Author | Birman, M. S. author |
---|---|
Title | Spectral Theory of Self-Adjoint Operators in Hilbert Space [electronic resource] / by M. S. Birman, M. Z. Solomjak |
Imprint | Dordrecht : Springer Netherlands, 1987 |
Connect to | http://dx.doi.org/10.1007/978-94-009-4586-9 |
Descript | XVI, 302 p. online resource |
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 Neumannโs 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 Measures whose 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 Schrรถdinger Operator -- 11 Classes of Compact Operators -- 1. Canonical Representation and Singular Numbers of Compact Operators -- 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. Lidskiiโs 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