Norming vectors of linear operators between \(L_ p\) spaces
type: Item
instance of: scholarly article
title: Norming vectors of linear operators between \(L_ p\) spaces
zbMATH Open document ID: 0746.47011
author: Charn-Huen Kan
published in: Pacific Journal of Mathematics
publication date: 1992-06-28T00:00:00Z
review text: For a bounded linear operator \(T\) from an \(L_ p\) to an \(L_ q\) space (\(1\leq p,q<\infty\)), we study its norming vectors, i.e. those, including the zero vector, on which \(T\) attains its norm. The scalar field may be the reals or the complex numbers. Our first two main results are the characterization of the set of norming vectors for a positive \(T\) when both \(p>1\) and either (i) \(p=q\) or (ii) \(p>q\). In case (i) the set is a Banach sublattice of \(L_ p\) isometrically vector-lattice isomorphic to another \(L_ p\) space over essentially a measure subspace of the original underlying one, with a change of scale. The descriptions of the sets may not hold if \(T\) is not positive, but they do in modified forms if \(| T|\) exists with norm \(\| T\|\). We also prove that if \(p>q\) and one of the two underlying measures is purely atomic, then every regular \(T\) is norm-attaining. Sufficient conditions for \(T\) (of norm 1) to be an extreme contraction in the case \(p>q>1\) are derived from properties of its norming vectors. All results extend to the case of quaternion scalars with little change of the proofs.
Mathematics Subject Classification ID: 47B38
Mathematics Subject Classification ID: 47A65
zbMATH DE Number: 32730
zbMATH Keywords: decomposition system
zbMATH Keywords: norming vectors
zbMATH Keywords: Banach sublattices of \(L_ p\)
zbMATH Keywords: quaternion scalars
MaRDI profile type: Publication
full work available at URL: pjm.1991.150.309
OpenAlex ID: W2062198957
DOI: 10.2140/PJM.1991.150.309
description: scientific article; zbMATH DE number 32730