Spectral mapping theorems for essential spectra and regularized functional calculi

Gramsch and Lay [10] gave spectral mapping theorems for the Dunford-Taylor calculus of a closed linear operator $T$, $$\widetilde{\sigma}_i(f(T)) = f(\widetilde{\sigma}_i(T)), $$ for several extended essential spectra $\widetilde{\sigma}_i$. 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.