International Journal of Mathematics and Mathematical Sciences 
Volume 2003 (2003), Issue 37, Pages 2375-2378
doi:10.1155/S0161171203302315

DP1 and completely continuous operators

Elizabeth M. Bator1 and Dawn R. Slavens2

 1Department of Mathematics, University of North Texas, P.O. Box 311400, Denton 76203-1400, TX, USA
2Department of Mathematics, Midwestern State University, 3410 Taft Blvd, Wichita Falls 76308, TX, USA
Received 11 February 2003

Abstract

W. Freedman introduced an alternate to the Dunford-Pettis property, called the DP1 property, in 1997. He showed that for 1≤p<∞, (⊕α∈𝒜Xα)p has the DP1 property if and only if each Xα does. This is not the case for (⊕α∈𝒜Xα)∞. In fact, we show that (⊕α∈𝒜Xα)∞ has the DP1 property if and only if it has the Dunford-Pettis property. A similar result also holds for vector-valued continuous function spaces.