balls

Details Hyperbolic geometry on noncommutative balls Hyperbolic geometry on noncommutative balls

2009-11-29

arxiv.org/abs/0911.5489v1

Documenta Math. 14 (2009), 595--651

Mathematics

Functional Analysis

3es9pag

Scientific paper

In this paper, we study the hyperbolic geometry of noncommutative balls generated by the joint operator radius $\omega_\rho$, $\rho\in (0,\infty]$, 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 $\omega_\infty$ 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 $\omega_\rho$.

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