Let $H$ be a Hilbert space and consider $$A:D(A) \subset H \to H $$ to be unbounded linear monotone operator.
Is there any relation between $\mathrm{Range}(I+A)$ and $\mathrm{Range}(A)$, generally?
Definition of monotone operator: A is said to be monotone if it satisfies $$ (Au,u) \geq 0, \quad \forall u \in D(A). $$
Some hint: When the Range of $I+A$ is $H$, then $A$ is called maximal monotone operator which enjoys the following properties:
I) $D(A)$ is dense in $H$.
II) $A$ is a closed operator.
III) For any $\lambda>0$, $(I+\lambda A)$ is bijective from $D(A)$ onto $H$, $(I+\lambda A)^{-1}$ is a bounded operator, and its operator norm is equal or less than $1$.
I think from a continuity or a fixed point argument, we could prove that $I+A$ is surjective iff $A$ is surjective.