Let \(V\) be a finite-dimensional inner product space and \(T\) a linear operator on \(V\). If \(T\) is invertible, show that \(T^*\) is invertible and \((T^*)^{-1} = (T^{-1})^*\).
Solution
1. \((T^{-1})^* T^* = (T T^{-1})^* = I^* = I\).
\((AB)^* = B^* A^*\), and \(T T^{-1} = I\).
2. \(T^* (T^{-1})^* = (T^{-1} T)^* = I^* = I\).
\((AB)^* = B^* A^*\), and \(T^{-1} T = I\).
3. Hence \(T^*\) is invertible with inverse \((T^{-1})^*\).
1 and 2 exhibit a two-sided inverse.
4. Q.E.D.
3.