Hyperbolic geometry on noncommutative balls

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Popescu, Gelu
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Abstract
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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 $\omega_\rho$.
Comment: 3es9pag
Keywords
Mathematics - Functional Analysis, Mathematics - Operator Algebras, 46L52, 46T25, 47A20
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