Non-Archimedean operator theory / by Toka Diagana, François Ramaroson.

Format:

Book

Author/Creator:

Diagana, Toka, Author.

Ramaroson, François., Author.

Series:

SpringerBriefs in Mathematics, 2191-8198

Language:

English

Subjects (All):

Operator theory.

Functional analysis.

Algebra.

Field theory (Physics).

Functions of real variables.

Operator Theory.

Functional Analysis.

Field Theory and Polynomials.

Real Functions.

Local Subjects:

Operator Theory.

Functional Analysis.

Field Theory and Polynomials.

Real Functions.

Physical Description:

1 online resource (163 p.)

Edition:

1st ed. 2016.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2016.

Language Note:

English

Summary:

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. .

Contents:

Preface; Acknowledgments; Contents; 1 Non-Archimedean Valued Fields; 1.1 Valuation; 1.1.1 Definitions and First Properties; 1.1.2 The Topology Induced by a Valuation on K; 1.1.3 Non-Archimedean Valuations; 1.1.4 Some Analysis on a Complete Non-Archimedean Valued Field; 1.1.5 The Order Function for a Discrete Valuation; 1.2 Examples; 1.2.1 Examples of Archimedean Valuation; 1.2.2 Examples of Non-Archimedean Valued Fields; 1.3 Additional Properties of Non-ArchimedeanValued Fields; 1.4 Some Remarks on Krull Valuations; 1.5 Bibliographical Notes; 2 Non-Archimedean Banach Spaces

2.1 Non-Archimedean Norms2.2 Non-Archimedean Banach Spaces; 2.3 Free Banach Spaces; 2.4 The p-adic Hilbert Space Eω; 2.5 Bibliographical Notes; 3 Bounded Linear Operators in Non-Archimedean Banach Spaces; 3.1 Bounded Linear Operators; 3.1.1 Definitions and Examples; 3.1.2 Basic Properties; 3.1.3 Bounded Linear Operators in Free Banach Spaces; 3.2 Additional Properties of Bounded Linear Operators; 3.2.1 The Inverse Operator; 3.2.2 Perturbations of Orthogonal Bases Using the Inverse Operator; 3.2.3 The Adjoint Operator; 3.3 Finite Rank Linear Operators; 3.3.1 Basic Definitions

3.3.2 Properties of Finite Rank Operators3.4 Completely Continuous Linear Operators; 3.4.1 Basic Properties; 3.4.2 Completely Continuous Linear Operators on Eω; 3.5 Bounded Fredholm Linear Operators; 3.5.1 Definitions and Examples; 3.5.2 Properties of Fredholm Operators; 3.6 Spectral Theory for Bounded Linear Operators; 3.6.1 The Spectrum of a Bounded Linear Operator; 3.6.2 The Essential Spectrum of a Bounded Linear Operator; 3.7 Bibliographical Notes; 4 The Vishik Spectral Theorem; 4.1 The Shnirel'man Integral and Its Properties; 4.1.1 Basic Definitions; 4.1.2 The Shnirel'man Integral

4.2 Distributions with Compact Support4.3 Cauchy-Stieltjes and Vishik Transforms; 4.4 Analytic Bounded Linear Operators; 4.5 Vishik Spectral Theorem; 4.6 Bibliographical Notes; 5 Spectral Theory for Perturbations of Bounded Diagonal Linear Operators; 5.1 Spectral Theory for Finite Rank Perturbations of Diagonal Operators; 5.1.1 Introduction; 5.1.2 Spectral Analysis for the Class of Operators T = D + K; 5.1.3 Spectral Analysis for the Class of Operators T = D + F; 5.2 Computation of σe(D); 5.3 Spectrum of T = D + F; 5.4 Examples; 5.5 Bibliographical Notes; 6 Unbounded Linear Operators

6.1 Unbounded Linear Operators on a Non-archimedean Banach Space6.2 Closed Linear Operators; 6.3 The Spectrum of an Unbounded Operator; 6.4 Unbounded Fredholm Operators; 6.5 Bibliographical Notes; 7 Spectral Theory for Perturbations of Unbounded Linear Operators; 7.1 Introduction; 7.2 Spectral Analysis for the Class of Operators T = D + K; 7.3 Spectral Analysis for the Class of Operators T = D + F; 7.4 Computation of σe(D); 7.5 Main Result; 7.6 Bibliographical Notes; A The Shnirel'man Integral; A.1 Distributions with Compact Support; A.2 Cauchy-Stieltjes and Vishik Transforms; References

Index

Notes:

Description based upon print version of record.

Includes bibliographical references and index.

ISBN:

3-319-27323-X