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.