Spectral mapping theorems for essential spectra and regularized functional calculi

Abstract

Gramsch and Lay [10] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$, $${\tilde{\sigma }}_i(f(T))=f({\tilde{\sigma }}_i(T)),$$ for several extended essential spectra ${\tilde{\sigma }}_i$. In this work, we extend such theorems for the natural functional calculus introduced by Haase [12,13]. We use the model case of bisectorial operators. The proofs presented here are generic, and are valid for similar functional calculus.