The Daugavet equation for operators not fixing a copy of $C(S)$

Banach spaces with the Daugavet property

A Banach space $X$ is said to have the Daugavet property if every operator $T: X\to X$ of rank~$1$ satisfies $\|Id+T\| = 1+\|T\|$. We show that then every weakly compact operator satisfies this equation as well and that $X$ contains a copy of $\ell_{1}$. However, $X$ need not contain a copy of $L_{1}$. We also study pairs of spaces $X\subset Y$ and operators $T: X\to Y$ satisfying $\|J+T\|=1+\|T\|$, where $J: X\to Y$ is the natural embedding. This leads to the result that a Banach space with the Daugavet property does not embed into a space with an unconditional basis. In another direction, we…

Geometric aspects of the Daugavet property

Let X be a closed subspace of a Banach space Y and J be the inclusion map. We say that the pair (X,Y) has the Daugavet property if for every rank one bounded linear operator T from X to Y the following equality \|J+T\|=1+\|T\| holds. A new characterization of the Daugavet property in terms of weak open sets is given. It is shown that the operators not fixing copies of l_1 on a Daugavet pair satisfy the Daugavet equation. Some hereditary properties are found: if X is a Daugavet space and Y is its subspace, then Y is also a Daugavet space provided X/Y has the Radon-Nikodym property; if Y is refl…

Thickness of the unit sphere, $\ell_1$-types, and the almost Daugavet property

We study those Banach spaces $X$ for which $S_X$ does not admit a finite $\eps$-net consisting of elements of $S_X$ for any $\eps < 2$. We give characterisations of this class of spaces in terms of $\ell_1$-type sequences and in terms of the almost Daugavet property. The main result of the paper is: a separable Banach space $X$ is isomorphic to a space from this class if and only if $X$ contains an isomorphic copy of $\ell_1$.

一个具有无条件基的巴拿赫空间，它不是切线可数确定的

A Banach space with an unconditional basis which is not slicely countably determined

本文研究了一个由无限二叉树分支的适当族生成的巴拿赫空间的单位球几何，并回答了关于切线可数确定巴拿赫空间的相关问题。主要结果是二叉树空间是一个具有无条件基但不切线可数确定的巴拿赫空间，且不包含具有达乌瓦特性质的空间的同构副本。