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

"Two major themes drive this article: identifying the minimal structure necessary to formulate quaternionic operator theory and revealing a deep relation between complex and quaternionic operator theory. The theory for quaternionic right linear operators is usually formulated under the assumption that there exists not only a right- but also a left-multiplication on the considered Banach space V . This has technical reasons, 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 on the space and in certain settings, for instance on quaternionic 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 the left multiplication on the space. We show that results derived from functional calculi involving intrinsic slice functions can be formulated without the assumption of a left multiplication. We develop the S-functional calculus in this setting and also a new approach to spectral integration of intrinsic slice functions. 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. The presented techniques only apply to intrinsic slice functions, but this is a negligible restriction. Indeed, due to the symmetry of the S-spectrum of T, only these functions are compatible with the very basic intuition of a functional calculus, namely that f(T) should be defined by letting f act on the spectral values of T. Using the above tools we develop a theory of spectral operators and obtain results analogue to those of the complex theory. In particular we show the existence of a canonical decomposition of a spectral operator and discuss its behavior under the S-functional calculus. Finally, we show a beautiful relation with complex operator theory: if we embed the complex numbers into the quaternions and consider the quaternionic vector space as a complex one, then complex and quaternionic operator theory are consistent. Again it is the symmetry of intrinsic slice functions that guarantees that this compatibility is true for any possible imbedding of the complex numbers"--