Monotone gradients on Banach lattices

Abstract:

It is well known that a differentiable real valued function on the real line is convex iff its derivative is nondecreasing. This characterization of differentiable convex functions does not extend if the domain of the function is a Banach lattice of $\dim \geqslant 2$. In this paper a complete characterization of a subclass of differentiable convex functions on a given Banach lattice is obtained in terms of the monotonicity and natural orthogonality properties of the gradients. It is well known that a differentiable real valued function on the real line is convex iff its derivative is nondecreasing. It is also known that a ${C^1}$-real valued function $F$ on an arbitrary Banach space $E$ is convex iff the gradient of $F$, say $f$, satisfies the condition (*) $(f(x) - f(y),x - y) \geqslant 0$ for all $x$, $y \in E$. Condition (*) is referred to in the literature as a monotonicity condition. However when $E$ is a Banach lattice, it is natural to define a function $f$ on $E$ into another Banach lattice ${E_1}$ as nondecreasing if $f(x) \geqslant f(y)$ whenever $x$, $y$ in $E$ are such that $x \geqslant y$. The present investigation originated out of the surprising fact that, as shown by suitable examples, the characterization of ${C^1}$-convex real valued functions on the real line recalled above, does not extend if the domain of the function is a Banach lattice of $\dim \geqslant 2$, and the monotonicity of the gradient function is interpreted as in the preceding sentence. More specifically, if $E$ is a Banach lattice of $\dim \geqslant 2$, there are ${C^1}$-convex real valued functions on $E$ with gradients not nondecreasing and there are ${C^1}$-real valued functions on $E$ with gradients nondecreasing which fail to be convex. The main result here provides a complete characterization of ${C^1}$-convex real valued functions $\Phi$ on an arbitrary Banach lattice $E$ of which the translates $\Phi (x + y) - \Phi (x)$ are orthogonally additive in $y$ for all $x$ in $E$, in terms of the monotonicity and natural orthogonality properties of the gradient map of $\Phi$. The characterization of the special class of ${C^1}$-convex functions on Banach lattices stated in the main result here appears to be a reasonable substitute of the characteristic property of ${C^1}$-convex real valued functions on the real line recalled above. The plan of the paper is as follows. In the introductory section, §1, the basic definitions and known results are recalled for convenience of reference, and the main results are stated and proved in §2.