The lattice Schäffer constant

Abstract

For a Banach lattice $X$, its lattice Schäffer constant is defined by: \begin{gather*} λ^+(X)=\inf\{\max\{\|x+y\|,\|x-y\|\}\,\colon\,\|x\|=\|y\|=1,x,y\geq{\bf0}\}. \end{gather*} In this paper, we investigate this constant, as well as the companion parameter \begin{gather*} β(X)=\inf\{\|x\vee y\|\,\colon\,\mbox{$\Vert x\Vert =\Vert y\Vert =1$, $x,y\ge 0$ and $x\wedge y=0$}\}. \end{gather*} Our main results fall into two groups. (1) We link the behavior of the parameters $\hat{I}{»}^{+}$ and ${\hat{I}}^2$ to the global properties of the lattice $X$. For instance, we prove that (i) if $\hat{I}{»}^{+}(X)>1$, then the Banach lattice $X$ is a KB-space, and moreover, it satisfies a lower $q$-estimate for some $q\in (1,\infty )$; (ii) $\hat{I}{»}^{+}(X)=1$ if and only if $X$ contains lattice-almost isometric copies of ${\ell }_{\infty }^2$; and (iii) that $\hat{I}{»}^{+}(X)=2$ if and only if $X$ is an abstract $L$-space. (2) We establish inequalities relating $\hat{I}{»}^{+}(X)$ to the characteristics of monotonicity, ${\epsilon }_{0,m}(X)$ and ${\tilde{\epsilon }}_{0,m}(X)$. Along the way, we compute $\hat{I}{»}^{+}(X)$ and ${\hat{I}}^2(X)$ for various Banach lattices $X$.