Operator minimax theorems in Banach lattices

Abstract:

Let $\psi :X \times Y \to (E, \leqslant ,\left \| \cdot \right \|)$, where X and Y are convex, X is compact, and E is a dedekind complete Banach lattice with unit e. If each $\psi ( \cdot ,y)$ is continuous $\leqslant$-concave on X, $\{ \psi ( \cdot ,y):y \in Y\}$ is convex, and $\psi (X \times Y)$ is minorized in E, then $\sup \inf \psi (x,y) = \inf \sup \psi (x,y)$. Similar theorems are included.