Norm equalities for operators

Abstract

A Banach space $X$ has the Daugavet property if the Daugavet equation $\|\Id + T\|= 1 + \|T\|$ holds for every rank-one operator $T:X⟶X$. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like $\Vert g(T)\Vert =f(\Vert T\Vert )$ 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 $\|\Id + T\|= \|\Id - T\|$) that lead to new, strictly weaker properties of Banach spaces.

21 pages