How to calculate adjoint operator in Hilbert space.

Question

I have the following question. Let $T\in B(\mathbb{H},\mathbb{H})$ with $\|T\|=1$ and $T$ attains its norm at $f_1\in\mathbb{H}$ with $\|f_1\|=1$. Also let $T(f_1)=f_2$. I have to show that $T^{\ast}(f_2)=f_1$. I did the following but can't conclude anything. We have $\|T\|=1=\|Tf_1\|=\|f_1\|$,$~$ Also $\|f_2\|^2=\langle f_2,f_2\rangle=\langle f_2,Tf_1\rangle=\langle T^{\ast}f_2,f_1\rangle=\|Tf_1\|^2=\|f_1\|^2$.

Answer 1

$\|T^{*}f_2-f_1\|^{2}=\|T^{*}f_2\|^{2}+\|f_1\|^{2}-2 \Re \langle T^{*}f_2, f_1\rangle$. Recall that $\|T^{*}\|=\|T\|$. Hence we get $\|T^{*}f_2-f_1\|^{2} \leq \|f_2\|^{2} -2\langle f_2, T(f_1)\rangle+\|f_1\|^{2}=1-2+1=0$.