Monotone operator

Remarks

A set-valued map $T$ from $X$ to subsets of $X^*$ is monotone if $\ip{u^* - v^*}{u - v} \ge 0$ whenever $u^* \in T(u)$ and $v^* \in T(v)$. It is maximal monotone if its graph is not properly contained in the graph of another monotone map. For a real Hilbert space one identifies $X^*$ with $X$. A linear operator $T$ is then monotone if and only if $\ip{Tu}{u} \ge 0$ for all $u$, that is, if and only if $-T$ is dissipative. The subdifferential of a proper convex lower semicontinuous function is maximal monotone.

Example: let $f \colon \R^n \to \R$ be a differentiable convex function. Adding the inequalities $f(v) \ge f(u) + \nabla f(u) \cdot (v - u)$ and $f(u) \ge f(v) + \nabla f(v) \cdot (u - v)$ gives $(\nabla f(u) - \nabla f(v)) \cdot (u - v) \ge 0$, so $\nabla f$ is monotone.

Non-example: $T(u) = -u$ on $\R$ gives $(T(1) - T(0))(1 - 0) = -1 < 0$.

On $X = \R$, a map $T \colon \R \to \R$ is monotone exactly when it is a nondecreasing monotone function.