Fixed point of an adjoint operator

Let $H$ Hilbert space and $T: H → H$ linear operator continuous. We define $T^*: H → H$ such as $\langle T^*x\,,y\rangle = \langle x\,,Ty\rangle $

We suppose $||T|| \lt 1 $, I want to proof that $T$ and $T^*$ has the same fixed points, (for $x_0 \in H$, $Tx_0 = x_0 \iff T^*x_0 = x_0$).

It seems simple but I do not see it clearly, I would also like to see an example of when this is not true (for example, when $||T|| \ge 1 $)

Ideas and suggestions are appreciated, many thanks!

If $\|T\|<1$ and $Tx=x$ with $x\ne0$, then $$ \|x\|=\|Tx\|\leq\|T\|\,\|x\|<\|x\|. $$ This is a contradiction, so the only fixed point is $x=0$. As $\|T^*\|=\|T\|$, the same reasoning applies to $T^*$.

When $\|T\|\geq1$, this is not true anymore. For instance consider $$ T=\begin{bmatrix} 1&0\\1&0\end{bmatrix}. $$ Then the fixed points of $T$ are $$ \left\{\begin{bmatrix} t\\ t\end{bmatrix}:\ t\in\mathbb C\right\}, $$ while the fixed points of $T^*$ are $$ \left\{\begin{bmatrix} t\\ 0\end{bmatrix}:\ t\in\mathbb C\right\}. $$