---
tag: 47NSS1R24Y
kind: definition
name: Monotone operator
msc: [47H05, 47J20]
defines: Y1M03JQKKR
uses: [06GZ956657, 07JBSZE0V4]
sources:
  - url: https://en.wikipedia.org/wiki/Monotonic_function
    locator: Monotonicity in functional analysis
---

## Definition

Let $X$ be a real [[Banach space|06GZ956657]] with [[dual space|07JBSZE0V4]] $X^*$, and write $\ip{\ell}{u} = \ell(u)$ for $\ell \in X^*$ and $u \in X$.
Let $D \subseteq X$ and let $T \colon D \to X^*$ be a map, not necessarily linear.
The map $T$ is a *monotone operator* if
$$\ip{T(u) - T(v)}{u - v} \ge 0 \quad \text{for all } u, v \in D.$$
It is *strictly monotone* if equality holds only for $u = v$.

## 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|3S7HJ5GQRF]].
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|07Q7CNM5WF]].
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|0A7KV2WB7A]].