Given two non-commuting linear operators prove that at least one is discontinuous

Asked May 28 '20 at 14:01

Active May 28 '20 at 14:01

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Given a normed space $X \neq \{0\}$ and $P, Q : X \rightarrow X$ linear such that $$PQ - QP = \mathbb{1}_X$$ Prove that at least one of P and Q is discontinuous.

Hint: Prove and use that $PQ^n - Q^nP = nQ^{n-1}$.

I was able to prove the hint, but I'm not able to use it. My idea was to try to prove this by contradiction, assume that both P and Q are bounded, then try to calculate the norm of Q by using the hint and recursively replacing the $Q^n$s in hopes of coming up with some non-converging series using the 'n terms', but this only leads to a mess. Can someone give some help please?

asked May 28 '20 at 14:01

eager2learn

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https://math.stackexchange.com/a/54413/588038 – cqfd May 28 '20 at 14:14
Thanks. Why does the last inequality hold? $|xy^{n} -y^{n}x| \leq 2|x|. |y| . |y^{n-1} |$ – eager2learn May 28 '20 at 14:20
Use the triangle inequality. – cqfd May 28 '20 at 14:22
yes, but I dont see how you can apply it here – eager2learn May 28 '20 at 14:29
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$|xy^{n} -y^{n}x| \leq |xy^{n}| +|y^{n}x| \leq |x| \cdot|y| \cdot |y^{n-1} | + |x| \cdot|y| \cdot|y^{n-1} | =2|x| \cdot|y| \cdot|y^{n-1} |$. Recall that the operator norm is submultiplicative. – cqfd May 28 '20 at 14:33
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Yes it did, thanks for the help. – eager2learn May 28 '20 at 16:49

Given two non-commuting linear operators prove that at least one is discontinuous

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