Spectral mapping theorems for essential spectra and regularized functional calculi

TL;DR

This paper extends spectral mapping theorems for essential spectra to the natural functional calculus for bisectorial operators, broadening the understanding of spectral behavior in operator theory.

Contribution

It generalizes spectral mapping theorems from Dunford-Taylor calculus to Haase's natural calculus, applicable to bisectorial operators.

Findings

Extended spectral mapping theorems for essential spectra.

Proved generic, broadly applicable proofs.

Applicable to similar functional calculi.

Abstract

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.