Example for an operator that is strictly monotone but not maximally monotone (or the other way)

Question

While the definition of strictly monotone = nowhere constant operators seems intuitive, I find it hard to picture in which way maximal monotone operators ($\forall (u,u') \in X \times X', \langle u'-v', u-v \rangle \geqslant 0 \forall (v,v') \in G(A) \Rightarrow (u,u') \in G(A)$) differ; I would much appreciate a clarifying example!

Answer 1

The operator

$$\mathbb{R} \rightrightarrows \mathbb{R} : x \mapsto \begin{cases}\{x\}, &\text{if $x\neq 0$;}\\ \\ \varnothing, &\text{if $x\neq 0$;} \end{cases} $$ is strictly monotone, but not maximally monotone.

The operator $$\mathbb{R} \rightrightarrows \mathbb{R} : x \mapsto \{0\} $$ is maximally monotone, but not strictly monotone.