A commutator identity for bounded linear maps and the identity operator of a non-zero normed space is never a commutator
Let $ \mathcal{X} $ be a normed linear space and $ S,T: \mathcal{X} \to \mathcal{X} $ be linear operators such that $ S \circ T- T \circ S=1 $.
Show that $ S \circ T^{n+1}- T^{n+1} \circ S=(n+1)T^n $ for $ n=0,1,2,... $
Deduce that if $ S$ is bounded then $ T$ is unbounded.
For the first part I thought of applying the principle of mathematical induction.Is it alright to get the result like that or is there any other method to get that result?And for the second part, Since $ S \circ T- T \circ S=1 $ is the commutator operator,the result is obvious,but how can I give a proof of this?Please help!! Thanks!!
Solution 1:
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Yes, induction is the way to go. The case $n=0$ is the hypothesis $ST - TS = 1$, and for the induction step assume $(n+1)T^{n} = ST^{n+1} - T^{n+1}S$ and use $ST = TS + 1$ to calculate $$\begin{align*}(n +1) T^{n+1} &= (n+1)T^{n}T = (ST^{n+1} - T^{n+1}S)T = ST^{n+2} - T^{n+1}(TS+1) \\&= ST^{n+2} - T^{n+2}S - T^{n+1}\end{align*}$$ which after rearranging is the identity you want to prove.
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I do not see how the second part is obvious "[s]ince $S \circ T - T\circ S = 1$ is the commutator", unless you are in positive finite dimension and apply the trace to reach the contradiction $0 = \dim\mathcal{X}$ or you know about the Wielandt-Wintner theorem, which is a consequence of the exercise you're solving.
Anyway, exclude the case $\mathcal{X} = 0$ and look at the identity from part 1. If $T^{N} = 0$ for some $N$, let $n$ be the smallest $n$ such that $T^{n+1} = 0$. Then $ST^{n+1} - T^{n+1}S = (n+1)T^{n} = 0$ hence $T^{n} = 0$ contradicting the choice of $n$. So, suppose $T^{n} \neq 0$ for all $n$ and assume that $T$ is bounded. Take the norm on both sides. Leave the right hand side as it is and for the left hand side you get the estimate $$\lVert S T^{n+1} - T^{n+1}S\rVert \leq 2 \lVert S\rVert \lVert T^{n} \rVert \lVert T\rVert.$$ Then you can deduce that $2 \lVert S \rVert \lVert T \rVert \geq n+1$ by dividing by $\lVert T^n\Vert$ on both sides. This quickly leads to a contradiction.