Let $H$ be a Hilbert space , If a continuous linear operator $T:H \to H$ has a bounded inverse $T^{-1}$ , then $T^{*}$ has a bounded inverse and $(T^*)^{-1}=(T^{-1})^*$
This theorem was in Michael Reed's functional analysis page $186$ .
I'm quite confused why we need to assume $T$ has a bounded inverse . If $T$ has an inverse $T^{-1}$ , then $T$ is necessarily bijective . In particular , $T$ is surjective . So by open mapping theorem on banach space , we conclude $T$ is an open mapping which means $T^{-1}$ is continuous . But on banach space , continuous is equivalent to bounded . So I don't understand why we need this assumption .
Best Answer
The assumptions make the proof easy. If $TS=ST=I$ where $T,S$ are bounded operators, then $S^*T^*=T^*S^*=I^*=I$ gives $(T^*)^{-1}=S^*=(T^{-1})^*$.