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The subject of this monograph is the quaternionic spectral theory
based on the notion of S-spectrum. With the purpose of giving a
systematic and self-contained treatment of this theory that has
been developed in the last decade, the book features topics like
the S-functional calculus, the F-functional calculus, the
quaternionic spectral theorem, spectral integration and spectral
operators in the quaternionic setting. These topics are based on
the notion of S-spectrum of a quaternionic linear operator. Further
developments of this theory lead to applications in fractional
diffusion and evolution problems that will be covered in a separate
monograph.
This book presents a new theory for evolution operators and a new
method for defining fractional powers of vector operators. This new
approach allows to define new classes of fractional diffusion and
evolution problems. These innovative methods and techniques, based
on the concept of S-spectrum, can inspire researchers from various
areas of operator theory and PDEs to explore new research
directions in their fields. This monograph is the natural
continuation of the book: Spectral Theory on the S-Spectrum for
Quaternionic Operators by Fabrizio Colombo, Jonathan Gantner, and
David P. Kimsey (Operator Theory: Advances and Applications, Vol.
270).
This book presents a new theory for evolution operators and a new
method for defining fractional powers of vector operators. This new
approach allows to define new classes of fractional diffusion and
evolution problems. These innovative methods and techniques, based
on the concept of S-spectrum, can inspire researchers from various
areas of operator theory and PDEs to explore new research
directions in their fields. This monograph is the natural
continuation of the book: Spectral Theory on the S-Spectrum for
Quaternionic Operators by Fabrizio Colombo, Jonathan Gantner, and
David P. Kimsey (Operator Theory: Advances and Applications, Vol.
270).
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.
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