Norm equalities for operators

A Banach space $X$ has the Daugavet property if the Daugavet equation $|\Id

• T|= 1 + |T|$holdsforeveryrank-oneoperator$T:X \longrightarrow X$. We show that the most natural attempts to introduce new properties by considering other norm equalities for operators (like $\mathrm{\parallel }g(T)\mathrm{\parallel }=f(\mathrm{\parallel }T\mathrm{\parallel })$ 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 $\mathrm{\parallel }\text{Id}+T\mathrm{\parallel }=\mathrm{\parallel }\text{Id}-T\mathrm{\parallel }$) that lead to new, strictly weaker properties of Banach spaces.