Operators which preserve the spectrum

It is a classical result that a linear surjection which preserves the set of eigenvalues. The project is to investigate what happens more generally in the context of continuous linear operators and Banach algebras.

Requirements: Functional analysis, complex variable. A knowledge of French would be an advantage, but is not mandatory. References: B. Aupetit, Spectrum-preserving linear mappings between Banach algebras and JordanBanach algebras, J. London Math. Soc. 62 (2000), 917-924.