Proving an orthogonal projection of the Hilbert adjoint is just the adjoint

Question

I'm facing the following problem:

let $ H_0 \subset H $ be a $ T$-invariant closed subspace of Hilbert space $ H $ (i.e. $ T(H_0) \subset H_0 $) and $ P$ - an orthogonal projection of $ H $ onto $ H_0 $. Prove that

$$ \left(\left.T \right|_{H_0}\right)^* = \left. (P \circ T^*) \right|_{H_0}$$

with $ ^*$ meaning the Hilbert adjoint.

It's easy to prove that $H_0^{\bot}$ is $ T^* $-invariant. There would be no question if so was $ H_0 $, but I'm not sure that it's true. Or maybe there's some subtelty with taking the adjoint after restricting $ T $ to $ H_0$? I'm rather new to this topic, I would be thankful for help

Answer 1

I seem to have found the solution. Correct me if I'm wrong, but if $ h_1, h_2 \in H_0 $ we can say that:

$$\langle Th_1, h_2 \rangle = \langle h_1, T^*h_2 \rangle = \langle Ph_1, T^*h_2 \rangle = \langle h_1, PF^*h_2 \rangle$$