This thesis is about the study of the norm ||G+T|| of a compact perturbation of an operator G acting between Banach spaces. We first use the Daugavet property's point of view: an operator G is a Daugavet center if every rank-1 operator (or equivalently every compact operator) T satisfies ||G+T||=||G||+||T||. In the first chapter we exhibit some examples of Daugavet centers among the set of weighted composition operators acting on function spaces, such as the space C(K) of continuous functions on a perfect compact space K, the disk algebra, or the space of Lipschitz functions on a complete metric space. In the second chapter we study a weaker property, that is to say the equation ||G+T||=||G||+||T|| is now fulfilled for a smaller class of rank-1 operators, and such an operator G is called an almost Daugavet center. We give a characterization of almost Daugavet centers in terms of canonical l^1-type and thickness of the operator G. This leads to a characterization of the class of operators fixing a copy of l^1. The last chapter's point of view is quite different: we do not look anymore for a G that "maximizes" the norm of G+T for every compact operator T, but we try to find a compact operator T that minimizes ||G+T||. In other words, we want to estimate the essential norm of G. We complete some results concerning weighted composition operators acting between different Hardy spaces.
Daugavet centres and weighted composition opertors