On the coincidence of two crossnorms connected with the order

V. T. Khudalov

Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part IX, Tome 92 (1979), pp. 307-311

Résumé

Let $E$ be an ordered normed space and $X$ be an arbitrary normed space. The following two crossnorms are considered: for \begin{gather*} n_E(z)=\inf\biggl\{\|u\|:\sum_{k=1}^ne_k\langle x_k,x^*\rangle\le u,\ z=\sum_{k=1}^ne_k\otimes x_k,\ x^*\in X,\ \|x^*\|\le1\biggr\}, \\ k_E(z)=\inf\biggl\{\biggl\|\sum_{k=1}^ne_k\|x_k\|\biggr\|:z=\sum_{k=1}^ne_n\otimes x_k,\ e_k\ge0\biggr\}. \end{gather*} Theorem 1. The following conditions are equivalent: 1) for every normed space $X$ and every $z\in E\otimes X$ we have $n_E(z)=k_E(z)$. 2) $E$ has the Riesz interpolation property.