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| Volume I Preface Table of Contents Volume I Volume II Introduction Part 1 Mathematical Foundations of Linear Hydrodynamics . . Chapter 1: Operators on Hilbert Spaces 1.1 General Facts 1.1.1 The Concept of a Hilbert Space 1.1.2 The Space L2 1.1.3 Orthogonality. Projection onto a Subspace 1.1.4 Equivalent Norms 1.1.5 Linear Functionals. Riesz Theorem 1.1.6 Embeddings of Spaces. Riesz Theorem for Equipments 1.1.7 Orthonormal Systems and Bases 1.1.8 Bounded Linear Operators 1.1.9 Adjoint Operators 1.1.10 Self-Adjoint Operators 1.1.11 Self-Adjoint Compact Operators 1.1.12 Compact operators, s-numbers 1.1.13 Riesz Bases and p-Bases 1.1.14 Direct Sum of Subspaces. Invariant Subspaces 1.1.15 Eigen- and Associated (Root) Elements. Root Subspaces . 1.1.16 Unbounded Linear Operators 1.1.17 Resolvent and Spectrum of a Linear Operator 1.1.18 Classification of Points in the Spectrum of a Linear Operator 1.1.19 Spectrum of a Self-Adjoint Operator. Weyl Theorem . . . 1.1.20 Riesz Projections 1.1.21 Symmetric and Self-Adjoint Operators 1.1.22 Spectral Decomposition of Self-Adjoint Operators. Functions of Operators 1.1.23 Spaces with Degenerate Scalar Products. Seminorms . . . 1.1.24 Equivalent Corrections of Seminorms 1.2 Sobolev Spaces 1.2.1 Finite Functions 1.2.2 Generalized Derivatives 1.2.3 The Definition of Sobolev Spaces 1.2.4 The Space L2. Regions of the First Type . 1.2.5 The Subspace H01 1.2.6 Embedding H1 into L2 Regions of the Second Type 1.2.7 The Trace Operator. Regions of the Third Type 1.3 Spaces With Indefinite Metrics 1.3.1 J-Spaces 1.3.2 Uniformly Definite Subspaces 1.3.3 J-Orthonormal Systems and Bases 1.3.4 Linear Operators on J-Spaces 1.3.5 Invariant Subspaces of J-Self-Adjoint Operators 1.3.6 Pontryagin Spaces 1.3.7 On Completeness and Basicity for the System of Root Elements of a J-Self-Adjoint Operator 1.4 Eigenvalue Problems 1.4.1 Operator B is the Identity Operator 1.4.2 Operator B is Positive Definite 1.4.3 Positivity Condition for a Matrix Operator 1.4.4 Simplifying Equations with an Alternating Operator …… Chapter 2:Fundamental Spaces and Operators of Linear Hydrodynamics Part 2: Motion of Bodies With Cavities Containing Ideal Fluids Chapter 3:Oscillations of a Heavy Ideal Fluid in Stationary and Nonstationay Contatiners …… |
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