Del 5 i serien Mathematics and its Applications

Spectral Theory of Self-Adjoint Operators in Hilbert Space

Inbunden, Engelska, 1987

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Further, the kind and level of sophistication of mathematics applied in various sciences has changed drastically in recent years: measure theory is used (non trivially) in regional and theoretical economics;

Produktinformation

Utgivningsdatum1987-05-31
Mått155 x 235 x 23 mm
Vikt610 g
FormatInbunden
SpråkEngelska
SerieMathematics and its Applications
Antal sidor302
Upplaga1987
FörlagSpringer
ISBN9789027721792

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1. Preliminaries.- 1. Metric Spaces. Normed Spaces.- 2. Algebras and ?-Algebras of Sets.- 3. Countably Additive Functions and Measures.- 4. Measurable Functions.- 5. Integration.- 6. Function Spaces.- 2. Hilbert Space Geometry. Continuous Linear Operators.- 1. Hilbert Space. The Space L2.- 2. Orthonormal Systems.- 3. Projection Theorem. Orthogonal Expansions and Orthogonal Sums.- 4. Linear Functionals and Sesqui-linear Forms. Weak Convergence.- 5. The Algebra of Continuous Operators on H.- 6. Compact Operators.- 7. Bounded Self-adjoint Operators.- 8. Orthogonal Projections.- 9. Examples of Hilbert Spaces and Orthonormal Systems.- 10. Examples of Continuous Functionals and Operators.- 3. Unbounded Linear Operators.- 1. General Notions. Graph of an Operator.- 2. Closed Operators. Closable Operators.- 3. Adjoint Operator.- 4. Domination of Operators.- 5. Invariant Subspaces.- 6. Reducing Subspaces.- 7. Defect Number, Spectrum, and Resolvent of a Closed Operator.- 8. Skew Decompositions. Skew Reducibility.- 9. Spectral Theory of Compact Operators.- 10. Connection between the Spectral Properties of TS and ST.- 4. Symmetric and Isometric Operators.- 1. Symmetric and Self-adjoint Operators. Deficiency Indices.- 2. Isometric and Unitary Operators.- 3. Cayley Transform.- 4. Extensions of Symmetric Operators. Von Neumann’s Formulae.- 5. The Operator T*T. Normal Operators.- 6. Classification of Spectral Points.- 7. Multiplication by the Independent Variable.- 8. Differentiation Operator.- 5. Spectral Measure. Integration.- 1. Basic Notions.- 2. Extension of a Spectral Measure. Product Measures.- 3. Integral with Respect to a Spectral Measure. Bounded Functions.- 4. Integral with Respect to a Spectral Measure. Unbounded Functions.- 5. An Example of Commuting Spectral Measureswhose Product is not Countably Additive.- 6 Spectral Resolutions.- 1. Statements of Spectral Theorems. Functions of Operators.- 2. Spectral Theorem for Unitary Operators.- 3. Spectral Theorem for Self-adjoint Operators.- 4. Spectral Resolution of a One-parameter Unitary Group.- 5. Joint Spectral Resolution for a Finite Family of Commuting Self-adjoint Operators.- 6. Spectral Resolutions of Normal Operators.- 7 Functional Model and the Unitary Invariants of Self-adjoint Operators.- 1. Direct Integral of Hilbert Spaces.- 2. Multiplication Operators and Decomposable Operators.- 3. Generating Systems and Spectral Types.- 4. Unitary Invariants of Spectral Measure.- 5. Unitary Invariants of Self-adjoint Operators.- 6. Decomposition of a Spectral Measure into the Absolutely Continuous and the Singular Part.- 8 Some Applications of Spectral Theory.- 1. Polar Decomposition of a Closed Operator.- 2. Differential Equations of Evolution on Hilbert Space.- 3. Fourier Transform.- 4. Multiplications on L2 (Rm, Cm).- 5. Differential Operators with Constant Coefficients.- 6. Examples of Differential Operators.- 9 Perturbation Theory.- 1. Essential Spectrum. Compact Perturbations.- 2. Compact Self-adjoint and Normal Operators.- 3. Finite-dimensional Perturbations and Extensions.- 4. Continuous Perturbations.- 10 Semibounded Operators and Forms.- 1. Closed Positive Definite Forms.- 2. Semibounded Forms.- 3. Friedrichs Method of Extension of a Semibounded Operator to a Self-adjoint Operator.- 4. Fractional Powers of Operators. The Heinz Inequality.- 5. Examples of Quadratic Forms. The Sturm-Liouville Operator on [?1, 1].- 6. Examples of Quadratic Forms. One-dimensional Schrödinger Operator.- 11 Classes of Compact Operators.- 1. Canonical Representation and Singular Numbers of CompactOperators.- 2. Nuclear Operators. Trace of an Operator.- 3. Hilbert-Schmidt Operators.- 4. Sp Classes.- 5. Additional Information on Singular Numbers of Compact Operators.- 6. ?p Classes.- 7. Lidskii’s Theorem.- 8. Examples of Compact Operators.- 12 Commutation Relations of Quantum Mechanics.- 1. Statement of the Problem. Auxiliary Material.- 2. Properties of (B)-systems and (C)-systems.- 3. Representations of the Bose Relations. The Case m = 1.- 4. Representations of the Bose Relations. General Case.- 5. Representations of the Canonical Relations.