Operator Theory on One-Sided Quaternion Linear Spaces: Intrinsic S -Functional Calculus and Spectral Operators

This article explores the minimal structure required for quaternionic operator theory and establishes a connection between complex and quaternionic operator theory. It also discusses the assumption of a right- and left-multiplication on Banach spaces for quaternionic operators and the independence of spectral properties from the left multiplication.

Format: Paperback / softback
Length: 101 pages
Publication date: 01 July 2021
Publisher: American Mathematical Society

This article delves into two fundamental themes: the identification of the minimal structure required to develop quaternionic operator theory, and the revelation of a profound connection between complex and quaternionic operator theory. Typically, the theory of quaternionic right linear operators assumes the existence of both a right- and a left-multiplication on the Banach space V under consideration. This assumption arises from technical considerations, as the space of bounded operators on V is not automatically a quaternionic linear space unless both multiplications are defined. However, a right linear operator is solely associated with the right multiplication on the space, and in certain contexts, such as quaternionic Hilbert spaces, the left multiplication may not be defined a priori and must be chosen randomly. It is crucial that the spectral properties of an operator remain unaffected by the choice of left multiplication on the space.

The minimal structure necessary for quaternionic operator theory is explored in this article. It is demonstrated that the theory can be formulated without the need for a left multiplication on the Banach space V. This is achieved by considering a specific class of operators known as quaternionic right linear operators. These operators are defined on a quaternionic vector space and satisfy certain properties, such as the commutativity and associativity of the multiplication.

The article also highlights the deep relation between complex and quaternionic operator theory. It is shown that quaternionic operators can be viewed as complex operators with additional degrees of freedom. This connection enables the study of quaternionic functions and their properties using techniques developed in complex analysis.

Furthermore, the article discusses the importance of the left multiplication on the space in the context of quaternionic operator theory. It is shown that the choice of left multiplication can have significant implications for the spectral properties of the operators. In particular, the article explores the relationship between the spectrum of a quaternionic right linear operator and the spectrum of its complex conjugate.

Overall, this article provides a comprehensive exploration of the minimal structure necessary for quaternionic operator theory, the connection between complex and quaternionic operator theory, and the role of the left multiplication on the space in the development of the theory.

Weight: 210g
ISBN-13: 9781470442385