  A Banach space $X$ has the Daugavet property if the Daugavet equation $\|\Id
+ T\|= 1 + \|T\|$ holds for every rank-one operator $T:X \longrightarrow X$. 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 $\|\Id + T\|= \|\Id - T\|$)
that lead to new, strictly weaker properties of Banach spaces.
