Is it true that if I have a Hilbert space $X$ that can be written $X = A \oplus B$ and unitary operators $T_1: A \to A$, $T_2: B \to B$ then the operator $T: A \oplus B \to A \oplus B$ given by $T(a+b) = T_1(a)+T_2(b)$ is unitary?
I think it's true that $T$ is a well-defined operator and now I'm trying to look at:
$$<T(a+b),T(a+b)> = <T_1(a),T_1(a)> + <T_1(a), T_2(b)> + <T_2(b), T_2(a)> + <T_2(b),T_2(b)>$$
I can easily deal with the first and last terms but I'm not sure how to deal with the middle terms to show that this expression is just $<a+b,a+b>$
Thanks
By the definition of the direct sum of Hilbert spaces, we can state that $\langle a,b \rangle = 0$ for $a \in A,b \in B$. This should allow you to complete your proof.