2024/10/08 by Fabrizio Colombo, Colombo, Fabrizio, Antonino De Martino +7
Mathematics · #Advanced Operator Algebra Research #Advanced Algebra and Geometry #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.2410.07251
Holomorphic functions are fundamental in operator theory and their Cauchy formula is a crucial tool for defining functions of operators. The Fueter-Sce extension theorem (often called Fueter-Sce mapping theorem) provides a two-step procedure for extending holomorphic functions to hyperholomorphic functions. In the first step, slice hyperholomorphic functions are obtained, and their associated Cauchy formula establishes the S-functional calculus for noncommuting operators on the S-spectrum. The second step produces axially monogenic functions, which lead to the development of the monogenic functional calculus. In this review paper we discuss the second operator in the Fueter-Sce mapping theorem that takes slice hyperholomorphic to axially monogenic functions. This operator admits several factorizations which generate various function spaces and their corresponding functional calculi, thereby forming the so-called fine structures of spectral theories on the S-spectrum.