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Two themes drive this article: identifying the structure necessary toformulate quaternionic operator theory and revealing the relation betweencomplex and quaternionic operator theory.  The theory of quaternionic right linear operators is usually formulatedassuming the existenc of both a right- and a left-multiplication on the Banachspace $V$, as the space of bounded operators on $V$ is otherwise not aquaternionic linear space. A right linear operator is however only associatedwith 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 leftmultiplication.  We show that results derived from functional calculi for intrinsic slicefunctions can be formulated without the assumption of a left multiplication. Wedevelop the S-functional calculus in this setting and a new approach tospectral integration. This approach has a clear interpretation in terms of theright linear structure on the space and allows to formulate the spectraltheorem without using any randomly chosen structure. Our techniques only applyto intrinsic slice functions, but only these functions are compatible with thebasic intuition of a functional calculus that $f(T)$ should be defined byletting $f$ act on the spectral values of $T$.  Using these tools, we develop a theory of quaternionic spectral operators. Inparticular, we show the existence of a canonical decomposition of such operatorand discuss its behavior under the S-functional calculus.  Finally, we show a relation with complex operator theory: if we embed thecomplex numbers into the quaternions, then complex and quaternionic operatortheory are consistent. The symmetry of intrinsic slice functions guaranteesthat this compatibility is true for any imbedding of the complex numbers.

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