2018/03/28 by Jonathan Gantner, Gantner, Jonathan · 4 citations
Mathematics · #Algebraic and Geometric Analysis #Mathematical Analysis and Transform Methods #Advanced Topics in Algebra
paper · pdf · doi:10.48550/arxiv.1803.10524
Two themes drive this article: identifying the structure necessary to\nformulate quaternionic operator theory and revealing the relation between\ncomplex and quaternionic operator theory.\n The theory of quaternionic right linear operators is usually formulated\nassuming the existenc of both a right- and a left-multiplication on the Banach\nspace V, as the space of bounded operators on V is otherwise not a\nquaternionic linear space. A right linear operator is however only associated\nwith the right-multiplication and in certain settings, e.g. on Hilbert spaces,\nthe left-multiplication is not defined a priori but must be chosen randomly.\nSpectral properties of an operator should hence be independent of this left\nmultiplication.\n We show that results derived from functional calculi for intrinsic slice\nfunctions can be formulated without the assumption of a left multiplication. We\ndevelop the S-functional calculus in this setting and a new approach to\nspectral integration. This approach has a clear interpretation in terms of the\nright linear structure on the space and allows to formulate the spectral\ntheorem without using any randomly chosen structure. Our techniques only apply\nto intrinsic slice functions, but only these functions are compatible with the\nbasic intuition of a functional calculus that f(T) should be defined by\nletting f act on the spectral values of T.\n Using these tools, we develop a theory of quaternionic spectral operators. In\nparticular, we show the existence of a canonical decomposition of such operator\nand discuss its behavior under the S-functional calculus.\n Finally, we show a relation with complex operator theory: if we embed the\ncomplex numbers into the quaternions, then complex and quaternionic operator\ntheory are consistent. The symmetry of intrinsic slice functions guarantees\nthat this compatibility is true for any imbedding of the complex numbers.\n