Let $\displaystyle H$ be a Hilbert Space and $\displaystyle T$ be an operator in $\displaystyle H$.

Prove that the mapping $\displaystyle T\rightarrow T^*$ is an one to one and onto mapping of $\displaystyle B(H)$ into itself,where $\displaystyle T^*$ is the adjoint of $\displaystyle T$ and $\displaystyle B(H)$ is the set of bounded operator on $\displaystyle H$.

Since we know that $\displaystyle T$ uniquely determines the adjoint $\displaystyle T^*$ and by the fact that $\displaystyle T^*$ is an operator in $\displaystyle H$,hence,we can claim that the mapping from $\displaystyle T\rightarrow T^*$ is one to one and onto.

Is this the right way to prove?Can anyone help?