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On Additive Operators

Published online by Cambridge University Press:  20 November 2018

N. A. Friedman  and
A. E. Tong
N. A. Friedman
Affiliation:
State University of New York, Albany, New York
A. E. Tong
Affiliation:
State University of New York, Albany, New York
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Representation theorems for additive functional have been obtained in [2, 4; 6-8; 10-13]. Our aim in this paper is to study the representation of additive operators.

Let S be a compact Hausdorff space and let C(S) be the space of real-valued continuous functions defined on S. Let X be an arbitrary Banach space and let T be an additive operator (see § 2) mapping C(S) into X. We will show (see Lemma 3.4) that additive operators may be represented in terms of a family of “measures” {μh} which take their values in X**. If X is weakly sequentially complete, then {μh} can be shown to take their values in X and are vector-valued measures (i.e., countably additive in the norm) (see Lemma 3.7). And, if X* is separable in the weak-* topology, T may be represented in terms of a kernel representation satisfying the Carathéordory conditions (see [9; 11; §4]):

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 3 , June 1971 , pp. 468 - 480
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Berberian, S. K., Measure and integration (Macmillan, New York, 1965).Google Scholar
2. Chacon, R. V. and Friedman, N. A., Additive junctionals, Arch. Rational Mech. Anal. 18 (1965), 230–240.Google Scholar
3. Day, M. M., Normed linear spaces, second printing, corrected, Ergebnisseder Mathematik und ihrer Grenzgebiete, N. F., Heft 21 (Academic Press, New York; Springer-Verlag, Berlin-Go ttingen-Heidelberg, 1962).Google Scholar
4. Drewnowski, L. and Orlicz, W., On orthogonally additive Junctionals, Bull. Acad. Polon. Sci. 16 (1968), 883–888.Google Scholar
5. Dunford, N. and Schwartz, J. T., Linear operators, Part I: General theory, Interscience, (New York, 1958). Pure and Applied Mathematics, Vol. 7.Google Scholar
6. Friedman, N. A. and Katz, M., A representation theorem for additive junctionals, Arch. Rational Mech. Anal. 21 (1966), 49–57.Google Scholar
7. Friedman, N. A. and Katz, M., Additive junctionals on Lp spaces, Can. J. Math. 18 (1966), 1264–1271.Google Scholar
8. Friedman, N. A. and Katz, M., Qn additive junctionals, Proc. Amer. Math. Soc. 21 (1969), 557–561.Google Scholar
9. Krasnosel'skiï, M. A., Topological methods in the theory oj nonlinear integral equations, translated by Burlak, J. (Macmillan, New York, 1964).Google Scholar
10. Martin, A. D., and Mizel, V. J., A representation theorem jor certain nonlinear junctionals, Arch. Rational Mech. Anal. 15 (1964), 353–367.Google Scholar
11. Mizel, V. J., Representation oj nonlinear transformations on Lp spaces, Bull. Amer. Math. Soc. 75 (1969), 164–168.Google Scholar
12. Mizel, V. J. and Sundaresan, K., Representation oj additive and biadditive junctionals, Arch. Rational Mech. Anal. SO (1968), 102–126.Google Scholar
13. Sundaresan, K., Additive junctionals on Orlicz spaces, Studia Math. 32 (1969), 270–276.Google Scholar