Weak minimizing property on pairs of classical Banach spaces

TL;DR

This paper explores the weak minimizing property in pairs of classical Banach spaces, establishing which pairs satisfy or fail this property, thus advancing understanding of geometric properties in Banach space theory.

Contribution

It characterizes specific pairs of classical Banach spaces that satisfy or do not satisfy the weak minimizing property, extending the theory of geometric properties in Banach spaces.

Findings

Pairs $(\, ext{ell}_p, L^p[0,1])$ for $2 \,\leq p < \infty$ satisfy the property.

Pairs $(\ell_s \oplus_q \ell_q, \ell_r \oplus_p \ell_p)$ for specified $p, r, s, q$ satisfy the property.

Pairs $(\ell_1, \ell_p)$, $(\ell_1, c_0)$, $(\ell_1, \ell_1)$, and $(c_0, \ell_p)$ do not satisfy the property.

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.