Remark Concerning Ces\'aro Operator on the Hardy Space $H^p(\mathbb{C}_+)$ in the Upper Half-Plane

We consider Ces\'aro operator on the Hardy space $H^p(\mathbb{C}_+)$ in the upper half-plane for $1<p<\infty$. In \cite{AS} it was proved that for all $1<p<\infty$ the spectrum of the operator $V=\frac{2(p-1)}{p}C-I$ is located on the unit circle and in \cite{ABC1} the authors of this note showed that for $p=2$ operator $V$ is unitary. In the present note we show that for $1<p<\infty$, $p\ne 2$, the norm of the operator $V$ is strictly greater than one.