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Gentle Perturbations of with Application to

Published online by Cambridge University Press:  20 November 2018

N. A. Derzko
N. A. Derzko*
Affiliation:
University of Toronto, Toronto, Ontario
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The theory of gentle perturbations was introduced by Friedrichs [3] as a tool to study the perturbation theory of the absolutely continuous spectrum of a self-adjoint operator H0 and developed in an abstract form by Rejto [7; 8]. Two examples of gentle structures are well knowTn. In the first of these, the gentle operators have Hölder continuous complex or operator-valued kernels, and in the second, the kernels are Fourier transforms of L1 functions [4].

The gentle structure has traditionally been verified in the case when H0 is in its spectral representation, that is, when H0 is the simple differentiation operator. This is not the natural setting for the second example mentioned above where one should consider the simple differentiation operator in a suitable L2-space and perturbations with L1 kernels.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 5 , 01 October 1970 , pp. 1055 - 1070
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Dunford, N. and Schwartz, J. T., Linear operators, Part I. General theory, Pure and Applied Mathematics, Vol. 7 (Interscience, New York, 1958); Part II. Spectral theory. Self adjoint operators in Hilbert space (Interscience, New York, 1963).Google Scholar
2. A., Erdélyi, Editor, Tables of integral transforms (Bateman Project, McGraw-Hill, New York, 1954).Google Scholar
3. Friedrichs, K. O., On the perturbation of continuous spectra, Comm. Pure Appl. Math. 4 (1948), 361–406.Google Scholar
4. Friedrichs, K. O., Perturbation of spectra in Hilbert space, Lectures in Applied Mathematics, Proc. Summer Seminar, Boulder, Colorado, 1960, Vol. III (Amer. Math. Soc, Providence, R.I., 1965).Google Scholar
5. Hille, E. and Phillips, R. S., Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., Vol. 31 (Amer. Math. Soc, Providence, R.I., 1967).Google Scholar
6. Magnus, W. and Oberhettinger, F., Formulas and theorems for the special functions of mathematical physics (Chelsea, New York, 1949).Google Scholar
7. Rejto, P., On gentle perturbations. I, Comm. Pure Appl. Math. 16 (1963), 279–303.Google Scholar
8. Rejto, P., On gentle perturbations, II. Comm. Pure Appl. Math. 17 (1964), 257–292.Google Scholar
9. Rejto, P., On partly gentle perturbations. I, J. Math. Anal. Appl. 17 (1967), 435–462.Google Scholar
10. Rejto, P., On partly gentle perturbations. II, J. Math. Anal. Appl. 20 (1967), 145–187.Google Scholar
11. Rejto, P., On partly gentle perturbations. III, J. Math. Anal. Appl. 27 (1969), 21–67.Google Scholar