ACTA issues

Symmetric points in spaces of linear operators between Banach spaces

Divya Khurana, Saikat Roy, Debmalya Sain

Acta Sci. Math. (Szeged) 86:3-4(2020), 617-634

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 $\ell _1^n$, $n\geq 2$, to any Banach space $X$. This improves a previously obtained characterization of left-symmetric operators from $\ell _1^n$, $n\geq 2$, to a reflexive smooth Banach space~$X$.

DOI: 10.14232/actasm-020-420-6

AMS Subject Classification (1991): 47L05; 46B20

Keyword(s): Birkhoff--James orthogonality, left-symmetric point, right-symmetric point, supporting functional, symmetric operator

received 20.4.2020, revised 18.5.2020, accepted 20.5.2020. (Registered under 420/2020.)