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Unfortunately, spectral mapping is not stable under bounded perturbations. If the perturbed semigroup does not have “nice” properties such as eventual norm continuity, it can be very difficult to prove a suitable spectral mapping theorem. Furthermore, the spectral mapping theorem holds: () = (). When the Banach algebra A is the algebra L(X) of bounded linear operators on a complex Banach space X (e.g., the algebra of square matrices), the notion of the spectrum in A coincides.
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