The Cotype of Operators from C(K)

Abstract

In 1987, Jameson [J1] studied the relationship between the (2, 1)-summing norm and the 2-summing norm for operators from l N ∞. He showed that, in general, these norms are not equivalent. At the end of his paper, he observed that the Rademacher cotype 2 constant of operators from l N ∞ lay between these two summing norms, and he asked whether it was indeed equivalent to one of them. Answering this question proved to be very hard. By delicate averaging arguments, I managed to prove that the Rademacher cotype 2 constant for an operator from l N ∞ is very close to its (2, 1)-summing norm; they are within about log log N of each other, and hence, in general, the cotype 2 constant and the 2-summing norms are inequivalent. The techniques used also enabled me to compare the Rademacher and Gaussian cotype p constants for many operators from l N ∞, deducing that these are not the same. Studying this problem also led me to consider quite a different subject. I defined new spaces which are a common generalization of the Lorentz Lp,q and the Orlicz LΦ spaces. As well as rederiving results of Bennett and Rudnick, I sought to calculate the Boyd indices of these new spaces.