Question about the relation between the adjoin and inverse of linear operator on Hilbert space I am teaching myself functional analysis from a CS background. I am clueless about the following exercise problem of introductory functional analysis. Any hint or help is appreciated. Thanks a lot!
Let $H$ be a Hilbert space and $T: H \rightarrow H$ a bijective bounded linear operator whose inverse is bounded. Show that $(T^{*})^{-1}$ exists and
$$
(T^{*})^{-1} = (T^{-1})^{*}
$$
It is easy to show $T^{-1}$ exists, but I seem to not able to find any clue to find the relation between the $T^{-1}$ and $T^{*}$.

Hint: Since $T$ is a bounded bijective linear operator, you have the existence of $T, T^*, T^{-1}$ and $(T^{-1})^*$ (why?)
So then $\left< x, y\right> = \left<T^{-1}Tx, y\right> = \left<Tx, (T^{-1})^*y\right> = \left<x, T^*(T^{-1})^*)y\right>$, for all $x,y\in\mathcal{H}$.  What does this tell us about the relationship between $T^*$ and $(T^{-1})^*$?

Further hint, inverses are unique.

