Closedness of the set of extreme points of the unit ball in Orlicz spaces equipped with the p-Amemiya norm

Abstract

It is known (see the classical book “Linear Operators I” by N. Dunford and J.T. Schwarz [6]) that compact nice operators from a Banach space X into the space of continuous functions C(Z,R) are extreme points of the unit ball of compact operators B(K(X,C(Z,R))), where Z is a compact Hausdorff space. Recall, that an operator T is called nice if T⁎(Z)⊂ExtB(X), where the continuous mapping T⁎ is defined by T⁎(z)(x)=T(x)(z) for all x∈X and z∈Z. It is evident that if the set ExtB(X) is closed, then T is nice if and only if T⁎(Z0)⊂ExtB(X) on a dense subset Z0⊂Z. In general the set of extreme points ExtB(X) need not to be closed. In 1992 A. Suarez-Granero and M. Wisła presented the criteria for closedness of the set of extreme points of the unit ball in the case of Orlicz spaces equipped with the Luxemburg norm. The aim of this paper is to present the criteria for the closedness of the set of extreme points of the unit ball for the wide class of norms – the so-called p-Amemiya norms. This class includes both the Orlicz and the Luxemburg norm. The Orlicz functions that generates Orlicz spaces are assumed to be as much general as possible, in particular the Orlicz functions can vanish outside zero and jump to infinity what follows that the investigated Orlicz spaces can contain isomorphic copy of L∞ or be a subspace of L∞.