Weak minimizing property on pairs of classical Banach spaces

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 $2\le p<\infty$ and $({\ell }_s{\oplus }_q{\ell }_q,{\ell }_r{\oplus }_p{\ell }_p)$ for $1<p\le r\le s\le q<\infty$ 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.