Geometric Aspects of the Daugavet Property

✍ Scribed by R.V. Shvydkoy

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 149 KB
• Volume: 176
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2000.3626

✦ Synopsis

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 equality

&J+T&=1+&T&

(1) 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 (1). Some hereditary properties are found: if X is a Daugavet space and Y is its subspace, then Y is also a Daugavet space provided XÂY has the Radon Nikody m property; if Y is reflexive, then XÂY is a Daugavet space. Besides, we prove that if (X, Y) has the Daugavet property and Y/Z, then Z can be renormed so that (X, Z) possesses the Daugavet property and the equivalent norm coincides with the original one on Y. 2000 Academic Press

which is called the Daugavet equation, holds. If (2) is satisfied by operators from some class M we say that (X, Y) has the Daugavet property with respect to this class. Investigation of (2) was originated with the work of Daugavet [5], in which he establishes the equality for compact operators on C[0, 1]. This