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Book DetailsMemoirs of the American Mathematical SocietyVolume: 267; 2020; 101 ppMSC: Primary 47
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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Table of Contents
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Chapters
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1. Introduction
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2. Preliminaries
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3. Slice Hyperholomorphic Functions
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4. The S-Functional Calculus
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5. The Spectral Theorem for Normal Operators
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6. Intrinsic S-Functional Calculus on One-Sided Banach Spaces
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7. Spectral Integration in the Quaternionic Setting
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8. On the Different Approaches to Spectral Integration
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9. Bounded Quaternionic Spectral Operators
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10. Canonical Reduction and Intrinsic S-Functional Calculus for Quaternionic Spectral Operators
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11. Concluding Remarks
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Additional Material
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RequestsReview Copy – for publishers of book reviewsPermission – for use of book, eBook, or Journal contentAccessibility – to request an alternate format of an AMS title
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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.
-
Chapters
-
1. Introduction
-
2. Preliminaries
-
3. Slice Hyperholomorphic Functions
-
4. The S-Functional Calculus
-
5. The Spectral Theorem for Normal Operators
-
6. Intrinsic S-Functional Calculus on One-Sided Banach Spaces
-
7. Spectral Integration in the Quaternionic Setting
-
8. On the Different Approaches to Spectral Integration
-
9. Bounded Quaternionic Spectral Operators
-
10. Canonical Reduction and Intrinsic S-Functional Calculus for Quaternionic Spectral Operators
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11. Concluding Remarks
