This self-contained book provides the reader with a comprehensive presentation of recent investigations on operator theory over non-Archimedean Banach and Hilbert spaces. This includes, non-Archimedean valued fields, bounded and unbounded linear operators, bilinear forms, functions of linear operators and one-parameter families of bounded linear operators on free branch spaces.
This book focuses on the theory of linear operators on non-Archimedean Banach spaces. The topics treated in this book range from a basic introduction to non-Archimedean valued fields, free non-Archimedean Banach spaces, bounded and unbounded linear operators in the non-Archimedean setting, to the spectral theory for some classes of linear operators. The theory of Fredholm operators is emphasized and used as an important tool in the study of the spectral theory of non-Archimedean operators. Explicit descriptions of the spectra of some operators are worked out. Moreover, detailed background materials on non-Archimedean valued fields and free non-Archimedean Banach spaces are included for completeness and for reference. The readership of the book is aimed toward graduate and postgraduate students, mathematicians, and non-mathematicians such as physicists and engineers who are interested in non-Archimedean functional analysis. Further, it can be used as an introduction to the study of non-Archimedean operator theory in general and to the study of spectral theory in other special cases.
This book provides the reader with a self-contained treatment of the classical operator theory with significant applications to abstract differential equations, and an elegant introduction to basic concepts and methods of the rapidly growing theory of the so-called p-adic operator theory.
The purpose of this book is to introduce a new notion of analytic space over a non-Archimedean field. Despite the total disconnectedness of the ground field, these analytic spaces have the usual topological properties of a complex analytic space, such as local compactness and local arcwise connectedness. This makes it possible to apply the usual notions of homotopy and singular homology. The book includes a homotopic characterization of the analytic spaces associated with certain classes of algebraic varieties and an interpretation of Bruhat-Tits buildings in terms of these analytic spaces. The author also studies the connection with the earlier notion of a rigid analytic space. Geometrical considerations are used to obtain some applications, and the analytic spaces are used to construct the foundations of a non-Archimedean spectral theory of bounded linear operators. This book requires a background at the level of basic graduate courses in algebra and topology, as well as some familiarity with algebraic geometry. It would be of interest to research mathematicians and graduate students working in algebraic geometry, number theory, and -adic analysis.
The first book to assemble the wide body of theory which has rapidly developed on the dynamics of linear operators. Written for researchers in operator theory, but also accessible to anyone with a reasonable background in functional analysis at the graduate level.
This volume presents a systematic treatment of the theory of unbounded linear operators in normed linear spaces with applications to differential equations. Largely self-contained, it is suitable for advanced undergraduates and graduate students, and it only requires a familiarity with metric spaces and real variable theory. After introducing the elementary theory of normed linear spaces--particularly Hilbert space, which is used throughout the book--the author develops the basic theory of unbounded linear operators with normed linear spaces assumed complete, employing operators assumed closed only when needed. Other topics include strictly singular operators; operators with closed range; perturbation theory, including some of the main theorems that are later applied to ordinary differential operators; and the Dirichlet operator, in which the author outlines the interplay between functional analysis and "hard" classical analysis in the study of elliptic partial differential equations. In addition to its readable style, this book's appeal includes numerous examples and motivations for certain definitions and proofs. Moreover, it employs simple notation, eliminating the need to refer to a list of symbols.
