Free holomorphic functions on the unit ball of \(B(\mathcal H)^n\). II. (English) Zbl 1197.47014
The author continues his study of free holomorphic functions on the unit ball in \(B(\mathcal H)^n\), initiated in Part I [J. Funct. Anal. 241, No. 1, 268–333 (2006; Zbl 1112.47004)]. Specifically, if \(B(\mathcal H)\) is the algebra of all bounded linear operators on the Hilbert space \(\mathcal H\), the author considers free holomorphic functions, defined on the set \([B(\mathcal H)^n]_1\), consisting of those \(n\)-tuples \((X_1,\dots,X_n)\) of operators satisfying \(\| X_1X_1^*+\cdots+X_nX_n\|^{1/2}<1\).
Roughly speaking, a free holomorphic function on \([B(\mathcal H)^n]_1\) is a map of the form \((X_1,\dots,X_n)\mapsto\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha X_\alpha\), where the series is convergent in the operator norm topology, the indices \(\alpha\) are elements of the unital free group of \(n\) generators \(g_1,\dots ,g_n\), \(|\alpha|\) is the length of \(\alpha\), and \(X_\alpha=X_{i_1}\cdots X_{i_k}\) if \(\alpha=g_{i_1}\cdots g_{i_k}\).
Several classical results from complex analysis are recaptured in this setting. We mention a maximum principle, a Naimark type representation theorem, and a Vitaly convergence theorem. For a special class of free holomorphic functions, inequalities of Schwarz and Harnack type are obtained. A noncommutative generalization of an inequality due to Lindelöf turns out to be sharper than the noncommutative von Neumann inequality. Finally, by introducing a pseudohyperbolic metric on \([B(\mathcal H)^n]_1\), a Schwarz-Pick type lemma is obtained.
Roughly speaking, a free holomorphic function on \([B(\mathcal H)^n]_1\) is a map of the form \((X_1,\dots,X_n)\mapsto\sum_{k=0}^\infty\sum_{|\alpha|=k} a_\alpha X_\alpha\), where the series is convergent in the operator norm topology, the indices \(\alpha\) are elements of the unital free group of \(n\) generators \(g_1,\dots ,g_n\), \(|\alpha|\) is the length of \(\alpha\), and \(X_\alpha=X_{i_1}\cdots X_{i_k}\) if \(\alpha=g_{i_1}\cdots g_{i_k}\).
Several classical results from complex analysis are recaptured in this setting. We mention a maximum principle, a Naimark type representation theorem, and a Vitaly convergence theorem. For a special class of free holomorphic functions, inequalities of Schwarz and Harnack type are obtained. A noncommutative generalization of an inequality due to Lindelöf turns out to be sharper than the noncommutative von Neumann inequality. Finally, by introducing a pseudohyperbolic metric on \([B(\mathcal H)^n]_1\), a Schwarz-Pick type lemma is obtained.
MSC:
| 47A13 | Several-variable operator theory (spectral, Fredholm, etc.) |
| 46L52 | Noncommutative function spaces |
Keywords:
multivariable operator theory; noncommutative function theory; free holomorphic functions; Fock space; Poisson transform; Schwarz’s lemma; Julia’s lemma; Pick’s theorem; pseudohyperbolic metric; Borel-Carathódory theorem; von Neumann inequalityCitations:
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