MHB If an operator commutes, its inverse commutes

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If an operator T on a Hilbert space commutes with another operator S and T is invertible, then the inverse operator T^{-1} also commutes with S. The proof begins with the relationship TS = ST, leading to the equation T^{-1}TS = T^{-1}ST. By manipulating these equations, it can be shown that T^{-1}S = ST^{-1}. Ultimately, this establishes that T^{-1} commutes with S, confirming the original statement. The discussion effectively demonstrates the properties of commutation in the context of invertible operators on Hilbert spaces.
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Prove that if operator on a hilbert space $T$ commutes with an operator $S$ and $T$ is invertible, then $T^{-1}$ commutes with $S$.

$T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$
 
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Boromir said:
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 ...
 

Thread 'About the definition of topological manifold using closed sets'

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