Range of a unitary transformed orthogonal projection

Question

Let $X$ be a Hilbert space, $P$ an orthogonal projection in $X$ and $Q \in L(X)$ (i.e. $Q \colon X \to X$ is linear and continuous) a unitary linear transformation, i.e. $Q^*Q=Id_X= QQ^*$ ($Q^*$ denotes the Hilbert space adjoint of $Q$). I have shown that $$\tilde{P} = QPQ^*$$ is an orthogonal projection. What can we say about the range of $\tilde{P}$?

Answer 1

The range of $\tilde{P}$ is equal to $Q \mathrm{ran} (P) = \{ Qy : y \in \mathrm{ran}(P) \},$ which you can prove simply by set inclusion. (For one of the directions you should insert $Q^* Q = I$ somewhere.)