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 equality \( TS = ST \), leading to the conclusion that \( T^{-1}S = ST^{-1} \). This establishes that \( T^{-1} \) maintains the commutative property with \( S \), confirming the relationship between invertible operators and their inverses in the context of operator theory.
PREREQUISITESMathematicians, physicists, and students studying functional analysis or quantum mechanics who seek to deepen their understanding of operator commutation and its implications in Hilbert spaces.
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 $S$.
$T^{-1}S$=$T^{-1}T^{-1}TS$=$T^{-1}T^{-1}ST$