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Semigroups of operators and an application to spectral theory

Published online by Cambridge University Press:  24 October 2008

W. Ricker
W. Ricker
Affiliation:
University of Adelaide, Australia
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Extract

A problem of fundamental importance in Spectral Theory consists of finding criteria for an operator to be of scalar-type in the sense of N. Dunford [1]. One relatively general approach in determining such criteria is based on the method of integral transforms (see for example [4], [5], [6], [11], [12]). For example, if X is a Banach space and T is a continuous linear operator on X, then the group {eitT; t real} exists. As noted by several authors (e.g. [4], [6]), this group can then be effectively used for analysing the operator T.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 1 , July 1984 , pp. 143 - 149
Copyright
Copyright © Cambridge Philosophical Society 1984

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References

[1] Dunford, N. and Schwartz, J.. Linear operators. III (Interscience Publishers 1971).Google Scholar
[2] Hille, E.. On semi-groups of transformations in Hilbert space. Proc. Nat. Acad. Sci. U.S.A. 24 (1938), 159–161.CrossRefGoogle ScholarPubMed
[3] Hille, E. and Phillips, R. S.. Functional analysis and semi-groups (Amer. Math. Soc. Colloq. Publications, XXXI, 1957).Google Scholar
[4] Kantorovitz, S.. On the characterization of spectral operators. Trans. Amer. Math. Soc. 111 (1964), 152–181.CrossRefGoogle Scholar
[5] Kantorovitz, S.. Characterization of unbounded spectral operators with spectrum in a half-line. Comment. Math. Helv. 56 (1981), 163–178.CrossRefGoogle Scholar
[6] Kluv´nek, I.. Characterization of Fourier-Stieltjes transforms of vector and operator valued measures. Czechoslovak Math. J. 17 (92) (1967), 261–276.CrossRefGoogle Scholar
[7] Kluv´nek, I. and Knowles, G.. Vector Measures and Control Systems (North Holland, 1976).Google Scholar
[8] Lewis, D. R.. Integration with respect to vector measures. Pacific J. Math. 33 (1970), 157–165.Google Scholar
[9] Neubauer, G.. Zur Spektraltheorie in lokalkonvexen Algebren. Math. Ann. 142 (1961), 131–164.CrossRefGoogle Scholar
[10] Plesner, A. I.. Spectral Theory of Linear Operators vol. II (Frederick Ungar, 1969).Google Scholar
[11] Ricker, W.. Extended spectral operators. J. Operator Theory 9 (1983), 269–296.Google Scholar
[12] Ricker, W.. Characterization of Stieltjes transforms of vector measures and an application to spectral theory (preprint).Google Scholar
[13] Schaefer, H. H.. Spectral measures in locally convex algebras. Acta Math. 107 (1962), 125–173.CrossRefGoogle Scholar
[14] Walsh, B.. Structure of spectral measures on locally convex spaces. Trans. Amer. Math. Soc. 120 (1965), 295–326.CrossRefGoogle Scholar
[15] Whitford, A. K.. Laplace—Stieltjes transforms of vector-valued measures. Matematický Časopis 22 (1972), 156–163.Google Scholar
[16] Widder, D. V.. The Laplace Transform (Princeton University Press, 1946).Google Scholar