Let T an S be nilpotent operators (both from V to V), and TS=ST.
Prove:
1. TS nilpotent
2. T+S nilpotent
Thank you all!
Assume $\displaystyle T^n=S^m=0$.
1. Then what is $\displaystyle (TS)^{mn}$? (You need to use that $\displaystyle ST=TS$.)
2. Using the above result, what is $\displaystyle (S+T)^{2mn}$? To *see* what is going on, work out $\displaystyle (S+T)^4$, assuming $\displaystyle S^2=T^2=0$.