If a scalar function $f\colon \mathbb R \to \mathbb R$ is monotone and differentiable, then $f'\geq 0$.

Monotonicity is generalized for an operator $A\colon V \to V^*$, where $V$ is a Banach spaces with its dual space $V^*$, via:

The operator $A\colon V \to V^*$ is called monotone if $$ \langle A(u) - A(v), u-v \rangle \geq 0,$$ for all $u$, $v \in V$.

So my question is: Does monotonicity of $A$ imply, that the Frechet derivative of $A$ is positive, i.e. $\langle A'(w)v,v\rangle \geq 0$ for any $w \in V$ and for all $v\in V$.

Any idea or reference is appreciated.


If the operator $A$ is Fréchet differentiable, then its derivative is positive (semidefinite).

For suppose to the contrary there were a point $w$ and a $v\in V$ with

$$c := \langle A'(w)v,v\rangle < 0.$$

Then for small real $t \neq 0$, we have

$$t^{-2}\langle A(w + tv) - A(w), tv\rangle = t^{-2} \langle A'(w)(tv) + o(t), tv\rangle = c + \frac{t\cdot o(t)}{t^2} \to c < 0,$$

contradicting the monotonicity of $A$.


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