Irene Sabadini

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.

This book presents English translations of Michele Sce’s most important works, originally written in Italian during the period 1955-1973, on hypercomplex analysis and algebras of hypercomplex numbers. Despite their importance, these works are not very well known in the mathematics community because of the language they were published in. Possibly the most remarkable instance is the so-called Fueter-Sce mapping theorem, which is a cornerstone of modern hypercomplex analysis, and is not yet understood in its full generality. This volume is dedicated to revealing and describing the framework Sce worked in, at an exciting time when the various generalizations of complex analysis in one variable were still in their infancy. In addition to faithfully translating Sce’s papers, the authors discuss their significance and explain their connections to contemporary research in hypercomplex analysis. They also discuss many concrete examples that can serve as a basis for further research. The vast majority of the results presented here will be new to readers, allowing them to finally access the original sources with the benefit of comments from fellow mathematicians active in the field of hypercomplex analysis. As such, the book offers not only an important chapter in the history of hypercomplex analysis, but also a roadmap for further exciting research in the field.

This book presents English translations of Michele Sce’s most important works, originally written in Italian during the period 1955-1973, on hypercomplex analysis and algebras of hypercomplex numbers. Despite their importance, these works are not very well known in the mathematics community because of the language they were published in. Possibly the most remarkable instance is the so-called Fueter-Sce mapping theorem, which is a cornerstone of modern hypercomplex analysis, and is not yet understood in its full generality. This volume is dedicated to revealing and describing the framework Sce worked in, at an exciting time when the various generalizations of complex analysis in one variable were still in their infancy. In addition to faithfully translating Sce’s papers, the authors discuss their significance and explain their connections to contemporary research in hypercomplex analysis. They also discuss many concrete examples that can serve as a basis for further research. The vast majority of the results presented here will be new to readers, allowing them to finally access the original sources with the benefit of comments from fellow mathematicians active in the field of hypercomplex analysis. As such, the book offers not only an important chapter in the history of hypercomplex analysis, but also a roadmap for further exciting research in the field.

This work contributes to the study of quaternionic linear operators. This study is a generalization of the complex case, but the noncommutative setting of quaternions shows several interesting new features, see e.g. the so-called S-spectrum and S-resolvent operators. In this work, we study de Branges spaces, namely the quaternionic counterparts of spaces of analytic functions (in a suitable sense) with some specific reproducing kernels, in the unit ball of quaternions or in the half space of quaternions with positive real parts. The spaces under consideration will be Hilbert or Pontryagin or Krein spaces. These spaces are closely related to operator models that are also discussed. The focus of this book is the notion of characteristic operator function of a bounded linear operator A with finite real part, and we address several questions like the study of J-contractive functions, where J is self-adjoint and unitary, and we also treat the inverse problem, namely to characterize which J-contractive functions are characteristic operator functions of an operator. In particular, we prove the counterpart of Potapov's factorization theorem in this framework. Besides other topics, we consider canonical differential equations in the setting of slice hyperholomorphic functions and we define the lossless inverse scattering problem. We also consider the inverse scattering problem associated with canonical differential equations. These equations provide a convenient unifying framework to discuss a number of questions pertaining, for example, to inverse scattering, non-linear partial differential equations and are studied in the last section of this book.

This book presents the extensions to the quaternionic setting of some of the main approximation results in complex analysis. It also includes the main inequalities regarding the behavior of the derivatives of polynomials with quaternionic cofficients. With some few exceptions, all the material in this book belongs to recent research of the authors on the approximation of slice regular functions of a quaternionic variable. The book is addressed to researchers in various areas of mathematical analysis, in particular hypercomplex analysis, and approximation theory. It is accessible to graduate students and suitable for graduate courses in the above framework.

This book defines and examines the counterpart of Schur functions and Schur analysis in the slice hyperholomorphic setting. It is organized into three parts: the first introduces readers to classical Schur analysis, while the second offers background material on quaternions, slice hyperholomorphic functions, and quaternionic functional analysis. The third part represents the core of the book and explores quaternionic Schur analysis and its various applications. The book includes previously unpublished results and provides the basis for new directions of research.

This Briefs volume develops the theory of entire slice regular functions. It is the first self-contained, monographic work on the subject, offering all the necessary background information and detailed studies on several central topics, including estimates on the minimum modulus of regular functions, relations between Taylor coefficients and the growth of entire functions, density of their zeros, and the universality properties. The proofs presented here shed new light on the nature of the quaternionic setting and provide inspiration for further research directions. Also featuring an exhaustive reference list, the book offers a valuable resource for graduate students, postgraduate students and researchers in various areas of mathematical analysis, in particular hypercomplex analysis and approximation theory.

This book presents a functional calculus for n-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions. Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory, hypercomplex analysis, and mathematical physics.

* The main treatment is devoted to the analysis of systems of linear partial differential equations (PDEs) with constant coefficients, focusing attention on null solutions of Dirac systems * All the necessary classical material is initially presented * Geared toward graduate students and researchers in (hyper)complex analysis, Clifford analysis, systems of PDEs with constant coefficients, and mathematical physics

This book presents a functional calculus for n-tuples of not necessarily commuting linear operators. In particular, a functional calculus for quaternionic linear operators is developed. These calculi are based on a new theory of hyperholomorphicity for functions with values in a Clifford algebra: the so-called slice monogenic functions which are carefully described in the book. In the case of functions with values in the algebra of quaternions these functions are named slice regular functions. Except for the appendix and the introduction all results are new and appear for the first time organized in a monograph. The material has been carefully prepared to be as self-contained as possible. The intended audience consists of researchers, graduate and postgraduate students interested in operator theory, spectral theory, hypercomplex analysis, and mathematical physics.

The subject of Clifford algebras has become an increasingly rich area of research with a significant number of important applications not only to mathematical physics but to numerical analysis, harmonic analysis, and computer science. The main treatment is devoted to the analysis of systems of linear partial differential equations (PDEs) with constant coefficients, focusing attention on null solutions of Dirac systems. Knowledge from different fields of mathematics such as commutative algebra, Grobner bases, sheaf theory, cohomology, topological vector spaces, and generalized functions (distributions and hyperfunctions) is required of the reader. However, all the necessary classical material is initially presented. The book may be used by graduate students and researchers interested in (hyper)complex analysis, Clifford analysis, systems of PDEs with constant coefficients, and mathematical physics.