Norming vectors of linear operators between \(L_ p\) spaces
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.
scientific article; zbMATH DE number 32730