This English edition is almost identical to the German original Lineare Operatoren in Hilbertriiumen, published by B. G. Teubner, Stuttgart in 1976. A few proofs have been simplified, some additional exercises have been included, and a small number of new results has been added (e.g., Theorem 11.11 and Theorem 11.23). In addition a great number of minor errors has been corrected. Frankfurt, January 1980 J. Weidmann vii Preface to the German edition The purpose of this book is to give an introduction to the theory of linear operators on Hilbert spaces and then to proceed to the interesting applica­ tions of differential operators to mathematical physics. Besides the usual introductory courses common to both mathematicians and physicists, only a fundamental knowledge of complex analysis and of ordinary differential equations is assumed. The most important results of Lebesgue integration theory, to the extent that they are used in this book, are compiled with complete proofs in Appendix A. I hope therefore that students from the fourth semester on will be able to read this book without major difficulty. However, it might also be of some interest and use to the teaching and research mathematician or physicist, since among other things it makes easily accessible several new results of the spectral theory of differential operators.

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 (Friedrichs' 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 extensionsof 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 Schrödinger operators.- 10.5 Spectra of Schrödinger 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.