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Gramsch and Lay [8] 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 regularized functional calculus introduced by Haase [10, 11] assuming suitable conditions on $f$
. At the same time, we answer in the positive a question made by Haase [11, Remark 5.4] regarding the conditions on $f$
which are sufficient to obtain the spectral mapping theorem for the usual extended spectrum $\widetilde \sigma$
. We use the model case of bisectorial-like operators, although the proofs presented here are generic, and are valid for similar functional calculi.
