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An Introduction to Classical and p-Adic Theory of Linear Operators and Applications

Toka Diagana
Publisher: 
Nova Science
Publication Date: 
2006
Number of Pages: 
116
Format: 
Hardcover
Price: 
69.00
ISBN: 
1594544247
Category: 
Monograph
We do not plan to review this book.

 

Preface

1 Banach and Hilbert Spaces; pp, 1- 10
1.1 Banach Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1.2 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . .
1.1.3 Examples of Banach Spaces . . . . . . . . . . . . . . . .
1.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.1 Pre-Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . .
1.2.2 Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . . . . .
1.2.3 Projections . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . .


2 Bounded Linear Operators on Classical and p-adic Hilbert Spaces; pp, 11-46
2.1 Bounded Linear Operators on Hilbert Spaces . . . . . .
2.1.1 Basic Definitions and Examples . . . . . . . . . . . . .
2.1.2 The Adjoint of An Operator . . . . . . . . . . . . . . . .
2.1.3 The Inverse of An Operator . . . . . . . . . . . . . . . .
2.1.4 Normal and Self-adjoint Operators . . . . . . . . . .
XII Contents
2.1.5 Square Root of a Positive Self-adjoint Operator
2.1.6 Compact Operators . . . . . . . . . . . . . . . . . . . . . . .
2.1.7 Hilbert-Schmidt Operators . . . . . . . . . . . . . . . . .
2.2 Bounded Linear Operators on p-adic Hilbert Spaces Ew
2.2.1 Absolute Value on a Field . . . . . . . . . . . . . . . . . .
2.2.2 Construction of the Field of p-adic Numbers . . .............
2.2.3 Ultrametric Banach Spaces . . . . . . . . . . . . . . . . .
2.2.4 Free Banach Spaces . . . . . . . . . . . . . . . . . . . . . . .
2.2.5 The p-adic Hilbert Space Ew . . . . . . . . . . . . . . .
2.2.6 Bounded Linear Operators Ew . . . . . . . . . . . . . .
2.3 p-adic Hilbert-Schmidt Operators . . . . . . . . . . . . . . . .
2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.2 Hilbert-Schmidt Operators on Ew . . . . . . . . . . .
2.3.3 Further Properties of Hilbert-Schmidt
Operators on Ew . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . .
2.5 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . .


3 Unbounded Linear Operators on Classical and p-adic Hilbert Spaces; pp, 47-75
3.1 Basic Definitions and Examples . . . . . . . . . . . . . . . . . .
3.1.1 Examples of Unbounded Operators . . . . . . . . .
3.1.2 Closed and Closable Linear Operators . . . . . .
3.1.3 Invariant Subspaces for Unbounded Linear
Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Contents XIII
3.1.4 Semigroups of Linear Operators . . . . . . . . . . . .
3.1.5 Spectral Theory for Unbounded Linear
Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.6 Symmetric and Self-adjoint Linear Operators . ............
3.1.7 Maximal Linear Operators . . . . . . . . . . . . . . . .
3.1.8 Algebraic Sum of Linear Operators . . . . . . . . .
3.2 Unbounded Linear Operators on p-adic Hilbert
Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2.2 The p-adic Hilbert Space Ew1 × Ew2 × ...× Ewn
3.2.3 Unbounded Linear Operators On Ew . . . . . . . .
3.3 Closed Linear Operators on Ew . . . . . . . . . . . . . . . . . .
3.4 The Diagonal Operator on Ew . . . . . . . . . . . . . . . . . . .
3.5 The Equation Ax = y on Ew . . . . . . . . . . . . . . . . . . . .
3.5.1 The Bounded Case . . . . . . . . . . . . . . . . . . . . . . . .
3.5.2 Application to the Perturbation of Bases on Ew
3.5.3 The Unbounded Case . . . . . . . . . . . . . . . . . . . . .
3.6 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . .
3.7 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . .


4 Applications To Abstract Differential Equations; pp, 77-108
4.1Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Basic Definitions and Notations . . . . . . . . . . . . . . . . .
4.2.1 Almost Automorphic Functions . . . . . . . . . . . . .
4.2.2 Almost Periodic Functions . . . . . . . . . . . . . . . . .
4.2.3 Pseudo Almost Periodic Functions . . . . . . . . . .
4.3 The Equation u0(t) = Au(t) + Bu(t) + f(t) . . . . . . . .
XIV Contents
4.3.1 Almost Automorphic Mild Solutions . . . . . . . . .
4.3.2 Almost Periodic Mild Solutions . . . . . . . . . . . . .
4.4 The Method of Invariant Subspaces for Unbounded
Linear Operators . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.2 Almost Automorphic Solutions . . . . . . . . . . . . .
4.4.3 Applications to Some Second-Order
Differential Equations . . . . . . . . . . . . . . . . . . . . .
4.4.4 Almost Automorphic Solutions to Some
Second-Order Hyperbolic Equations . . . . . . . . .
4.5 Pseudo Almost Periodic Solutions to Some
Second-Order Equations . . . . . . . . . . . . . . . . . . . . . . . .
4.5.1 Pseudo Almost Periodic Solutions . . . . . . . . .
4.5.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Existence and Uniqueness of Almost Automorphic
Mild Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Existence and Uniqueness of Mild Solutions .
4.6.3 Case of Bounded Perturbations . . . . . . . . . . . .
4.6.4 Applications . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Bibliographical Notes . . . . . . . . . . . . . . . . . . . . . .
4.8 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . .

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Dummy View - NOT TO BE DELETED