This is a little variation of following question: https://math.stackexchange.com/posts/1998643/edit
I'd like to prove the following statement:
Let $H$ be a positive self-adjoint operator, not necessarily bounded. For each real positive number $\lambda$, define $H_{\lambda} = \lambda H(H+\lambda I)^{-1}$.
Prove that each $H_{\lambda}$ is a bounded self-adjoint operator and that $H_{\lambda} \to H$ as $\lambda \to \infty$ strongly (i.e. $|| H_{\lambda} \varphi - H \varphi || \to 0$ when $\lambda \to \infty$ for every $\varphi \in D(H)$, where $D(H)$ is domain.
I tried to use same procedure of this link https://math.stackexchange.com/posts/1998643/edit, but I couldn't do it, taking this modification $H_{\lambda} = \lambda H(H+\lambda I)^{-1}$.
Someone can help me please?
Any idea?