Let $\Omega \subset \mathbb{R}^n$ be an open bounded set.
Consider the following operator $R=(\lambda I-\Delta)^{-1},$ where $\lambda>0$ and $I$ denotes the identity operator. Then, by the Hille-Yosida theorem, it is known that $$ Rx=\int_{0}^{+\infty} e^{-\lambda t} S(t) x\;dt, \; \forall x \in L^2(\Omega)$$ where $S(t)$ is the semigroup of contractions generated in $L^2(\Omega)$ by the Laplace operator $\Delta$.
I have used the above property in order to evaluate the quantity $y Rx,$ where $y \in L^\infty(\Omega).$
Does the following identity $$ y Rx:=y\int_{0}^{+\infty} e^{-\lambda t} S(t) x\;dt =\int_{0}^{+\infty} e^{-\lambda t} S(t) (xy)\;dt ? $$ hold?