Some spectral mapping theorems through local spectral theory

The spectral mapping theorems for Browder spectrum and for semi-Browder spectra have been proved by several authors [14], [29] and [33], by using different methods. We shall employ a local spectral argument to establish these spectral mapping theorems, as well as, the spectral mapping theorem relative to some other classical spectra. We also prove that if T or T* has the single-valued extension property some of the more important spectra originating from Fredholm theory coincide. This result is extended, always in the case T or T* has the single valued extension property, to f(T), where f is an analytic function defined on an open disc containing the spectrum of T. In the last part we improve a recent result of Curto and Han [10] by proving that for every transaloid operator T a-Weyl's theorem holds for f(T) and f(T)*. © 2004 Springer.