For a self adjoint $P$, we have $$\|P\|=\sup_{||x||\leq 1}|\langle Px,x\rangle|.$$ Take $x\in H$ with $\|x\|=1$. Then
\begin{align*}
|\langle P x, x\rangle|^{2} &\leqslant\langle P x, x\rangle \cdot\langle P x, x\rangle\\
&\leqslant\langle (Q-(Q-P)) x, x\rangle \cdot\langle (Q-(Q-P)) x, x\rangle\\
&
Yes, I think that there may be a counterexample if the operator isn't linear or isn't self-adjoint? Unless I am interested in the proof where we don't use these two assumptions.
Discussion between user905942 and Shi…
Discussion between user905942 and Shi…