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MR2604547 (2011b:47014) 47A10 (34G10 47D06)
Nikolski, N. K. The spectral localization property for diagonal operators and semigroups. Algebra i Analiz 21 (2009), no. 6, 202--226; translation in St. Petersburg Math. J. 21 (2010), no. 6, 995–1013
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