Slice hyperholomorphic functions and the quaternionic functional calculus

Abstract

The discovery of slice hyperholomorphic functions gave great impact to the field of quaternionic functional analysis. Based on this notion of generalized holomorphicity, it was possible to develop the analogue of the Riesz-Dunford functional calculus for quaternionic linear operators. In the present master thesis an overview on the theory of quaternion-valued slice hyperholomorphic functions and the associated S-functional calculus for quaternionic linear operators is given. The author treats the relation to classical holomorphicity and shows the two main tools for working with these functions: the Splitting Lemma and the Representation Formula. Furthermore, he proves the generalizations of certain classical results, which are necessary to define the S-functional calculus: slice hyperholomorphic functions allow a power series expansion at points on the real axis and they satisfy a Runge-type approximation theorem as well as an integral formula of Cauchy-type with a modified kernel. In the second half of the thesis, the notion of S-spectrum and the S-functional calculus is defined, which is based on this notion of spectrum for quaternionic linear operators. Their main properties such as the boundedness of the S-spectrum, the Spectral Mapping Theorem and compatibility of the S-functional calculus with algebraic operations and uniform limits are proven. In particular, the analogue of the classical resolvent equation is shown and proofs of the product rule and for the existence of Riesz-projectors, which are based on this equation, are given.