Functional Analysis

12 General Theory of Unbounded Operators in Hilbert Spaces.- 1 Definition of an Unbounded Operator. The Graph of an Operator.- 1.1 Definitions.- 1.2 Graphs of Operators.- 2 Closed and Closable Operators. Differential Operators.- 2.1 Closed Operators.- 2.2 Closable Operators.- 2.3 Differential Operators.- 3 The Adjoint Operator.- 3.1 Definition and Properties of the Adjoint Operator.- 3.2 The Second Adjoint Operator.- 3.3 The Closed Graph Theorem.- 4 Defect Numbers of General Operators.- 4.1 Deficient Subspaces.- 4.2 Defect Numbers.- 5 Hermitian and Selfadjoint Operators. General Theory.- 5.1 Hermitian Operators.- 5.2 Criterion of Selfadjointness.- 5.3 Semibounded Operators.- 6 Isometric and Unitary Operators. Cayley Transformation.- 6.1 Defect Numbers of Isometric Operators.- 6.2 Direct Cayley Transformation.- 6.3 Inverse Cayley Transformation.- 7 Extensions of Hermitian Operators to Selfadjoint Operators.- 7.1 The Construction of Extensions.- 7.2 Von Neumann Formulas.- 13 Spectral Decompositions of Selfadjoint, Unitary, and Normal Operators. Criteria of Selfadjointness.- 1 The Resolution of the Identity and Its Properties.- 1.1 The Resolution of the Identity.- 1.2 Theorem on Extension.- 2 The Construction of Spectral Integrals.- 2.1 Integrals of Simple Functions.- 2.2 Integrals of Bounded Measurable Functions.- 2.3 Integrals of Unbounded Measurable Functions.- 2.4 Other Properties of Spectral Integrals.- 3 Image of a Resolution of the Identity. Change of Variables in Spectral Integrals. Product of Resolutions of the Identity.- 3.1 Image of a Resolution of the Identity.- 3.2 Product of Resolutions of the Identity.- 4 Spectral Decomposition of Bounded Selfadjoint Operators.- 4.1 The Spectral Theorem.- 4.2 Functions of Operators and Their Spectrum.- 5 Spectral Decompositions for Unitary and Bounded Normal Operators.- 5.1 Spectral Theorem for Unitary Operators.- 5.2 Spectral Theorem for Normal Operators.- 6 Spectral Decompositions of Unbounded Operators.- 6.1 Selfadjoint Operators.- 6.2 Stone’s Formula.- 6.3 Commuting Operators.- 6.4 The Function E?.- 6.5 The Case of Normal Operators.- 7 Spectral of Representation One-Parameter Unitary Groups and Operator Differential Equations.- 7.1 Stone’s Theorem.- 7.2 Operator Differential Equations.- 8 Evolutionary Criteria of Selfadjointness.- 8.1 The Schrödinger Criterion of Selfadjointness.- 8.2 The Hyperbolic Criterion of Selfadjointness.- 8.3 The Parabolic Criterion of Selfadjointness.- 9 Quasianalytic Criteria of Selfadjointness and Commutability.- 9.1 The Quasianalytic Criterion of Selfadjointness.- 9.2 Other Criteria of Selfadjointness.- 9.3 Commutability of Operators.- 10 Selfadjointness of Perturbed Operators.- 14 Rigged Spaces.- 1 Hilbert Riggings.- 1.1 Positive and Negative Norms.- 1.2 Operators in Chains.- 2 Rigging of Hilbert Spaces by Linear Topological Spaces.- 2.1 Topological Spaces.- 2.2 Projective Limits of Spaces.- 2.3 Riggings Constructed by Using Projective Limits.- 3 Sobolev Spaces in Bounded Domains.- 3.1 The ?-Function.- 3.2 Embeddings of Sobolev Spaces.- 4 Sobolev Spaces in Unbounded Domains. Classical Spaces of Test Functions.- 4.1 The ?-Function.- 4.2 Embeddings of Weighted Sobolev Spaces.- 4.3 The Classical Spaces of Test Functions.- 5 Tensor Products of Spaces.- 5.1 Tensor Products of Spaces.- 5.2 Tensor Products of Operators.- 5.3 Tensor Products of Chains.- 5.4 Projective Limits.- 6 The Kernel Theorem.- 6.1 Hilbert Riggings.- 6.2 Nuclear Riggings.- 6.3 Bilinear Forms.- 6.4 One More Kernel Theorem.- 7 Completions of a Space with Respect to Two Different Norms.- 7.1 Completions with Respect to Two Different Norms.- 7.2 Examples.- 8 Semibounded Bilinear Forms.- 8.1 Lemma on Hilbert Riggings.- 8.2 Positive Forms.- 8.3 Semibounded Forms.- 8.4 Form Sums of Operators.- 15 Expansion in Generalized Eigenvectors.- 1 Differentiation of Operator-Valued Measures and Resolutions of the Identity.- 1.1 Differentiation of Operator-Valued Measures.- 1.2 Differentiation of a Resolution of the Identity.- 1.3 The Case of a Nuclear Rigging.- 2 Generalized Eigenvectors and the Projection Spectral Theorem.- 2.1 The Case of a Selfadjoint Operator.- 2.2 The Case of a Normal Operator.- 2.3 Families of Commuting Operators.- 2.4 Cyclic Vectors.- 3 Fourier Transformation in Generalized Eigenvectors and the Direct Integral of Hilbert Spaces.- 3.1 Fourier Transformation.- 3.2 The Direct Integral of Hilbert Spaces.- 4 Expansion in Eigenfunctions of Càrleman Operators.- 4.1 The Inverse Theorem.- 4.2 Nonquasinuclear Riggings.- 4.3 Càrleman Operators.- 16 Differential Operators.- 1 Theorem on Isomorphisms for Elliptic Operators.- 1.1 Preliminary Information.- 1.2 The Principal Result.- 2 Local Smoothing of Generalized Solutions of Elliptic Equations.- 2.1 Generalized Solutions Inside a Domain.- 2.2 Smoothing Inside a Domain.- 2.3 Smoothing up to the Boundary.- 3 Elliptic Differential Operators in a Domain with Boundary.- 3.1 The Case of a Bounded Domain.- 3.2 The Case of an Unbounded Domain.- 4 Differential Operators in ?N.- 4.1 The Operator of Multiplication.- 4.2 Perturbation of an Operator.- 4.3 Expressions with Constant Coefficients.- 4.4 Semibounded Expressions.- 4.5 Nonsmooth Potentials.- 4.6 The Sochrödinger Operator as a Form Sum.- 5 Expansion in Eigenfunctions and Green’s Function of Elliptic Differential Operators.- 5.1 Generalized Eigenfunctions of Differential Operators.- 5.2 Green’s Function (Kernel of the Resolvent).- 5.3 The Càrleman Property of Elliptic Operators.- 5.4 The Laplace Operator.- 6 Ordinary Differential Operators.- 6.1 Theorem on Smoothing of Solutions.- 6.2 Selfadjointness of Differential Operators.- 6.3 Green’s Function.- 6.4 Expansion in Generalized Eigenfunctions.- 6.5 The Spectral Matrix.- 6.6 Classical Fourier Transformation.- Bibliographical Notes.- References.