Prove $Q$ is Idempotent $\iff$ $Qy=y$ for $y \in R(Q)$ where $R(Q)$ is the range of $Q$

Question

Let $H$ be a Hilbert Space, an operator $Q \in B(H)$ is said to be idempotent if $Q^2=Q$ for all $x \in H$. (B(H) is the vector space of bounded operators on $H$).

Prove $Q$ is Idempotent $\iff$ $Qy=y$ for $y \in R(Q)$ where $R(Q)$ is the range of $Q$

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I've been really struggling with this one for some time now.... Trying to use the geometric properties of a hilbert space but am really suffering.. Can somebody help me out here? Thanks I appreciate it so much!!

Answer 1

It is just a matter of definition: $$Q^2=Q \Leftrightarrow \forall x\in H\quad Q(Q(x))=Q(x) \Leftrightarrow \forall y\in R(Q)\quad Q(y)=y $$ since $R(Q)=\{y=Q(x)| x\in H\} $.