Hostname: page-component-848d4c4894-hfldf Total loading time: 0 Render date: 2024-05-08T20:10:48.327Z Has data issue: false hasContentIssue false
  • English
  • Français

The Spectra of Semi-Normal Singular Integral Operators

Published online by Cambridge University Press:  20 November 2018

C. R. Putnam
C. R. Putnam*
Affiliation:
Purdue University, Lafayette, Indiana
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.

Suppose that

(1.1)

and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by

(1.2)

the integral being a Cauchy principal value

It is seen that

(1.3)

or, equivalently,

(1.4)

Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 1 , 01 February 1970 , pp. 134 - 150
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Atkinson, F. V., The normal solubility of linear equations in normed spaces, Mat. Sb. (N.S.) 28 (70) (1951), 3–14.Google Scholar
2. Coburn, L. A., WeyVs theorem for nonnormal operators, Michigan Math. J. 13 (1966), 285–288.Google Scholar
3. Pincus, J. D., Commutators, generalized eigenfunction expansions and singular integral operators, Trans. Amer. Math. Soc. 121 (1966), 358–377.Google Scholar
4. Putnam, C. R., Commutation properties of Hilbert space operators and related topics Ergebnisse der Math, und ihrer Grenzgebiete, Band 36 (Springer-Verlag, New York, 1967).Google Scholar
5. Rosenblum, M., A spectral theory for self-adjoint singular integral operators, Amer. J. Math. 88 (1966), 314–328.Google Scholar
6. Schechter, M., Invariance of the essential spectrum, Bull. Amer. Math. Soc. 71 (1965), 365–367.Google Scholar
7. Schwartz, J., Some results on the spectra and spectral resolutions of a class of singular integral operators, Comm. Pure Appl. Math. 15 (1962), 75–90.Google Scholar
8. Stampfli, J. G., Hyponormal operators and spectral density, Trans. Amer. Math. Soc. 117 (1965), 469–476.Google Scholar
9. Stampfli, J. G., Minimal range theorems for operators with thin spectra, Pacific J. Math. 23 (1967), 601–612.Google Scholar
10. Wolf, F., On the invariance of the essential spectrum under a change of boundary conditions of partial differential boundary operators, Indag. Math. 21 (1959), 142–147.Google Scholar
11. Yoshino, T., Spectral resolution of a hyponormal operator with the spectrum on a curve, Töhoku Math. J. 19 (1967), 86–97.Google Scholar