# Reference request: Original source of Yosida approximation

Numerous papers/books(citation needed) refer to the operator $$A_\lambda := \lambda AR_\lambda (A) = \lambda^2 R_\lambda(A) - \lambda I$$ where $R_\lambda(A)=(I+\lambda A)^{-1}$ is the resolvent, as Yosida approximation without indicating the original source of it.

Can somebody provide the exact source to a Yosida's paper where this operator was originally introduced?

## 1 Answer

...It is well known that the celebrated Hille-Yosida theorem, discovered independently by Hille [1] and Yosida [2], gave the first characterization of the infinitesimal generator of a strongly continuous semigroup of contractions. This was the beginning of a systematic development of the theory of semigroups of bounded linear operators. The bounded linear operator $A_λ$ appearing in the sufficiency part of Yosida’s proof of this theorem is called the Yosida approximation of $A$... ([3] preface i)

[1] Functional Analysis and Semi-groups, 3rd Print, Amer. Math. Soc. Colloq. Publ. Vol. 31, NY, 1948.

[2] On the differentiability and representation of one parameter semi-groups of linear operators, J. Math. Soc. Japan, 1, 15–21, 1948.

[3] Govindan, T. E. "Yosida Approximations of Stochastic Differential Equations in Infinite Dimensions and Applications." Probability theory and stochastic modelling ( 79 (2016).