Weak minimizing property on pairs of classical Banach spaces
Manwook Han
TL;DR
The paper introduces the weak minimizing property (WmP) as the minimum modulus analogue of the weak maximizing property and studies when pairs of Banach spaces have this property. It develops sufficient conditions based on properties $(m)$ and $(o)$, along with modulus bounds, to ensure the WmP, and applies these to derive new examples and counterexamples. It proves that pairs such as $(\ell_p,L_p[0,1])$ for $2\le p<\infty$ and certain direct-sum constructions satisfy the WmP, while pairs like $(\ell_1,\ell_p)$, $(\ell_1,c_0)$, $(\ell_1,\ell_1)$, and $(c_0,\ell_p)$ fail. The work clarifies the landscape of WmP versus WMP and CPPm in classical spaces and raises open questions, including CPPm status for $(c_0,c_0)$.
Abstract
We investigate the minimum modulus analogue of the weak maximizing property, termed the \emph{weak minimizing property}. We establish that the pairs $(\ell_p, L^p[0, 1])$ for $2 \leq p < \infty$ and $(\ell_s \oplus_q \ell_q, \ell_r \oplus_p \ell_p)$ for $1 < p \leq r\leq s \leq q < \infty$ satisfy the weak minimizing property. Conversely, we prove that the pairs $(\ell_1, \ell_p)$, $(\ell_1, c_0)$, $(\ell_1, \ell_1)$ and $(c_0, \ell_p)$ fail to satisfy the weak minimizing property.