Norm equalities for operators

Details Norm equalities for operators Norm equalities for operators

2006-04-05

arxiv.org/abs/math/0604102v1

Indiana Univ. Math. J. 56 (2007), 2385--2412.

Mathematics

Functional Analysis

21 pages

Scientific paper

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.

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