Yosida approximation ($L_{\lambda }={\lambda }^2(\lambda I-L{)}^{-1}-\lambda I$)

For a contraction semigroup with semigroup generator $L$, the bounded operators $L_{\lambda }$ approximate $L$ on its generator domain. The Laplace-transform formula for a semigroup resolvent and strong continuity give $\lambda R_{\lambda }g\to g$ for every vector $g$: split ${\int }_0^{\infty }\lambda e^{-\lambda t}(P_{t}g-g)dt$ into a short interval and an exponentially small tail. For $f\in D(L)$, integration by parts gives $R_{\lambda }Lf=\lambda R_{\lambda }f-f$, so $L_{\lambda }f=\lambda R_{\lambda }Lf\to Lf$. No operator norm convergence of $\lambda R_{\lambda }$ is implied.