Free holomorphic functions on the unit ball of B(H)^n

Abstract

We develop a theory of holomorphic functions in several noncommuting (free) variables and thus provide a framework for the study of arbitrary n-tuples of operators. The main topics are the following: Free holomorphic functions and Hausdorff derivations; Cauchy, Liouville, and Schwartz type results for free holomorphic functions; Algebras of free holomorphic functions; Free analytic functional calculus and noncommutative Cauchy transforms; Weierstrass and Montel theorems for free holomorphic functions; Free pluriharmonic functions and noncommutative Poisson transforms; Hardy spaces of free holomorphic functions.

51 pages

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• Local theory of free noncommutative functions: germs, meromorphic functions and Hermite interpolation
• Hardy-space function theory, operator model theory, and dissipative linear systems: the multivariable, free-noncommutative, weighted Bergman-space setting
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