Let $U$ be a unitary operator and $H$ a Hilbert space. I := \{ v \in H | Uv = v\} and A := \{ Uw - w | w \in H\}.
I would like to show that $A$ is dense in the orthogonal $I^\bot$ of $I$. I think for this I need to show (i) $A \subset I^\bot$ and (ii) $\bar{A} = I^\bot$. I've done (i) and half of (ii). Now I'm stuck with $I^\bot \subset \bar{A}$. That is for $x \in I^\bot$ I want to find a sequence $a_n \in A$ such that $\lim_{n \rightarrow \infty}a_n = x$.
Now I'm not sure how to construct such a sequence. Can someone help me with this? Thanks for your help.