On Compact Sets of Compact Operators on Banach Spaces not Containing a Copy of $l^1$

F. Galaz-Fontes (Proc. AMS., 1998) has established a criterion for a subset of the space of compact linear operators from a reflexive and separable space X into a Banach space Y to be compact. F. Mayoral (Proc. AMS., 2000) has extended this criterion to the case of Banach spaces not containing a copy of $l^1$. The purpose of this note is to give a new proof of the result of F. Mayoral. In our proof, we use $l^{\mathrm{\infty }}$-spaces, a well known result of H. P. Rosenthal and L.E. Dor which characterizes the spaces without a copy of $l^1$ and a recent result obtained by G. Nagy in 2007 concerining compact sets in normed spaces. We point out that another proof of Mayoral’s result was given by E. Serrano, C. Pineiro and J.M. Delgado (Proc. AMS., 2006) by using a different method.