Spectral Theory of Self-Adjoint Operators in Hilbert Space

It isn't that they can't see the solution. It is Approach your problems from the right end that they can't see the problem. and begin with the answers. Then one day, perhaps you will find the final question. G. K. Chesterton. The Scandal of Father 'The Hermit Clad in Crane Feathers' in R. Brown 'The point of a Pin'. van Gulik's The Chinese Maze Murders. Growing specialization and diversification have brought a host of monographs and textbooks on increasingly specialized topics. However, the "tree" of knowledge of mathematics and related fields does not grow only by putting forth new branches. It also happens, quite often in fact, that branches which were thought to be com­ pletely disparate are suddenly seen to be related. 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; algebraic geometry interacts with physics; the Minkowsky lemma, coding theory and the structure of water meet one another in packing and covering theory; quantum fields, crystal defects and mathematical programming profit from homotopy theory; Lie algebras are relevant to filtering; and prediction and electrical engineering can use Stein spaces. And in addition to this there are such new emerging subdisciplines as "experimental mathematics", "CFD", "completely integrable systems", "chaos, synergetics and large-scale order" , which are almost impossible to fit into the existing classification schemes. They draw upon widely different sections of mathematics.

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.