Banach spaces with the Daugavet property Kadets, Vladimir Martín Suárez, Miguel Rueda Zoca, Abraham Werner, Dirk This manuscript is not in its final form, it is a preliminary version which is freely distributed under a CC BY-NC-ND 4.0 licence. We ask the readers for communicating any misprint or inaccuracy to us that they may find and to send us any suggestion which may help to improve the manuscript. 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. 2025-05-23T06:33:05Z 2025-05-23T06:33:05Z 2025 book https://hdl.handle.net/10481/104200 eng http://creativecommons.org/licenses/by-nc-nd/4.0/ open access Attribution-NonCommercial-NoDerivatives 4.0 Internacional