To show that $T^{-1}=T^{*}$

Question

This is given as exercise problem in Functional analysis by Conway.It says if T$\in B(H,K)$,(where H and K are Hilbert spaces and T is continuous linear operator), then T is an isomorphism if and only if T is invertible and $T^{-1}=T^{*}$. Initially I thought it will be very easy and I have proved everything but I just couldn't find the reason why $T^{-1}=T^{*}$. Basically, I have to prove that $<Tx,y>=<x,T^{-1}y> \forall x\in H,y\in K $. But I am stuck. Any hint. Thanks.

Answer 1

If $T\in B(H,K)$ is an isometric isomorphism, then $\langle x,x\rangle=\langle Tx,Tx\rangle =\langle T^*Tx,x\rangle$, or $\langle (I_H-T^*T)x,x\rangle=0$ for all $x\in H$. Since $I_H-T^*T$ is self-adjoint, Corollary 2.14 of chapter II in Conway's book implies that $T^*T=I_H$, and similarly we see that $TT^*=I_K$.