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Hyperbolic geometry on noncommutative balls. (English) Zbl 1186.47004

Summary: In this paper, we study the noncommutative balls
\[ \mathcal C_\rho:=\{(X_1,\dots, X_n)\in B(\mathcal H)^n: \;\omega_\rho(X_1,\dots, X_n)\leq 1\},\qquad \rho\in (0,\infty], \]
where \(\omega_\rho\) is the joint operator radius 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 introduce a Harnack type equivalence relation on \(\mathcal C_\rho\), \(\rho>0\), and use it to define a hyperbolic distance \(\delta_\rho\) on the Harnack parts (equivalence classes) of \(\mathcal C_\rho\). We prove that the open ball
\[ [\mathcal C_\rho]_{<1}:=\{(X_1,\dots, X_n)\in B(\mathcal H)^n: \omega_\rho(X_1,\dots, X_n)<1\},\qquad \rho>0, \]
is the Harnack part containing \(0\), and obtain a concrete formula for the hyperbolic distance in terms of the reconstruction operator associated with the right creation operators on the full Fock space with \(n\) generators. Moreover, we show that the \(\delta_\rho\)-topology and the usual operator norm topology coincide on \([\mathcal C_\rho]_{<1}\). While the open ball \([\mathcal C_\rho]_{<1}\) is not a complete metric space with respect to the operator norm topology, we prove that it is a complete metric space with respect to the hyperbolic metric \(\delta_\rho\). In the particular case when \(\rho=1\) and \(\mathcal H=\mathbb C\), the hyperbolic metric \(\delta_\rho\) coincides with the Poincaré-Bergman distance on the open unit ball of \(\mathbb C^n\). We introduce a Carathéodory type metric on \([\mathcal C_\infty]_{<1}\), the set of all \(n\)-tuples of operators with joint spectral radius strictly less then \(1\), by setting
\[ d_K(A,B)=\sup_p \|\operatorname{Re} p(A)-\mathfrak R p(B)\|,\qquad A,B\in [\mathcal C_\infty]_{<1}, \]
where the supremum is taken over all noncommutative polynomials with matrix-valued coefficients \(p\in \mathbb C[X_1,\dots, X_n]\otimes M_{m}\), \(m\in \mathbb N\), with \(\mathfrak R p(0)=I\) and \(\mathfrak R p(X)\geq 0\) for all \(X\in \mathcal C_1\). We obtain a concrete formula for \(d_K\) in terms of the free pluriharmonic kernel on the noncommutative ball \([\mathcal C_\infty]_{<1}\). We also prove that the metric \(d_K\) is complete on \([\mathcal C_\infty]_{<1}\) and its topology coincides with the operator norm topology. 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 Carathéodory metric \(d_K\), and the joint operator radius \(\omega_\rho\), \(\rho\in (0,\infty]\).

MSC:

47A13 Several-variable operator theory (spectral, Fredholm, etc.)
46L52 Noncommutative function spaces
46T25 Holomorphic maps in nonlinear functional analysis
47A20 Dilations, extensions, compressions of linear operators
32Q45 Hyperbolic and Kobayashi hyperbolic manifolds
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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