self-adjoint operators $T,P$ and their kernel

Question

Consider $T,P \in \mathcal{B}(H)$ to be self-adjoint operators, where $H$ is a Hilbert space. Show that $\ker(T^*T + P^*P) = \ker T \cap \ker P$.

I feel like this should be easy to prove. I can prove that $\ker(T^*T) = \ker T$ but not sure how to proceed.

Answer 1

Suppose that $x\in Ker(T^*T+P^*P)$, $\langle T^*T(x)+P^*P(x),x\rangle =\langle T(x),T(x)\rangle+\langle P(x),P(x)\rangle=0$ implies that $P(x)=T(x)=0$.

On the other hand, if $P(x)=T(x)=0$, $(T^*T+P^*P)(x)=T^*(T(x))+P^*(P(x))=0$.