Yosida approximants

Let $A:D(A)\subseteq X\to X$ be closed and densely defined on a Banach space $X$ with $(\omega ,\mathrm{\infty })\subseteq \rho (A)$, and let $R(\lambda ,A)=(\lambda I-A{)}^{-1}$ (Resolvent and spectrum of a closed operator on a Banach space). For real $\lambda >\omega$ the Yosida approximant of $A$ at $\lambda$ is $A_{\lambda }:=\lambda AR(\lambda ,A)={\lambda }^{2}R(\lambda ,A)-\lambda I\in \mathcal{B}(X)\mathrm{.}$ The equality with ${\lambda }^{2}R(\lambda ,A)-\lambda I$ uses $AR(\lambda ,A)=\lambda R(\lambda ,A)-I$; in particular $A_{\lambda }$ is a bounded everywhere defined operator. Distinct approximants commute, $A_{\lambda }{A}_{\mu }=A_{\mu }{A}_{\lambda }$, and each commutes with $R(\mu ,A)$.

The two formulae agree. Since $\lambda \in \rho (A)$, the resolvent identity $AR(\lambda ,A)=\lambda R(\lambda ,A)-I$ of Resolvent and spectrum of a closed operator on a Banach space gives $A_{\lambda }=\lambda AR(\lambda ,A)=\lambda (\lambda R(\lambda ,A)-I)={\lambda }^{2}R(\lambda ,A)-\lambda I,$ and the right-hand side is a sum of bounded everywhere defined operators (A bounded linear operator between normed spaces). Hence each $A_{\lambda }$ is bounded with $D(A_{\lambda })=X$, and it is meant as a bounded approximation of the possibly unbounded $A$; the sense in which $A_{\lambda }x\to Ax$ for $x\in D(A)$ is a theorem proved on this page, not part of the definition.

Commutativity. For real $\lambda ,\mu >\omega$ the resolvents commute, $R(\lambda ,A)R(\mu ,A)=R(\mu ,A)R(\lambda ,A)$, by Resolvent identity for closed operators. Therefore $A_{\lambda }{A}_{\mu }=({\lambda }^{2}R(\lambda ,A)-\lambda I)({\mu }^{2}R(\mu ,A)-\mu I)$ is symmetric in $\lambda$ and $\mu$: the only mixed term that is not obviously symmetric is ${\lambda }^2{\mu }^{2}R(\lambda ,A)R(\mu ,A)$, and that product is symmetric by the resolvent identity. Hence distinct approximants commute, and for the same reason $A_{\lambda }R(\mu ,A)=\lambda (\lambda R(\lambda ,A)-I)R(\mu ,A)={\lambda }^{2}R(\lambda ,A)R(\mu ,A)-\lambda R(\mu ,A)$ equals $R(\mu ,A)A_{\lambda }$, since $A_{\lambda }$ is the product of $R(\lambda ,A)$ with bounded operators and resolvents at $\lambda$ and $\mu$ commute. The closedness and density of $A$ are hypotheses, not conclusions drawn from a generator theorem. The resolvent vocabulary is Resolvent and spectrum of a closed operator on a Banach space, and the definition itself asserts no approximation property.