Spectral set

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In operator theory, a set X\subseteq\mathbb{C} is said to be a spectral set for an operator T if the spectrum of T is in X and von-Neumann's inequality holds for T on X - i.e. for all rational functions r(x) with no poles on X

\left\Vert r(T) \right\Vert \leq \left\Vert r \right\Vert_{X} = \sup \left\{\left\vert r(x) \right\vert : x\in X \right\}
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