Extreme points of the unit ball of L(X)_w^* and best approximation in L(X)_w

2022/03/19 by Mal, Arpita
#FOS: Mathematics #Functional Analysis (math.FA) #Primary 46B28 #Secondary 47A12

paper · doi:10.48550/arxiv.2203.10265

Abstract

We study the geometry of L(X)_w, the space of all bounded linear operators on a Banach space X, endowed with the numerical radius norm, whenever the numerical radius defines a norm. We obtain the form of the extreme points of the unit ball of the dual space of L(X)_w. Using this structure, we explore Birkhoff-James orthogonality, best approximation and deduce distance formula in L(X)_w. A special attention is given to the case of operators satisfying a notion of smoothness. Finally, we obtain an equivalence between Birkhoff-James orthogonality in L(X)_w and that in X.

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