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

TL;DR

This paper develops a quaternionic operator theory framework that avoids the need for a left-multiplication structure, introduces an intrinsic S-functional calculus, and relates quaternionic and complex operator theories.

Contribution

It formulates quaternionic operator theory without assuming a left-multiplication, introduces a new spectral calculus, and establishes compatibility with complex operator theory.

Findings

Spectral properties are independent of left-multiplication.

The S-functional calculus is developed for intrinsic slice functions.

A canonical spectral decomposition for quaternionic operators is established.

Abstract

Two themes drive this article: identifying the structure necessary to formulate quaternionic operator theory and revealing the relation between complex and quaternionic operator theory. The theory of quaternionic right linear operators is usually formulated assuming the existenc of both a right- and a left-multiplication on the Banach space $V$, as the space of bounded operators on $V$ is otherwise not a quaternionic linear space. A right linear operator is however only associated with the right-multiplication and in certain settings, e.g. on Hilbert spaces, the left-multiplication is not defined a priori but must be chosen randomly. Spectral properties of an operator should hence be independent of this left multiplication. We show that results derived from functional calculi for intrinsic slice functions can be formulated without the assumption of a left multiplication. We develop the S-functional calculus in this setting and a new approach to spectral integration. This approach has a clear interpretation in terms of the right linear structure on the space and allows to formulate the spectral theorem without using any randomly chosen structure. Our techniques only apply to intrinsic slice functions, but only these functions are compatible with the basic intuition of a functional calculus that $f(T)$ should be defined by letting $f$ act on the spectral values of $T$. Using these tools, we develop a theory of quaternionic spectral operators. In particular, we show the existence of a canonical decomposition of such operator and discuss its behavior under the S-functional calculus. Finally, we show a relation with complex operator theory: if we embed the complex numbers into the quaternions, then complex and quaternionic operator theory are consistent. The symmetry of intrinsic slice functions guarantees that this compatibility is true for any imbedding of the complex numbers.