Banach spaces with the Daugavet property

Abstract

The aim of this manuscript is to study Banach spaces with the Daugavet property: Banach spaces $X$ satisfying that the norm equality $ ||Id + T||=1+ ||T|| $ (known as the Daugavet equation, (DE) for short) holds for every bounded linear operator $T: X\longrightarrow X$ of rank one. Its starting point is a review of Daugavet's result from 1963 showing that (DE) holds for compact linear operators on $C[0,1]$ and of related results which were established in the XX~Century. Next, a chapter on those results from Banach space theory and topology that are used in the book is included. The core part of the text deals with the ``geometrical'' treatment of the subject developed in the XXI~Century using slices, narrow operators, and slicely countably determined sets. It presents the main consequences, the main examples, and some generalisations such as Daugavet centres, the almost Daugavet property, and a Lipschitz version of (DE). Finally, some geometric properties related to the Daugavet property are commented on: other possible norm equalities for operators, the so-called big slice phenomena, the alternative Daugavet property, and alternatively convex or smooth spaces. Each chapter ends with some notes, remarks and open questions.