Let $\mathcal{H}$ be a Hilbert space and $\mathcal{C}\subset\mathcal{H}$ a closed convex subspace. Let $P:H\to \mathcal{C}$ be a projection, $A_1,A_2:\mathcal{H} \to \mathcal{H}$ be bounded linear operators and $Id$ the identity operator.
What would be the an approach to show that:
$P(A_1+\mu Id)^{-1}A_1f=P(A_2+\mu Id)^{-1}A_2f$
where $f\in\mathcal{H}$ and $\mu>0$. It is obvious that the equation holds for $\mu=0$. I know that this is not a very specific question but maybe someone can point me in the right direction.