PreprintCC BY 4.0arXiv
On the simultaneous zero inclusion property and characterization of Inner product space
Abstract
We show that if a Banach space X is smooth and almost locally uniformly rotund and if satisfies the following 0∈ W(T) ⟹ 0∈ W(T−1) for all invertible bounded linear map on X then Birkhoff-James orthogonality in X is symmetric. This result, therefore, characterizes the inner product spaces out of all smooth and almost locally uniformly rotund spaces having three or more dimension via simultaneous zero inclusion property, introduced by Bračič et al. [J. Math. Anal. Appl., 449 (2017) 1413-1423]. Further, this gives answers to some open questions asked by E. Poon [J. Math. Anal. Appl., 534(2024), 128052].
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BibTeX
@misc{ghosh2026simultaneous,
title = {{On the simultaneous zero inclusion property and characterization of Inner product space}},
author = {Souvik Ghosh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13378},
eprint = {2610.13378},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13378},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - On the simultaneous zero inclusion property and characterization of Inner product space
AU - Souvik Ghosh
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13378
UR - https://arxiv.org/abs/2610.13378
AB - We show that if a Banach space X is smooth and almost locally uniformly rotund and if satisfies the following 0∈ W(T) ⟹ 0∈ W(T^−1) for all invertible bounded linear map on X then Birkhoff-James orthogonality in X is symmetric. This result, therefore, characterizes the inner product spaces out of all smooth and almost locally uniformly rotund spaces having three or more dimension via simultaneous zero inclusion property, introduced by Bračič et al. [J. Math. Anal. Appl., 449 (2017) 1413-1423]. Further, this gives answers to some open questions asked by E. Poon [J. Math. Anal. Appl., 534(2024), 128052].
ER -