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This excercise was part of an exam of a second course on linear algebra. I'm studing for my exam so I'm trying to do the old ones. It goes as follows:

Let $V$ be a real finite-dimensional vector space. Let $S$ and $T$ be operators in $V$ such that $$TS-ST=T$$ a) Prove that for any integer $k \geq 1$ the following equTion holds: $$T^kS-ST^k=kT^k$$ b)Prove $T$ is nilpotent (hint: consider the linear operator $\phi:End(V)\to End(V)$ given by $\phi(H)=TH-HT$)

I've managed to prove part a) by induction but I'm completly stuck with b) because I don't understand how to use the hint or any other fact. At first I tried to prove there exist some non-zero $k$ such that $T^kS=ST^k$ but I think the proof of b) should be independent of the order of nilpotence of the operator so that proof would not be a direct proof and reasoning by contradiction took me nowhere. It doesn't feel neither there's some polynomial approach. Of course I also tried to use the hint and made the following computations:

$\phi(S)=T$

$\phi^2(H)=T^2H-2THt+HT^2$

$T[\phi(H)]=T^2H-THT=\phi[T(H)]$

(and some more that I belive are useless)

The last equality seems to be an important fact but I just don't know where to go from here. I also tried to prove that $\phi$ is nilpotent, which I don't know if its true, but even if it were true I don't know how I would use it.

I feel terrible because I've studied the whole content of the program and I've done most of the excercises from the problem sets but with this excercise it seems I'm missing something important and don't know what it is. Any guidence of hint will be much appreciated.

la flaca
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    I think the hint is wrong. It should be $\phi(H) = HS-SH$. Now look at $\phi(T^k)$ – Lukas Betz Nov 28 '16 at 21:16
  • @LeBtz that makes completly sense, now I can conclude that for any $k\geq 1$ if $T^k\not=0$ then $k$ is an eigenvalue of $\phi$ and because $V$ is finite-dimensional $T^k$ must be zero for some $k$. Sadly the question was closed, otherwise I would have accepted your answer (if you had posted it as an answer) because none of the answers given to the question linked uses this approach. Thanks! – la flaca Nov 29 '16 at 00:07

1 Answers1

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In question a., you have proved that for all $k\in\mathbb{N}$, $\phi(T^k)=kT^k$. Hence, if for all $k\in\mathbb{N}$, $T^k$ is nonzero, then all the integers are eigenvalues of $\phi$. However, since $\textrm{End}(V)$ is finite dimensional, $\phi$ as only a finite number of eigenvalues.

As pointed out by LeBtz in the comments, I am considering $\phi(H)=HS-SH$.

C. Falcon
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