2015/06/03 by Fabrizio Colombo, Colombo, Fabrizio, Jonathan Gantner +1 · 2 citations
Mathematics · Physics and Astronomy · #Algebraic and Geometric Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Noncommutative and Quantum Gravity Theories #Relativity and Gravitational Theory
paper · pdf · doi:10.48550/arxiv.1506.01266
openalex publication_date 2015/06/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce fractional powers of quaternionic operators. Their\ndefinition is based on the theory of slice-hyperholomorphic functions and on\nthe S-resolvent operators of the quaternionic functional calculus. The\nintegral representation formulas of the fractional powers and the quaternionic\nversion of Kato's formula are based on the notion of S-spectrum of a\nquaternionic operator.\n The proofs of several properties of the fractional powers of quaternionic\noperators rely on the S-resolvent equation. This equation, which is very\nimportant and of independent interest, has already been introduced in the case\nof bounded quaternionic operators, but for the case of unbounded operators some\nadditional considerations have to be taken into account. Moreover, we introduce\na new series expansion for the pseudo-resolvent, which is of independent\ninterest and allows to investigate the behavior of the S-resolvents close to\nthe S-spectrum.\n The paper is addressed to researchers working in operator theory and in\ncomplex analysis.\n