A two-dimensional Hahn-Banach theorem

Abstract:

Let $\tilde T=\sum _{i=1}^n \tilde { u}_i\otimes v_i:\;V\rightarrow V=[v_1,...,v_n]\subset X$, where $\tilde { u}_i\in V^{*}$ and $X$ is a Banach space. Let $T= \sum _{i=1}^n u_i\otimes v_i:\;X\rightarrow V$ be an extension of $\tilde T$ to all of $X$ (i.e., $u_i\in X^{*}$) such that $T$ has minimal (operator) norm. In this paper we show in particular that, in the case $n=2$ and the field is R, there exists a rank-$n$ $\tilde T$ such that $\|T\|=\|\tilde T\|$ for all $X$ if and only if the unit ball of $V$ is either not smooth or not strictly convex. In this case we show, furthermore, that, for some $\|T\|=\|\tilde T\|$, there exists a choice of basis $v=v_1, v_2$ such that $\|u_i\| = \|\tilde { u}_i\|,\;i=1,2$; i.e., each $u_i$ is a Hahn-Banach extension of $\tilde { u}_i$.