= Spectrum of a nontrivial bounded projection
{title2=$P^2=P,\quad P\ne0,I\quad\Longrightarrow\quad\sigma(P)=\{0,1\}$}
Let $P$ be a <bounded linear operator> on a complex <Banach space> with $P^2=P$, $P\ne0$ and $P\ne I$. Its <image of a linear map> and <kernel> contain nonzero <vectors>, giving <eigenvalues> one and zero. For $\lambda\notin\{0,1\}$,
$$
(\lambda I-P)^{-1}=\lambda^{-1}(I-P)+(\lambda-1)^{-1}P.
$$
Direct multiplication proves the formula, so the <spectrum> is exactly $\{0,1\}$ and its <spectral radius> is one. Since $P^j=P$ and $\|P\|\geq1$, the <spectral radius formula> is explicit: $\inf_{j\geq1}\|P^j\|^{1/j}=\lim_{j\to\infty}\|P\|^{1/j}=1$.
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