I found the following theorem:
Given $H$ Hilbert space and a monotone operator $A\colon H\rightarrow H$, then A is maximal monotone if and only if $\operatorname{Range}(A+I)=H$.
Note that: $A$ monotone (multivalued) means that $\forall u,v \in H$ and $\forall f\in Au, g \in Av$, then $(u-v,f-g) \geq 0$.
Moreover a monotone operator is said to be maximal in the sense of inclusion of graphs (i.e. the graph of A has no proper monotone extension).
Where can I find a detailed proof of this fact?
Thank you in advance, I'm really clueless.
This result is known as Minty's Theorem. A modern reference and proof can be found in Bauschke & Combettes' book, volume 2, Section 21.1 (Theorem 21.1)