The Fine Structure of the Spectral Theory on the S-Spectrum in Dimension Five

Abstract Holomorphic functions play a crucial role in operator theory and the Cauchy formula is very important tool to define of operators. The Fueter–Sce–Qian extension theorem two-step procedure extend holomorphic hyperholomorphic setting. first step gives class slice functions; their allows so-called S -functional calculus for noncommuting operators based on -spectrum. In second this generates monogenic related functional calculus, spectrum, contains Weyl as particular case. paper we show that from admits various possible factorizations induce different function spaces. integral representations such spaces allow associated calculi fine structure spectral theories S-spectrum . Among structures there are harmonic polyharmonic calculi. study depends dimension considered detail case five, describe all them. five-dimensional importance because it determine almost will also appear greater than but with orders.