Eigenspace of \lambda = 1 for a bounded linear operator and its adjoint

$\displaystyle H$ is a Hilbert space and $\displaystyle T \in B(H)$ is such that $\displaystyle ||T||_{H} \leq 1$. Show that $\displaystyle Tx = x \iff T^{*}x = x $.

I have tried looking at this problem in several ways and I just cannot seem to see the significance of $\displaystyle ||A|| \leq 1$ nor how to go about proving it. I understand how the adjoint is defined for a Hilbert space as a consequence of Reisz representation. I have tried to attack it using fixed point theorems. Any nudge in the right direction would be appreciated.