Spectral Theory of Operators in Hilbert Space
Published on 1. January 1973
Book
Paperback/Softback
IX, 245 pages
978-0-387-90076-6 (ISBN)
Description
The present lectures intend to provide an introduction to the spectral analysis of self-adjoint operators within the framework of Hilbert space theory. The guiding notion in this approach is that of spectral representation. At the same time the notion of function of an operator is emphasized. The formal aspects of these concepts are explained in the first two chapters. Only then is the notion of Hilbert space introduced. The following three chapters concern bounded, completely continuous, and non-bounded operators. Next, simple differential operators are treated as operators in Hilbert space, and the final chapter deals with the perturbation of discrete and continuous spectra. The preparation of the original version of these lecture notes was greatly helped by the assistance of P. Rejto. Various valuable suggestions made by him and by R. Lewis have been incorporated. The present version of the notes contains extensive modifica tions, in particular in the chapters on bounded and unbounded operators. February, 1973 K.O.F. PREFACE TO THE SECOND PRINTING The second printing (1980) is a basically unchanged reprint in which a number of minor errors were corrected. The author wishes to thank Klaus Schmidt (Lausanne) and John Sylvester (New York) for their lists of errors. v TABLE OF CONTENTS I. Spectral Representation 1 1. Three typical problems 1 12 2. Linear space and functional representation.
More details
Series
Edition
1st ed. 1973. Corr. 2nd printing 1981
Language
English
Place of publication
New York
United States
Target group
Professional and scholarly
Research
Illustrations
IX, 245 p.
Dimensions
Height: 242 mm
Width: 170 mm
Thickness: 15 mm
Weight
451 gr
ISBN-13
978-0-387-90076-6 (9780387900766)
DOI
10.1007/978-1-4612-6396-8
Schweitzer Classification
Content
I. Spectral Representation.- 1. Three typical problems.- 2. Linear space and functional representation. Linear operators.- 3. Spectral representation.- 4. Functional calculus.- 5. Differential equations.- II. Norm and Inner Product.- 6. Normed spaces.- 7. Inner product.- 8. Inner products in function spaces.- 9. Formally self-adjoint operators.- 10. Adjoint operators in function spaces.- 11. Orthogonality.- 12. Orthogonal projection.- 13. Remarks about the role of self-adjoint operators in physics.- III. Hilbert Space.- 14. Completeness.- 15. First extension theorem. Ideal functions.- 16. Fourier transformation.- 17. The projection theorem.- 18. Bounded forms.- IV. Bounded Operators.- 19. Operator inequalities, operator norm, operator convergence.- 20. Integral operators.- 21. Functions of bounded operators.- 22. Spectral representation.- 23. Normal and unitary operators.- V. Operators with Discrete Spectra.- 24. Operators with partly discrete spectra.- 25. Completely continuous operators.- 26. Completely continuous integral operators.- 27. Maximum-minimum properties of eigenvalues.- VI. Non-Bounded Operators.- 28. Closure and adjointness.- 29. Closed forms.- 30. Spectral resolution of self-adjoint operators.- 31. Closeable forms.- VII. Differential Operators.- 32. Regular differential operators.- 33. Ordinary differential operators in a semi-bounded domain.- 34. Partial differential operators.- 35. Partial differential operators with boundary conditions.- 36. Partial differential operators with discrete spectra.- VIII. Perturbation of Spectra.- 37. Perturbation of discrete spectra.- 38. Perturbation of continuous spectra.- 39. Scattering.- References.