= Holomorphic spectral mapping theorem
{title2=$\sigma(f(x))=f(\sigma(x))$}
For a <holomorphic function> near the spectrum, the spectrum of its functional-calculus value is exactly the image of the original spectrum. If a scalar is absent from that image, the reciprocal of the shifted function is holomorphic on a smaller spectral neighborhood and supplies an inverse. If it is attained at $\lambda$, a divided difference factors $f(x)-f(\lambda)$ through $x-\lambda$. These factors commute, so invertibility of their product would contradict $\lambda\in\sigma(x)$.
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