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Strictly singular and strictly cosingular operators on spaces of continuous functions

Published online by Cambridge University Press:  24 October 2008

Catherine Abbott ,
Elizabeth Bator  and
Paul Lewis
Catherine Abbott
Affiliation:
Francis Marion College, Florence, S.C., U.S.A.
Elizabeth Bator
Affiliation:
University of North Texas, Denton, Texas, U.S.A.
Paul Lewis
Affiliation:
University of North Texas, Denton, Texas, U.S.A.
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Extract

In this paper we will be concerned with studying operators T: C(K, X) → Y defined on Banach spaces of continuous functions. We will be particularly interested in studying the classes of strictly singular and strictly cosingular operators. In the process, we obtain answers to certain questions recently raised by Bombal and Porras in [5]. Specifically, we study Banach space X and Y for which an operator T: C(K, X) → Y with representing measure m is strictly singular (strictly cosingular) whenever m is strongly bounded and m(A) is strictly singular (strictly cosingular) for each Borel subset A of K. Along the way we establish several results dealing with non-compact operators on continuous function spaces, and we consolidate numerous results concerning extension theorems for operators defined on these same spaces. Also, we join Saab and Saab [25] in demonstrating that if l1 does not embed in X* then the adjoint T* of a strongly bounded map must be weakly precompact, thereby presenting an alternative solution to a question raised in [2].

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 110 , Issue 3 , November 1991 , pp. 505 - 521
Copyright
Copyright © Cambridge Philosophical Society 1991

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References

REFERENCES

[1]Abbott, C.. Weakly. precompact and GSP operators on continuous function spaces. Bull. Polish Acad. Sci. Math. 37 (1989), 467–476.Google Scholar
[2]Abbott, C., Bator, E., Bilyeu, R. and Lewis, P.. Weak precompactness, strong boundedness, and weak complete continuity. Math. Proc. Cambridge Philos. Soc. 108 (1990), 325–335.CrossRefGoogle Scholar
[3]Batt, J. and Berg, E. J.. Linear bounded transformations on the space of continuous functions. J. Funct. Anal. 4 (1969), 215–239.CrossRefGoogle Scholar
[4]Bombal, F. and Cembranos, P.. Characterization of some classes of operators on spaces of vector-valued continuous functions. Math. Proc. Cambridge Philos. Soc. 97 (1985), 137–146.CrossRefGoogle Scholar
[5]Bombal, F. and Porras, B.. Strictly singular and strictly cosingular operators on C(K, E). Math Nachr. 143 (1989), 355–364.CrossRefGoogle Scholar
[6]Bourgain, J.. An averaging result for l1-sequences and applications to weakly conditionally compact sets in L x1. Israel J. Math. 32 (1979), 289–298.CrossRefGoogle Scholar
[7]Brooks, J.. Equicontinuous sets of measures and applications to Vitali's integral convergence theorem and control measures. Adv. in Math. 10 (1973), 165–171.CrossRefGoogle Scholar
[8]Brooks, J. and Lewis, P.. Linear operators and vector measures. Trans. Amer. Math. Soc. 192 (1974), 139–162.CrossRefGoogle Scholar
[9]Brooks, J. and Lewis, P.. Linear operators and vector measures. II. Math. Z. 144 (1975), 45–53.CrossRefGoogle Scholar
[10]Diestel, J.. Sequences and Series in Banach Spaces. Graduate Texts in Math. no. 92 (Springer-Verlag, 1984).CrossRefGoogle Scholar
[11]Diestel, J. and Url, J. J. Jr. Vector Measures. Math. Surveys no. 15 (American Mathematical Society, 1977).CrossRefGoogle Scholar
[12]Dinculeanu, N.. Vector Measures (Pergamon Press, 1967).CrossRefGoogle Scholar
[13]Dobrakov, I.. On integration in Banach spaces. I. Czech. Math. J. 20 (95) (1970), 511–536.CrossRefGoogle Scholar
[14]Grothendieck, A.. Sur les applications lineaires faiblement compactes d'éspaces du type C(K). Canad. J. Math. 5 (1953), 129–173.CrossRefGoogle Scholar
[15]Jameson, G. J. O.. Topology and Normed spaces (Chapman and Hall, 1974).Google Scholar
[16]Lacey, H. E.. The Isometric Theory of Classical Banach Spaces (Springer-Verlag, 1974).CrossRefGoogle Scholar
[17]Lewis, P.. Some regularity conditions on vector measures with finite semivariation. Rev. Roumaine Math. Pures Appl. 15 (1970), 375–384.Google Scholar
[18]Lindenstrauss, J. and Tzafriri, L.. Classical Banach Spaces, I (Springer-Verlag, 1977).CrossRefGoogle Scholar
[19]Pelczynski, A.. On strictly singular and strictly cosingular operators. I. Strictly singular and strictly cosingular operators in C(S)-spaces. Bull. Polish Acad. Sci. Math. 13 (1965), 31–36.Google Scholar
[20]Pelczynski, A.. On strictly singular and strictly cosingular operators. II. Strictly singular and strictly cosingular operators in L(v)-spaces. Bull. Polish Acad. Sci. Math. 13 (1965), 37–41.Google Scholar
[21]Pietsch, A.. Operator Ideals (VEB Deutscher Verlag der Wissenschaften, 1978).Google Scholar
[22]Royden, H. L.. Real Analysis (Macmillan Publishing Company, 1988).Google Scholar
[23]Saab, P.. Weak compact, unconditionally converging, and Dunford–Pettis operators on spaces of vector-valued continuous functions. Math. Proc. Cambridge Philos. Soc. 95 (1984), 101–108.CrossRefGoogle Scholar
[24]Saab, E. and Saab, P.. A stability property of a class of Banach spaces not containing a complemented copy of l1. Proc. Amer. Math. Soc. 84 (1982), 44–46.Google Scholar
[25]Saab, E. and Saab, P.. On unconditionally converging and weakly precompact operators. Preprint.Google Scholar
[26]Swartz, C.. Unconditionally converging operators on the space of continuous functions. Rev. Roumaine Math. Pures Appl. 17 (1972), 1695–1702.Google Scholar