Norm of the solution of a Poisson equation

Question

Let $\kappa$ be a linear contraction (operator norm at most $1$) on a $\mathbb R$-Hilbert space $H$, $L:=1-\kappa$ and $f,g,\tilde g\in H$ with $$Lg=f=L\tilde g\tag1.$$ Are we able to show that $\left\|g\right\|_H^2-\left\|\kappa g\right\|_H^2=\left\|\tilde g\right\|_H^2-\left\|\kappa\tilde g\right\|_H^2$?

Clearly, the claim is trivial if $L$ is injective. If necessary, feel free to assume that $\kappa$ is self-adjoint. In that case, $$\left\|g\right\|_H^2-\left\|\kappa g\right\|_H^2=\langle f,(1+\kappa)g\rangle_H\tag2.$$

Answer 1

Let me use $K$ instead of $\kappa$. Assume $K$ self-adjoint. Then $$\begin{split} \|g\|^2-\|Kg\|^2&=\langle f,(I+K)g\rangle\\ & =\langle (I-K)\tilde g, (I+K)g\rangle\\ &=\langle (I+K) \tilde g,(I-K)g \rangle\\ & =\langle (I+K)\tilde g, f\rangle\\ & =\|\tilde g\|^2 - \|K\tilde g\|^2 \end{split}$$