Here is the proposition:
The part I'm not understanding is (as usual with most of Pederson's proofs) the last implication. The last equality just seems to imply that $S$ commutes with the power series $$\sum_{n=0}^{\infty} \frac{\lambda^n (T^{*})^n}{n!}$$ in the limit. I don't see how this implies that $S$ nessiarily commutes with $T^{*}$. Even if $S$ commuted with a finite partial sum of the power series, I don't see how this would imply what he's claiming is true.