2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/142331We 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 pagesFunctional AnalysisOperator Algebras47A56; 47A13; 47A60; 46T25; 46E15text