Math. Z. 197, 21-32 (1988) Mathematische Zeitschrift Springer-Verlag 1988 Uniform Convergence of Operators and Grothendieck Spaces with the Dunford-Pettis Property* Denny Leung Department of Mathematics, The Catholic University of America, Washington, D.C. 20064, USA 1. Introduction In this paper, we deal with conditions on a Banach space which turn the strong convergence of certain classes of operators into uniform convergence. The start- ing point is the following theorem of Dean [1]. Theorem 1. Let E be a Grothendieck space with the Dunford-Pettis property, then E admits no Schauder decompositions Further results along the same lines were also obtained by Lotz [6, 7]. Theorem 2. Let E be a Grothendieck space with the Dunford-Pettis property. Then (1) if T is a strongly ergodic operator on E such that IITn/ni]-~0, then T is uniformly ergodic ; and (2) if (Tt)t_> o is a strongly continuous semi-group of operators on E, then (Tt)t >= o is uniformly continuous. The main goal of this paper is to extend the above theorems to a wider class of Banach spaces. In w 2, we define the surjective Dunford-Pettis property. As its name implies, it is closely related to the Dunford-Pettis property. From the definition, we will be able to see that it is a formally weaker property. The surjective Dunford- Pettis property is then made use of in the main theorem. As consequences of the main theorem, we obtain the extensions of Theorems 1 and 2 to Grothen- dieck spaces with the surjective Dunford-Pettis property. w is devoted entirely to examples. It culminates in the Examples 13 and 15, which show that the surjective Dunford-Pettis property is a genuinely weaker condition that the Dunford-Pettis property. Our notation follows that of [11]. Throughout, E denotes a real Banach space or Banach lattice unless otherwise stated, and E' denotes its dual. If E, * This paper is taken from the author's Ph.D. thesis written under Professor H.P. Lotz and submitted to the University of Illinois September, 1986