$\vec{v^*}$ is a fixed point

$$\begin{eqnarray*}0 & \leq & \Vert T\vec{v^*}-\vec{v}^{\ *}\Vert\\ & \leq & \Ver... ... \ \Vert\underbrace{\vec{v}^{n}-\vec{v^*}}_{\rightarrow 0}\Vert \end{eqnarray*}$$

Since $\vec{v^*}$ is a limit of $\vec{v}_n$,

$$\begin{eqnarray*}\lim_{n\rightarrow\infty}\Vert\vec{v}_n-\vec{v^*}\Vert = 0 \end{eqnarray*}$$

hence

$$\begin{eqnarray*}\Vert T\vec{v^*}-\vec{v^*}\Vert = 0 \end{eqnarray*}$$

thus $\vec{v^*}$ is a fixed point of the operator L.