Norm-attaining tensors and nuclear operators

2020/06/17 by Dantas, Sheldon, Jung, Mingu, Roldán, Óscar +1
#FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2006.09871

Abstract

Given two Banach spaces X and Y, we introduce and study a concept of norm-attainment in the space of nuclear operators N(X,Y) and in the projective tensor product space X \widehat⊗_πY. We exhibit positive and negative examples where both previous norm-attainment hold. We also study the problem of whether the class of elements which attain their norms in N(X,Y) and in X\widehat⊗_πY is dense or not. We prove that, for both concepts, the density of norm-attaining elements holds for a large class of Banach spaces X and Y which, in particular, covers all classical Banach spaces. Nevertheless, we present Banach spaces X and Y failing the approximation property in such a way that the class of elements in X\widehat⊗_πY which attain their projective norms is not dense. We also discuss some relations and applications of our work to the classical theory of norm-attaining operators throughout the paper.

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