Author | Weidmann, Joachim. author |
---|---|
Title | Linear Operators in Hilbert Spaces [electronic resource] / by Joachim Weidmann |
Imprint | New York, NY : Springer US, 1980 |
Connect to | http://dx.doi.org/10.1007/978-1-4612-6027-1 |
Descript | XIV, 402 p. online resource |
1 Vector spaces with a scalar product, pre-Hilbert spaces -- 1.1 Sesquilinear forms -- 1.2 Scalar products and norms -- 2 Hilbert spaces -- 2.1 Convergence and completeness -- 2.2 Topological notions -- 3 Orthogonality -- 3.1 The projection theorem -- 3.2 Orthonormal systems and orthonormal bases -- 3.3 Existence of orthonormal bases, dimension of a Hilbert space -- 3.4 Tensor products of Hilbert spaces -- 4 Linear operators and their adjoints -- 4.1 Basic notions -- 4.2 Bounded linear operators and functionals -- 4.3 Isomorphisms, completion -- 4.4 Adjoint operator -- 4.5 The theorem of Banach-Steinhaus, strong and weak convergence -- 4.6 Orthogonal projections, isometric and unitary operators -- 5 Closed linear operators -- 5.1 Closed and closable operators, the closed graph theorem -- 5.2 The fundamentals of spectral theory -- 5.3 Symmetric and self-adjoint operators -- 5.4 Self-adjoint extensions of symmetric operators -- 5.5 Operators defined by sesquilinear forms (Friedrichsโ{128}{153} extension) -- 5.6 Normal operators -- 6 Special classes of linear operators -- 6.1 Finite rank and compact operators -- 6.2 Hilbert-Schmidt operators and Carleman operators -- 6.3 Matrix operators and integral operators -- 6.4 Differential operators on L2(a, b) with constant coefficients -- 7 The spectral theory of self-adjoint and normal operators -- 7.1 The spectral theorem for compact operators, the spaces Bp (H1H2) -- 7.2 Integration with respect to a spectral family -- 7.3 The spectral theorem for self-adjoint operators -- 7.4 Spectra of self-adjoint operators -- 7.5 The spectral theorem for normal operators -- 7.6 One-parameter unitary groups -- 8 Self-adjoint extensions of symmetric operators -- 8.1 Defect indices and Cayley transforms -- 8.2 Construction of self-adjoint extensions -- 8.3 Spectra of self-adjoint extensions of a symmetric operator -- 8.4 Second order ordinary differential operators -- 8.5 Analytic vectors and tensor products of self-adjoint operators -- 9 Perturbation theory for self-adjoint operators -- 9.1 Relatively bounded perturbations -- 9.2 Relatively compact perturbations and the essential spectrum -- 9.3 Strong resolvent convergence -- 10 Differential operators on L2(?m) -- 10.1 The Fourier transformation on L2(?m) -- 10.2 Sobolev spaces and differential operators on L2(?m) with constant coefficients -- 10.3 Relatively bounded and relatively compact perturbations -- 10.4 Essentially self-adjoint Schrรถdinger operators -- 10.5 Spectra of Schrรถdinger operators -- 10.6 Dirac operators -- 11 Scattering theory -- 11.1 Wave operators -- 11.2 The existence and completeness of wave operators -- 11.3 Applications to differential operators on L2(?m) -- A.1 Definition of the integral -- A.2 Limit theorems -- A.3 Measurable functions and sets -- A.4 The Fubini-Tonelli theorem -- A.5 The Radon-Nikodym theorem -- References -- Index of symbols -- Author and subject index