Symmetric points in spaces of linear operators between Banach spaces

Abstract

We explore the relation between left-symmetry (right-symmetry) of elements in a real Banach space and right-symmetry (left-symmetry) of their supporting functionals. We obtain a complete characterization of symmetric functionals on a reflexive, strictly convex and smooth Banach space. We also prove that a bounded linear operator from a reflexive, Kadets–Klee and strictly convex Banach space to any Banach space is symmetric if and only if it is the zero operator. We further characterize left-symmetric operators from l n 1 , n ≥ 2, to any Banach space X. This improves a previously obtained characterization of left-symmetric operators from l n 1 , n ≥ 2, to a reflexive smooth Banach space X.