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If an operator commutes, its inverse commutes

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Boromir

Banned
Feb 15, 2014
38
Prove that if operator on a hilbert space $T$ commutes with an operator $S$ and $T$ is invertible, then $T^{-1}$ commutes with $S$.

$T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$
 

ThePerfectHacker

Well-known member
Jan 26, 2012
236
Prove that if operator on a hilbert space $T$ commutes with an operator $S$ and $T$ is invertible, then $T^{-1}$ commutes with $S$.

$T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$
Start with $TS = ST$ so $T^{-1}TS = T^{-1}ST$. This simplifies to $S = T^{-1}ST$ so ...