Hostname: page-component-848d4c4894-sjtt6 Total loading time: 0 Render date: 2024-06-27T15:07:10.459Z Has data issue: false hasContentIssue false
  • English
  • Français

Adjoint Abelian Operators on Banach Space

Published online by Cambridge University Press:  20 November 2018

J. G. Stampfli
J. G. Stampfli*
Affiliation:
Indiana University, Bloomington, 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.

I. In the first part of this paper we introduce a new class of operators, mentioned in the title. It is easy to say that these are a generalization of self-adjoint operators for Hilbert space. This is deceptive since it implies that the definition of self-adjointness is forced into the unnatural setting of a Banach space. We feel that the definition of adjoint abelian preserves the obvious distinction between a space and its dual. Certain attractive properties of self-adjoint operators have already been singled out and carried over to Banach space. Specifically, we mention the notion of hermitian (see 17; 11), and spectral type operators (see 4). There is some comparison of these concepts later.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 505 - 512
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Bartle, R. G., Spectral localization of operators in Banach spaces, Math. Ann. 153 (1964), 261–269.Google Scholar
2. Bartle, R. G., Dunford, N., and Schwartz, J., Weak compactness and vector measures, Can. J. Math. 7 (1955), 289–305.Google Scholar
3. Berkson, E., A characterization of scalar operators on reflexive Banach spaces, Pacific J. Math. 13 (1963), 365–373.Google Scholar
4. Dunford, N., Spectral operators, Pacific J. Math. 4 (1954), 321–354.Google Scholar
5. Fixman, U., Problems in spectral operators, Pacific J. Math. 9 (1959), 1029–1059.Google Scholar
6. Foguel, S. R., The relations between a spectral operator and its scalar part, Pacific J. Math. 8 (1958), 51–65.Google Scholar
7. Gelfand, I. M., Abstrakte Funktionen und lineare Operatoren, Mat. Sb. (N.S.) (4) 46 (1938), 235–286.Google Scholar
8. Grothendieck, A., Sur les applications linéaires faiblement compactes d'espace du type C(K), Can. J. Math. 5 (1953), 129–173.Google Scholar
9. Kantorovitz, S., Classification of operators by means of their operational calculus, Trans. Amer. Math. Soc. 115 (1965), 194–224.Google Scholar
10. Loomis, L. A., An introduction to abstract harmonic analysis (Van Nostrand, New York, 1953).Google Scholar
11. Lumer, G., Semi-inner-product spaces, Trans. Amer. Math. Soc. 100 (1961), 29–43.Google Scholar
12. Lumer, G., Spectral operators, hermitian operators and bounded groups, Acta Sci. Math. 25 (1964), 75–85.Google Scholar
13. Maeda, F.-Y., Generalized scalar operators whose spectra are contained in a Jordan curve, Illinois J. Math. 10 (1966), 431–459.Google Scholar
14. Pettis, B. J., Absolutely continuous functions in vector spaces, Bull. Amer. Math. Soc. 45 (1939), 677.Google Scholar
15. Stampfli, J. G., Roots of scalar operators, Proc. Amer. Math. Soc. 13 (1962), 796–798.Google Scholar
16. Tillmann, H. G., Eine Erweiterung des Funktionalkalkùls fur lineare Operatoren, Math. Ann. 151 (1963), 424–430.Google Scholar
17. Vidav, I., Eine metrische Kennzeichnung der selbstadjungierten Operatoren, Math. Z. 66 (1956), 121–128.Google Scholar
18. Williams, J. P., Schwarz norms for operators, Pacific J. Math. 24 (1968), 181–188.Google Scholar