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Free holomorphic functions on the unit ball of \(B(\mathbb H)^n\). (English) Zbl 1112.47004

The aim of the paper under review is to develop a theory of holomorphic functions in several noncommuting free variables and to apply this theory to the study of tuples of Hilbert space bounded operators. A free analytic functional calculus is introduced and noncommutative Cauchy and Poisson transform, as well as Hausdorff derivations and von Neumann type inequalities are studied in this framework. In fact, several classical results, as for instance the Weierstrass and Montel theorems as well as the concept of a Hardy spaces, have free analogues in this noncommutative setting.

MSC:

47A13 Several-variable operator theory (spectral, Fredholm, etc.)
47A60 Functional calculus for linear operators
47L10 Algebras of operators on Banach spaces and other topological linear spaces
46L54 Free probability and free operator algebras
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