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Spaces on which unconditionally converging operators are weakly completely continuous. (English) Zbl 0807.46034

Summary: Let \(\Omega\) be a compact Hausdorff space, and let \(E\) be a Banach space with unconditional reflexive decomposition, then every unconditionally converging operator \(T\) on \(C(\Omega, E)\), the space of \(E\)-valued continuous functions on \(\Omega\), is weakly completely continuous, i.e., \(T\) sends weakly Cauchy sequences into sequences that converge weakly.

MSC:

46E40 Spaces of vector- and operator-valued functions
46G10 Vector-valued measures and integration
47B38 Linear operators on function spaces (general)
28B05 Vector-valued set functions, measures and integrals
28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections
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References:

[1] J. Batt and E.J. Berg, Linear bounded transformations on the space of continuous functions , J. Functional Analysis 4 (1969), 215-239. · Zbl 0183.13502 · doi:10.1016/0022-1236(69)90012-3
[2] F. Bombal and P. Cembranos, Dieudonné operators on \(C(K,E)\) , Bull. Polish Acad. Sci. Math. 34 (1986), 301-305. · Zbl 0605.47035
[3] P. Cembranos, N.J. Kalton, E. Saab and P. Saab, Pelczynski’s Property (V) on \(C(\O,E)\) spaces , Math. Ann. 271 (1985), 91-97. · Zbl 0546.46029 · doi:10.1007/BF01455797
[4] J. Diestel, Sequences and series in Banach Spaces , Graduate Texts in Math. 92 , Springer-Verlag, New York, 1984. · Zbl 0542.46007
[5] J. Diestel and J.J. Uhl, Jr., Vector measures , Math. Surveys 15 , Amer. Math. Soc., Providence, RI, 1977. · Zbl 0369.46039
[6] I. Dobrakov, On representation of linear operators on \(C_0(T,X)\) , Czechoslovak Math. J. 21 (1971), 13-30. · Zbl 0225.47018
[7] W.B. Johnson and M. Zippin, Separable \(L_1\) preduals are quotients of \(C(\D)\) , Israel J. Math. 16 (1973), 198-202. · Zbl 0283.46007 · doi:10.1007/BF02757870
[8] J. Lindenstraus and L. Tzafriri, Classical Banach spaces \(II\) 97 , Berlin-Heidelberg, New York, Springer-Verlag, 1979. · Zbl 0403.46022
[9] A. Pelczynski, Banach spaces on which every unconditionally converging operator is weakly compact , Bull. Polish Acad. Sci. Math. 10 (1962), 641-648. · Zbl 0107.32504
[10] Z. Semadeni, Banach spaces of continuous functions , Vol. I, Monografic Mat. 55 , PWN, Warsaw, 1971. · Zbl 0225.46030
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