Prove bounded linear operator equal 0

Question

Let $A$ and $B$ be bounded linear operators on a Hilbert Space $H$. Show if $A^*A+B^*B=0$ then $A=B=0$.

So far I have shown that $A^*A+B^*B$ is self-adjoint.

This means that $A^*A+B^*B = \overline{(A^*A)^T}+\overline{(B^*B)^T}$.

It is also obvious that $A^*A = -B^*B$ which gives that $||A|| = ||B||$ with some proof.

I'm fairly certain this should be an easy proof to show, but I am stuck and not sure the next steps to take. Any advice would be greatly appreciated.

Answer 1

Hint

Let $x \in H$. Then: $$\langle A^*A x,x \rangle=\langle A x, A x \rangle=\|Ax \|² \geq 0$$ using for example $A^*A=-B^* B$ you can deduce that $\forall x \in H$: $$ 0 \leq \|Ax\|^2 = -\|Bx\|^2 \leq 0$$ which gives the result.