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\in W(T) \implies 0\in 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].