IUMJ

Title: Norm equalities for operators on Banach spaces

Authors: Miguel Martin, Vladimir Kadets and Javier Meri

Issue: Volume 56 (2007), Issue 5, 2385-2412

Abstract:

A Banach space $X$ has the Daugavet property if the Daugavet equation $\| \mathrm{Id} + T \| = 1 + \| T \|$ holds for every rank-one operator $T : X \to X$. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like $\| g(T) \| = f(\| T \|)$ for some functions $f$ and $g$) lead in fact to the Daugavet property of the space. On the other hand, there are equations (for example $\| \mathrm{Id} + T \| = \| \mathrm{Id} - T \|$) that lead to new, strictly weaker properties of Banach spaces.