Self adjoint operators leaves invariant some unidimensional subspace

Question

Suppose $A$ is a positive self adjoint operator in some Hilbert space $\mathcal{H}$ and $\psi$ a nonzero vector of $\mathcal{H}$. Let $\mathcal{H}=\mathbb{C}_\psi\oplus\mathcal{H}_1$ be the orthogonal sum decomposition. I want to show that if $A\big|_{\mathcal{H}_1}=0$ and Tr$A=1$, then $A$ is the orthogonal projection onto $\mathbb{C}_\psi$. My idea is to prove that $A$ maps $\mathbb{C}_\psi$ into itself, then WLOG assume $\vert \psi \vert=1$, and we can find an orthonormal basis of $\mathcal{H}_1$ so that Tr$A=a\psi=1\Rightarrow A\psi=\psi$, which proves our assertion. But I cannot prove that $A$ leaves $\mathbb{C}_\psi$ invariant.

Answer 1

If $\phi\in \mathcal{H}_1$, then $\langle A\psi,\phi\rangle=\langle \psi,A\phi\rangle=0$. Thus $A\psi\in \mathcal{H}_1^\perp=\mathbb{C}_\psi$.