SOME SPECTRAL INEQUALITIES INVOLVING GENERALIZED SCALAR OPERATORS

In 1971, Allan Sinclair proved that for a hermitian element h of a Ban ach algebra and lambda complex we have parallel-to lambda + h parallel -to = r(lambda + h), where r denotes the spectral radius. Using Levin' s subordination theory for entire functions of exponential type, we ex tend this result locally to a much larger class of generalized spectra l operators. This fundamental result improves many earlier results due to Gelfand, Hille, Colojoara-Foias, Vidav, Dowson, Dowson-Gillespie-S pain, Crabb-Spain, I. & V. Istratescu, Barnes, Pytlik, Boyadzhiev and others.