Non-Archimedean linear operators and applications

Contents......Page 8
Preface......Page 12
1.1 Introduction......Page 16
1.2.1 Basic Definitions......Page 17
1.2.2 The t-Vector Space Kt......Page 20
1.3.2 The Field Qp......Page 21
1.3.3 Convergence of Power Series overQp......Page 23
1.4 Construction of K((x))......Page 24
1.5 Bibliographical Notes......Page 25
2.1.1 Basic Definitions......Page 26
2.1.2 Examples of Non-Archimedean Banach Spaces......Page 27
2.2.2 Examples......Page 28
2.3.2 Non-Archimedean Hilbert Spaces......Page 29
2.3.3 The Hilbert Space Ew1 ×Ew2 ×...×Ewt......Page 31
2.4 Bibliographical Notes......Page 33
3.2 Bounded Linear Operators on Non-Archimedean Banach Spaces......Page 34
3.2.1 Basic Definitions......Page 35
3.2.2 Examples......Page 36
3.2.4 Further Properties of Bounded Linear Operators......Page 37
3.3.2 Representation of Bounded Operators By Infinite Matrices......Page 40
3.3.3 Existence of the Adjoint......Page 41
3.3.4 Examples of Bounded Operators with no Adjoint......Page 43
3.4 Perturbation of Bases......Page 44
3.5.1 Basic Definitions......Page 47
3.5.2 Further Properties of Hilbert-Schmidt Operators......Page 53
3.5.3 Completely Continuous Operators......Page 54
3.5.5 Examples......Page 55
3.7 Bibliographical Notes......Page 57
4.2 Basic Definitions......Page 58
4.2.1 Example......Page 59
4.2.3 Examples of Unbounded Operators With no Adjoint......Page 60
4.3 Closed Linear Operators on Ew......Page 61
4.4 Diagonal Operators on Ew......Page 64
4.6 Bibliographical Notes......Page 66
5.2.1 Continuous Linear Functionals on Ew......Page 68
5.2.2 Bounded Bilinear Forms on Ew×Ew......Page 70
5.2.3 Unbounded Bilinear Forms on Ew×Ew......Page 71
5.3 Closed and Closable non-Archimedean Bilinear Forms......Page 73
5.3.1 Closedness of the Form Sum......Page 74
5.3.2 Construction of a non-Archimedean Hilbert Space Using Quadratic Forms......Page 76
5.3.3 Further Properties of the Closure......Page 77
5.4 Representation of Bilinear Forms on Ew×Ew by Linear Operators......Page 78
5.5 Bibliographical Notes......Page 80
6.2 Products and Sums of Diagonal Operators......Page 82
6.3 Integer Powers of Diagonal Operators......Page 84
6.4 Functions of Self-Adjoint Operators......Page 87
6.5 Functions of Some Symmetric Square Matrices Over Qp×Qp......Page 88
6.5.2 Exponential of the Matrix T......Page 90
6.7 Bibliographical Notes......Page 91
7.1 Introduction......Page 92
7.2 Basic Definitions......Page 93
7.3 Properties of non-Archimedean C0-Groups......Page 94
7.4 Existence of Solutions to Some p-adic Differential Equations......Page 98
7.6 Bibliographical Notes......Page 101
References......Page 102
Index......Page 106