  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.
