# Thread: Nilpotent operator once again.

1. ## Nilpotent operator once again.

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!

2. Originally Posted by Also sprach Zarathustra
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 $T^n=S^m=0$.

1. Then what is $(TS)^{mn}$? (You need to use that $ST=TS$.)

2. Using the above result, what is $(S+T)^{2mn}$? To *see* what is going on, work out $(S+T)^4$, assuming $S^2=T^2=0$.