NON-ARCHIMEDEAN LINEAR OPERATORSAND APPLICATIONS -- Contents -- Preface -- Non-Archimedean Valued Fields -- 1.1 Introduction -- 1.2 Non-Archimedean Valued Fields -- 1.2.1 Basic Definitions -- 1.2.2 The t-Vector Space Kt -- 1.3 Construction of Qp -- 1.3.1 Introduction -- 1.3.2 The Field Qp -- 1.3.3 Convergence of Power Series overQp -- 1.4 Construction of K((x)) -- 1.5 Bibliographical Notes -- Non-Archimedean Banach and HilbertSpaces -- 2.1 Non-Archimedean Banach Spaces -- 2.1.1 Basic Definitions -- 2.1.2 Examples of Non-Archimedean Banach Spaces -- 2.2 Free Banach Spaces -- 2.2.1 Definitions -- 2.2.2 Examples -- 2.3 Non-Archimedean Hilbert Spaces -- 2.3.1 Introduction -- 2.3.2 Non-Archimedean Hilbert Spaces -- 2.3.3 The Hilbert Space Ew1 ×Ew2 ×...×Ewt -- 2.4 Bibliographical Notes -- Non-Archimedean Bounded LinearOperators -- 3.1 Introduction -- 3.2 Bounded Linear Operators on Non-Archimedean Banach Spaces -- 3.2.1 Basic Definitions -- 3.2.2 Examples -- 3.2.3 The Banach Algebra B(X) -- 3.2.4 Further Properties of Bounded Linear Operators -- 3.3 Bounded Linear Operators on Hilbert Spaces Ew -- 3.3.1 Introduction -- 3.3.2 Representation of Bounded Operators By Infinite Matrices -- 3.3.3 Existence of the Adjoint -- 3.3.4 Examples of Bounded Operators with no Adjoint -- 3.4 Perturbation of Bases -- 3.5 Hilbert-Schmidt Operators -- 3.5.1 Basic Definitions -- 3.5.2 Further Properties of Hilbert-Schmidt Operators -- 3.5.3 Completely Continuous Operators -- 3.5.4 Trace -- 3.5.5 Examples -- 3.6 Open Problems -- 3.7 Bibliographical Notes -- Non-Archimedean Unbounded LinearOperators -- 4.1 Introduction -- 4.2 Basic Definitions -- 4.2.1 Example -- 4.2.2 Existence of the Adjoint -- 4.2.3 Examples of Unbounded Operators With no Adjoint -- 4.3 Closed Linear Operators on Ew -- 4.4 Diagonal Operators on Ew -- 4.5 Open Problems -- 4.6 Bibliographical Notes.

Non-Archimedean Bilinear Forms -- 5.1 Introduction -- 5.2 Basic Definitions -- 5.2.1 Continuous Linear Functionals on Ew -- 5.2.2 Bounded Bilinear Forms on Ew×Ew -- 5.2.3 Unbounded Bilinear Forms on Ew×Ew -- 5.3 Closed and Closable non-Archimedean Bilinear Forms -- 5.3.1 Closedness of the Form Sum -- 5.3.2 Construction of a non-Archimedean Hilbert Space Using Quadratic Forms -- 5.3.3 Further Properties of the Closure -- 5.4 Representation of Bilinear Forms on Ew×Ew by Linear Operators -- 5.5 Bibliographical Notes -- Functions of Some Self-adjoint LinearOperators on Ew -- 6.1 Introduction -- 6.2 Products and Sums of Diagonal Operators -- 6.3 Integer Powers of Diagonal Operators -- 6.4 Functions of Self-Adjoint Operators -- 6.5 Functions of Some Symmetric Square Matrices Over Qp×Qp -- 6.5.1 The Powers of the Matrix T -- 6.5.2 Exponential of the Matrix T -- 6.6 Open Problems -- 6.7 Bibliographical Notes -- One-Parameter Family of Bounded LinearOperators on Free Banach Spaces -- 7.1 Introduction -- 7.2 Basic Definitions -- 7.3 Properties of non-Archimedean C0-Groups -- 7.4 Existence of Solutions to Some p-adic Differential Equations -- 7.5 Open Problems -- 7.6 Bibliographical Notes -- References -- Index.