Abstract
A normed space X has the simultaneous zero inclusion (S0I) property if, for every invertible bounded linear operator T on X, 0 lies in the spatial numerical range of T if and only if 0 lies in the spatial numerical range of T−1. It is shown that the only absolute symmetric norms on C2 with the S0I property are scalar multiples of the Euclidean norm. However, the S0I property does not characterize inner products in a slightly wider class of norms—for example, there is a family of absolute polygonal norms on C2 that have the S0I property. There are two keys: (1) The S0I property is a weak version of symmetric Birkhoff-James orthogonality. (2) Complex spaces of dimension two are constructed, based on Radon curves, in which Birkhoff-James orthogonality is symmetric.
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