Let $H$ be a Hilbert space and $A(\neq \emptyset)\subset H$. If $T\in BL(H)$ is unitary, then $$T(A^\perp)=T(A)^\perp.$$
Let $y\in T(A^\perp)\implies \exists x\in A^\perp\ \text{s.t}\ T(x)=y. $ Since, $x\in A^\perp\implies \forall\ a\in A,\ \langle x,a\rangle=0$.
Since $T$ is unitary, implies $TT^*=I=T^*T$. \begin{align*} & T(x)=y\\ \implies& T^*T(x)=T^*(y)\\ \implies& T^*(y)=x \end{align*} Since, $\langle x,a\rangle=0\implies \langle T^*(y),a\rangle=0$.
Now I'm stuck, what to do. How to proceed further. Please help me.
Thanks.