THE THEORY OF LINEAR OPERATORS FROM THE STANDPOINT OF DIFFEREN TIAL EQUATIONS OF INFINITE ORDER By HAROLD T. DAVIS INDIANA UNIVERSITY AND THE COWLES COMMISSION FOR RESEARCH IN ECONOMICS THE PRINCIPIA PRESS Bloommgton, Indiana 1936 MONOGRAPH OF THE WATERMAN INSTITUTE OF INDIANA UNIVERSITY CONTRIBUTION NO. 72 THE THEORY OF LINEAR OPERATORS To Agnes, who endured so patiently the writing of it, this boo is affectionately dedicated. TABLE OF CONTENTS CHAPTER I LINEAR OPERATORS 1. The Nature of Operators ------------1 2. Definition of an Operator -----.--3 3. A Classification of Operational Methods --------7 4. The Formal Theory of Operators ----------g 5. Generalized Integration and Differentiation - - 16 6. Differential and Integral Equations of Infinite Order -----23 7. The Generatrix Calculus - - 28 8. The Heaviside Operational Calculus ---------34 9. The Theory of Functionals ------------33 10. The Calculus of Forms in Infinitely Many Variables -----4 CHAPTER II PARTICULAR OPERATORS 1. Introduction ----------------51 2. Polynomial Operators --------53 3. The Fourier Definition of an Operator ---------53 4. The Operational Symbol of von Neumann and Stone -----57 5. The Operator as a Laplace Transform ---------59 6. Polar Operators ...-60 7. Branch Point Operators ------------64 8. Note on the Complementary Function ---------70 9. Riemanns Theory - .--.--72 10. Functions Permutable with Unity ----------76 11. Logarithmic Operators ------------78 12. Special Operators --------------85 13. The General Analytic Operator ----------99 14. The Differential Operator of Infinite Order -------100 15. Differential Operators as a Cauchy Integral -------103 16. The Generatrix of Differential Operators--------104 17. Five Operators of Analysis ------------105 CHAPTER III THE THEORY OF LINEAR SYSTEMS OF EQUATIONS 1. Preliminary Remarks -------------108 2. Types of Matrices --------------109 3. The Convergence of an Infinite Determinant -------114 4. The Upper Bound of a Determinant. Hadamards Theorem - - 116 5. Determinants which do not Vanish - - - - - - - - - 123 6. The Method of the Liouville-Neumann Series -------126 7. The Method of Segments ------------130 8. Applications of the Method of Segments. --------132 9. The Hilbert Theory of Linear Equations in an Infinite Number of Variables - - - - 137 10. Extension of the Foregoing Theory to Holder Space 149 vii Vlll THE THEORY OF LINEAR OPERATORS CHAPTER IV OPERATIONAL MULTIPLICATION AND INVERSION 1. Algebra and Operators -------.. --153 2. The Generalized Formula of Leibnitz ---------154 3. Bourlets Operational Product --. 155 4. The Algebra of Functions of Composition --------159 5. Selected Problems in the Algebra of Permutable Functions - - - - 164 G. The Calculation of a Function Permutable with a Given Function - 166 7. The Transformation of Peres -----------171 8. The Permutability of Functions Permutable with a Given Function - 173 9. Permutable Functions of Second Kind - --176 10. The Inversion of Operators Bourlets Theory ------177 It. The Method of Successive Substitutions --------181 12. Some Further Properties of the Resolvent Generatrix - 185 13. The Inversion of Operators by Infinite Differentiation - 188 14. The Permutability of Linear PilYeiential Operators -----190 15. A Class of Non-permutable Operators ---------194 16. Special Examples Illustrating the Application of Operational Processes 200 CHAPTER V GRADESDEFINED BY SPECIAL OPERATORS 1. Definition ----------------211 2. The Grade of an Unlimitedly Differentiable Function - 212 3. Functions of Finite Grade ------------215 4. Asymptotic Expansions --- 222 5. The Summability of Differential Operators with Constant Coefficients 230 6. The Summability of Operators of Laplace Type ------235 CHAPTER VI DIFFERENTIAL EQUATIONS OF INFINITE ORDER WITH CONSTANT COEFFICIENTS 1. Introduction ---------------238 2. Expansion of the Resolvent Generatrix --------239 3. The Method of Cauchy-Bromwich ----------250 4...
Most books on linear operators are not easy to follow for students and researchers without an extensive background in mathematics. Self-contained and using only matrix theory, Invitation to Linear Operators: From Matricies to Bounded Linear Operators on a Hilbert Space explains in easy-to-follow steps a variety of interesting recent results on linear operators on a Hilbert space. The author first states the important properties of a Hilbert space, then sets out the fundamental properties of bounded linear operators on a Hilbert space. The final section presents some of the more recent developments in bounded linear operators.
This volume contains the Proceedings of the 13th International Conference on p-adic Functional Analysis, held from August 12–16, 2014, at the University of Paderborn, Paderborn, Germany. The articles included in this book feature recent developments in various areas of non-Archimedean analysis, non-Archimedean functional analysis, representation theory, number theory, non-Archimedean dynamical systems and applications. Through a combination of new research articles and survey papers, this book provides the reader with an overview of current developments and techniques in non-Archimedean analysis as well as a broad knowledge of some of the sub-areas of this exciting and fast-developing research area.
