Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-15T08:48:41.563Z Has data issue: false hasContentIssue false
  • English
  • Français

On a Class of Non-Self-Adjoint Differential Operators

Published online by Cambridge University Press:  20 November 2018

R. R. D. Kemp
R. R. D. Kemp*
Affiliation:
Queen's University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The problem of spectral analysis of non-self-adjoint (and non-normal) operators has received considerable attention recently. Livšic (5), and more recently Brodskii and Livšic (1) have considered operators on Hilbert space with completely continuous imaginary parts. Dunford (3) has generalized the notion of spectral measure and defined a class of spectral operators on Hilbert and Banach space. Schwartz (8) and Rota (7) have investigated conditions under which a differential operator will be spectral. The work of Naimark (6) and the author (4) on non-self-adjoint differential operators leads to an expansion theorem which implicitly defines a type of spectral measure. However the projections involved in this will not in general be bounded, much less uniformly bounded.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 12 , 1960 , pp. 641 - 659
Copyright
Copyright © Canadian Mathematical Society 1960

References

1. Brodskiï, M. S. and Livsic, M.S., Spectral analysis of non-self-adjoint operators and intermediate systems, Usp. Mat. Nauk. (N. S.), 13, no. 1 (79) (1958), 3–85.Google Scholar
2. Coddington, E.A. and Levinson, N., Theory of ordinary differential equations (New York, 1955).Google Scholar
3. Dunford, N., Spectral operators, Pac. J. Math., 4 (1954), 321–354.Google Scholar
4. Kemp, R.R.D., A singular boundary value problem for a non-self-adjoint differential operator, Can. J. Math., 10 (1958), 447–462.Google Scholar
5. Livsic, M.S., On spectral decomposition of linear non-s elf-adjoint operators, Amer. Math. Soc. Translations (2), 5 (1957), 67–114.Google Scholar
6. Naimafk, M.A., Investigation of the spectrum and expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, Trudy Moskov Mat. Obsc, 3 (1954), 181–270.Google Scholar
7. Rota, G.C., Extension theory of differential operators I, Comm. Pure and App. Math., 11 (1958), 23–66.Google Scholar
8. Schwartz, J., Perturbations of spectral operators, and applications I, bounded perturbations, Pac. J. Math., 4 (1954), 415–458.Google Scholar