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.)
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