The purpose of the present book is to develop the counterparts of Banach and Hilbert spaces in the setting of slice hyperholomorphic functions. Banach and Hilbert spaces of analytic functions, in one or several complex variables, play an important role in analysis and related fields. Besides their intrinsic interest, such spaces have numerous applications.
The book is divided into three parts. In the first part, some foundational material on quaternionic functions and functional analysis are introduced. The second part is the core of the book and contains various types of functions spaces ranging from the Hardy spaces, also in the fractional case, to the Fock space extended to the case of quaternions. The third and final part present some further generalization.
Researchers in functional analysis and hypercomplex analysis will find this book a key contribution to their field, but also researchers in mathematical physics, especially in quantum mechanics, will benefit from the insights presented.

lgli/978-3-031-73430-4.pdf

Springer Nature Switzerland AG

Springer Basel AG

Switzerland, Switzerland

Prologue
Acknowledgments
Contents
Part I Quaternionic Functional Analysis and Related Topics
1 Functions of a Quaternionic Variable
1.1 Quaternions
1.2 Quaternionic Matrices and Quantum Mechanics: A Glimpse
1.3 Slice Hyperholomorphic Functions
1.4 Rational Functions and Realizations
1.5 Realizations of Unitary Rational Functions
2 Quaternionic Banach and Hilbert Spaces
2.1 Quaternionic Topological Vector Spaces
2.2 Quaternionic Banach and Hilbert Spaces
2.3 Fréchet Spaces
2.4 Strong Algebras
2.5 Examples in One Variable
2.6 The Case of Several Quaternionic Variables
2.7 Tensor Products of Quaternionic Hilbert Spaces
3 Quaternionic Linear Operators on a Hilbert Space
3.1 Preliminary Results
3.2 The S-Spectrum and the Spectral Theorem
3.3 Singular Values of Compact Operators and p-Schatten Classes
3.4 Tensor Product of Operators in 2(N0,H) and Second Quantization
4 Reproducing Kernel Hilbert Spaces
4.1 Positive Functions
4.2 Reproducing Kernel Hilbert Spaces and Operator Ranges
4.3 Other Examples
4.4 Rational Functions with Positive Real Part and the Associated Positive Kernel
4.5 Positive Definite Functions and Covariances of Gaussian Processes
4.6 Miscellanea
Part II Spaces of Slice Hyperholomorphic Functions
5 Slice Hyperholomorphic Hardy Spaces
5.1 Hardy Space of the Unit Ball
5.2 The Hardy Space of the Half-Space H+
5.3 Fractional Hardy Space: The Half-Space Case
5.4 Fractional Hardy Space: The Unit Ball Case
5.5 Hardy-Sobolev Spaces
6 de Branges Spaces
6.1 The Structure Theorem: Half-Space Case
6.2 The Unit Ball Case
7 Slice Hyperholomorphic Bergman Spaces
7.1 Slice Hyperholomorphic Bergman Spaces
7.2 The Bergman Kernel of the First Kind
7.3 The Bergman Kernel of the Second Kind
7.4 Relation Between the Bergman Spaces of First and Second Kind
7.5 Schwarz Reflection Principle and Consequences
7.6 The Bergman-Fueter Transform on the Unit Ball
8 Slice Hyperholomorphic Bloch, Besov and Dirichlet Spaces
8.1 Bloch Spaces
8.2 Weighted Bergman Spaces
8.3 Besov Spaces
8.4 Dirichlet Space
9 Slice Hyperholomorphic Fock Space
9.1 The Fock Space in Complex Analysis
9.2 The Fock Space in the Slice Hyperholomorphic Case
10 Wiener Algebras
10.1 The Wiener Algebra
10.2 Toeplitz Operators
10.3 The Wiener Algebra in the Continuous Case
10.4 Hardy Spaces and Wiener-Hopf Operators
Part III Various Extensions
11 Slice Polyanalytic Functions
11.1 Polyanalytic Functions of a Complex Variable
11.2 Quaternionic Slice Polyanalytic Functions
11.3 Cauchy Formulas for Slice Polyanalytic Functions
11.3.1 Cauchy Formulas with Kernels P SL-1 and P SR-1
11.3.2 Cauchy Formulas with Kernels Π SL-1 and Π SR-1
11.4 The Quaternionic Slice Polyanalytic Fock Space
11.5 The Quaternionic Slice Polyanalytic Bergman Space
12 Several Quaternionic Variables
12.1 Noncommutative Monomials and Power Series
12.2 Schur Multipliers
12.3 Realization Theorem for Schur Multipliers
12.4 A Structure Theorem
12.5 Blaschke Factors and Interpolation in the Fock Space
13 Function Spaces and Spectral Theories
13.1 The Problem of the Spectrum for Quaternionic Operators
13.2 The Fueter-Sce-Qian Construction: Functions and Spectral Theories
13.3 Interaction of the Hyperholomorphic Spectral Theories
13.4 Fine Structures of the Spectral Theories on the S-Spectrum
13.5 Applications of the Hypercomplex Functions and Their Spectral Theories
Bibliography
Index

2024-12-12