Saab, Paulette; Smith, Brenda Spaces on which unconditionally converging operators are weakly completely continuous. (English) Zbl 0807.46034 Rocky Mt. J. Math. 22, No. 3, 1001-1009 (1992). 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. Cited in 2 Documents 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 Keywords:Banach space with unconditional reflexive decomposition; unconditionally converging operator PDFBibTeX XMLCite \textit{P. Saab} and \textit{B. Smith}, Rocky Mt. J. Math. 22, No. 3, 1001--1009 (1992; Zbl 0807.46034) Full Text: DOI 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.