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 ℓ1n, n ≥ 2, to any Banach space X. This improves a previously obtained characterization of left-symmetric operators from ℓ1n, n ≥ 2, to a reflexive smooth Banach space X.
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Communicated by L. Molnár
Acknowledgment.
Dr. Debmalya Sain feels elated to acknowledge the loving and motivating friendship of his childhood friend Mr. Arijeet Mukherjee. The research of Dr. Debmalya Sain and Dr. Divya Khurana is sponsored by Dr. D. S. Kothari Postdoctoral Fellowship under the mentorship of Professor Gadadhar Misra. The research of Mr. Saikat Roy is supported by CSIR MHRD in terms of Junior Research Fellowship under the mentorship of Professor Satya Bagchi.
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Khurana, D., Roy, S. & Sain, D. Symmetric points in spaces of linear operators between Banach spaces. ActaSci.Math. 86, 617–634 (2020). https://doi.org/10.14232/actasm-020-420-6
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DOI: https://doi.org/10.14232/actasm-020-420-6