= Yosida approximation
{c}
{title2=$L_\lambda=\lambda^2(\lambda I-L)^{-1}-\lambda I$}
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 $\int_0^\infty\lambda e^{-\lambda t}(P_tg-g)dt$ 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.
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