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

Research paper by **Jonathan Gantner**

Indexed on: **28 Mar '18**Published on: **28 Mar '18**Published in: **arXiv - Mathematics - Spectral Theory**

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#### 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.