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E-raamat: Non-Archimedean Operator Theory

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  • Formaat: PDF+DRM
  • Sari: SpringerBriefs in Mathematics
  • Ilmumisaeg: 07-Apr-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319273235
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Non-Archimedean Operator TheorySuurem pilt
  • Formaat: PDF+DRM
  • Sari: SpringerBriefs in Mathematics
  • Ilmumisaeg: 07-Apr-2016
  • Kirjastus: Springer International Publishing AG
  • Keel: eng
  • ISBN-13: 9783319273235
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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. 

Arvustused

This book presents some of the authors recent work on continuous linear operators on non-archimedean Banach space as well as their spectral theory. The book can be recommended to beginners as an introduction to non-archimendean operator theory. (Bertin Diarra, Mathematical Reviews, January, 2017)

The book is intended as an introduction to the non-Archimedean operator theory for graduate and postgraduate students, mathematicians, and non-mathematicians such as physicists and engineers who are interested in functional analysis in the non-Archimedean context. Of course, expecting the readership being so wide, the authors make the exposition as elementary as possible. (Anatoly N. Kochubei, zbMATH 1357.47002, 2017)

1 Non-Archimedean Valued Fields
1(40)
1.1 Valuation
1(15)
1.1.1 Definitions and First Properties
1(4)
1.1.2 The Topology Induced by a Valuation on K
5(3)
1.1.3 Non-Archimedean Valuations
8(5)
1.1.4 Some Analysis on a Complete Non-Archimedean Valued Field
13(2)
1.1.5 The Order Function for a Discrete Valuation
15(1)
1.2 Examples
16(12)
1.2.1 Examples of Archimedean Valuation
16(1)
1.2.2 Examples of Non-Archimedean Valued Fields
17(11)
1.3 Additional Properties of Non-Archimedean Valued Fields
28(8)
1.4 Some Remarks on Krull Valuations
36(3)
1.5 Bibliographical Notes
39(2)
2 Non-Archimedean Banach Spaces
41(20)
2.1 Non-Archimedean Norms
41(3)
2.2 Non-Archimedean Banach Spaces
44(6)
2.3 Free Banach Spaces
50(4)
2.4 The p-adic Hilbert Space Eω
54(6)
2.5 Bibliographical Notes
60(1)
3 Bounded Linear Operators in Non-Archimedean Banach Spaces
61(24)
3.1 Bounded Linear Operators
61(6)
3.1.1 Definitions and Examples
61(3)
3.1.2 Basic Properties
64(1)
3.1.3 Bounded Linear Operators in Free Banach Spaces
65(2)
3.2 Additional Properties of Bounded Linear Operators
67(8)
3.2.1 The Inverse Operator
67(1)
3.2.2 Perturbations of Orthogonal Bases Using the Inverse Operator
68(5)
3.2.3 The Adjoint Operator
73(2)
3.3 Finite Rank Linear Operators
75(2)
3.3.1 Basic Definitions
75(1)
3.3.2 Properties of Finite Rank Operators
75(2)
3.4 Completely Continuous Linear Operators
77(1)
3.4.1 Basic Properties
77(1)
3.4.2 Completely Continuous Linear Operators on Eω
77(1)
3.5 Bounded Fredholm Linear Operators
78(3)
3.5.1 Definitions and Examples
78(1)
3.5.2 Properties of Fredholm Operators
79(2)
3.6 Spectral Theory for Bounded Linear Operators
81(3)
3.6.1 The Spectrum of a Bounded Linear Operator
81(1)
3.6.2 The Essential Spectrum of a Bounded Linear Operator
82(2)
3.7 Bibliographical Notes
84(1)
4 The Vishik Spectral Theorem
85(22)
4.1 The Shnirel'man Integral and Its Properties
85(11)
4.1.1 Basic Definitions
85(6)
4.1.2 The Shnirel'man Integral
91(5)
4.2 Distributions with Compact Support
96(2)
4.3 Cauchy--Stieltjes and Vishik Transforms
98(4)
4.4 Analytic Bounded Linear Operators
102(2)
4.5 Vishik Spectral Theorem
104(1)
4.6 Bibliographical Notes
105(2)
5 Spectral Theory for Perturbations of Bounded Diagonal Linear Operators
107(16)
5.1 Spectral Theory for Finite Rank Perturbations of Diagonal Operators
107(8)
5.1.1 Introduction
107(2)
5.1.2 Spectral Analysis for the Class of Operators T = D + K
109(3)
5.1.3 Spectral Analysis for the Class of Operators T = D + F
112(3)
5.2 Computation of σe(D)
115(3)
5.3 Spectrum of T = D + F
118(1)
5.4 Examples
118(3)
5.5 Bibliographical Notes
121(2)
6 Unbounded Linear Operators
123(8)
6.1 Unbounded Linear Operators on a Non-archimedean Banach Space
123(1)
6.2 Closed Linear Operators
124(2)
6.3 The Spectrum of an Unbounded Operator
126(1)
6.4 Unbounded Fredholm Operators
127(2)
6.5 Bibliographical Notes
129(2)
7 Spectral Theory for Perturbations of Unbounded Linear Operators
131(10)
7.1 Introduction
131(1)
7.2 Spectral Analysis for the Class of Operators T = D + K
132(1)
7.3 Spectral Analysis for the Class of Operators T - D + F
133(2)
7.4 Computation of σe(D)
135(3)
7.5 Main Result
138(1)
7.6 Bibliographical Notes
139(2)
A The Shnirel'man Integral
141(10)
A.1 Distributions with Compact Support
146(3)
A.2 Cauchy-Stieltjes and Vishik Transforms
149(2)
References 151(4)
Index 155
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