Spectrum of a nontrivial bounded projection ($P^2=P,P\ne 0,I⟹\sigma (P)=\{0,1\}$)

Let $P$ be a bounded linear operator on a complex Banach space with $P^2=P$, $P\ne 0$ and $P\ne I$. Its image of a linear map and kernel contain nonzero vectors, giving eigenvalues one and zero. For $\lambda \notin \{0,1\}$,Direct multiplication proves the formula, so the spectrum is exactly $\{0,1\}$ and its spectral radius is one. Since $P^j=P$ and $\Vert P\Vert \ge 1$, the spectral radius formula is explicit: ${\inf }_{j\ge 1}\Vert P^j{\Vert }^{1/j}={\lim }_{j\to \infty }\Vert P{\Vert }^{1/j}=1$.