Definition 129 in these notes states:
The Yosida approximation to a semigroup $K_t$ with generator $G$ is given by $$ K_t^\lambda := e^{tG^\lambda}$$ $$ G^\lambda := \lambda GR_\lambda = \lambda (\lambda R_\lambda - I)$$
where $R_\lambda$ is the resolvent operator, given by $R_\lambda = \int_0^\infty e^{-\lambda t}K_tf(x) dt$.
I don't understand any of the equalities in this.
First, since $K_t = e^{tG}$, shouldn't $K_t^\lambda = e^{\lambda tG}$? Now I see this multiplication in the second equality, but why is there now a resolvent operator? And I also don't understand the factorization $\lambda GR_\lambda = \lambda (\lambda R_\lambda - I)$.