MHB If an operator commutes, its inverse commutes

[Boromir]

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]

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 ...