Author | Putnam, C. R. author |
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Title | Commutation Properties of Hilbert Space Operators and Related Topics [electronic resource] / by C. R. Putnam |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1967 |
Connect to | http://dx.doi.org/10.1007/978-3-642-85938-0 |
Descript | XII, 168 p. online resource |
I. Commutators of bounded operators -- 1.1 Introduction -- 1.2 Structure of commutators of bounded operators -- 1.3 Commutators C = AB?BA with AC = CA -- 1.4 Multiplicative commutators -- 1.5 Commutators and numerical range -- 1.6 Some results on normal operators -- 1.7 Operator equation BX?XA= Y -- II. Commutators and spectral theory -- 2.1 Introduction -- 2.2 Spectral properties -- 2.3 Absolute continuity and measure of spectrum -- 2.4 Absolute continuity and numerical range -- 2.5 Higher order commutators -- 2.6 Further results on commutators and normal operators -- 2.7 Half-bounded operators and unitary equivalence -- 2.8 Half-boundedness and absolute continuity -- 2.9 Applications -- 2.10 Commutators of self-adjoint operators -- 2.11 Examples -- 2.12 More on non-negative perturbations and spectra -- 2.13 Commutators of self-adjoint operators -- 2.14 An application to quantum mechanics -- III. Semi-normal operators -- 3.1 Introduction -- 3.2 Structure properties -- 3.3 Spectrum of a semi-normal operator -- 3.4 Further spectral properties -- 3.5 An integral formula -- 3.6 Isolated parts of sp (T) -- 3.7 Measure of sp (T) -- 3.8 Zero measure of sp (T) and normality -- 3.9 Special products of self-adjoint operators -- 3.10 Resolvents of semi-normal operators -- 3.11 Semi-normal operators and arc spectra -- 3.12 TT* ? T*T of one-dimensional range -- 3.13 An example concerning T2 -- 3.14 Subnormal operators -- IV. Commutation relations in quantum mechanics -- 4.1 Introduction -- 4.2 Unitary groups itP and eisQ -- 4.3 Von Neumannโs theorem -- 4.4 The equation AA* = A*A+I -- 4.5 The operators P and Q -- 4.6 Results of Rellich and Dixmier -- 4.7 Results of Tillmann -- 4.8 Results of Foia?, Gehรฉr and Sz.-Nagy -- 4.9 A result of Kato -- 4.10 Results of Kristensen, Mejlbo and Poulsen -- 4.11 Systems with n(< ?) degrees of freedom -- 4.12 Anticommutation relations -- 4.13 General systems -- 4.14 A uniqueness theorem -- 4.15 Existence of the vacuum state -- 4.16 Self-adjointness of ?A?*A? -- 4.17 Remarks on commutators and the equations of motion -- V. Wave operators and unitary equivalence of self-adjoint operators -- 5.1 Introduction and a basic theorem -- 5.2 Schmidt and trace classes -- 5.3 Some lemmas -- 5.4 One-dimensional perturbations -- 5.5 Perturbations by operators of trace class -- 5.6 Invariance of wave operators -- 5.7 Generalizations -- 5.8 Applications to differential operators -- 5.9 A sufficient condition for the existence of Wยฑ(H1, H0) -- 5.10 Hamiltonian operators -- 5.11 Existence of Wยฑ for the Hamiltonian case -- 5.12 A criterion for self-adjointness of perturbed operators -- 5.13 Existence and properties of wave and scattering operators -- 5.14 Stationary approach to scattering -- 5.15 Non-negative perturbations -- 5.16 Hamiltonians and non-negative perturbations -- 5.17 Remarks on unitary equivalence -- VI. Laurent and Toeplitz operators, singular integral operators and Jacobi matrices -- 6.1 Laurent and Toeplitz operators -- 6.2 A spectral inclusion theorem -- 6.3 A special Toeplitz matrix -- 6.4 Spectra of self-adjoint Toeplitz operators -- 6.5 Two lemmas -- 6.6 Analytic and coanalytic Toeplitz operators -- 6.7 Absolute continuity of Toeplitz operators -- 6.8 Spectral resolutions for certain Toeplitz operators -- 6.9 Some results for unbounded operators -- 6.10 Hilbert matrix -- 6.11 Singular integral operators -- 6.12 A(h, ?, E) with E bounded -- 6.13 The norm of A(0, ?, E) -- 6.14 An estimate of meas sp (A(h, ?, E)) -- 6.15 Remarks -- 6.16 Absolute continuity -- 6.17 Other singular integrals -- 6.18 Reducing spaces of A(0, ?, E) -- 6.19 Estimates for ?? and ?? -- 6.20 Spectral representation for A(0,1, (a, b)) -- 6.21 Remarks on the spectra of singular integral operators -- 6.22 Jacobi matrices and absolute continuity -- Symbol Index -- Author Index