On Some Properties of the Quaternionic Functional Calculus

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Abstract

In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup etA when A is a linear quaternionic operator.

Keywords

Slice regular functions Functional calculus Spectral theory Algebraic properties S-spectral radius theorem S-spectral mapping theorem Semigroup of a linear quaternionic operator 

Mathematics Subject Classification (2000)

47A10 47A60 30G35 

Copyright information

© Mathematica Josephina, Inc. 2009

Authors and Affiliations

  1. 1.Dipartimento di MatematicaPolitecnico di MilanoMilanoItaly

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