We prove the norm identity $\|Id+T\| = 1+\|T\|$, which is known as the
Daugavet equation, for operators $T$ on $C(S)$ not fixing a copy of $C(S)$,
where $S$ is a compact metric space without isolated points

Weis, Lutz

Werner, Dirk

English

arXiv

. We prove the norm identity kId + Tk = 1 + kTk, which is known as the Daugavet equation, for operators T on C(S) not fixing a copy of C(S), where S is a compact metric space without isolated points. 1. Introduction  An operator T : X ! X on a Banach space is said to satisfy the Daugavet equation if  kId + Tk = 1 + kTk; (1:1) this terminology is derived from Daugavet&apos;s theorem that a compact operator on C[0; 1] satisfies (1.1). Many authors have established the Daugavet equation for various classes of operators, e.g., the weakly compact ones, on various spaces; we refer to [1], [2], [4], [7], [10], [11], [13] and the references in these papers for more information. The most far-reaching result for operators on L  1  [0; 1] is due to Plichko and Popov [7, Th. 9.2 and Th. 9.8] who prove that an operator on L  1  [0; 1] which does not fix a copy of L  1  [0; 1] satisfies the Daugavet equation. (As usual, T : X ! X fixes a copy of a Banach space E if there is a subspace  F ae X isomorphic ..

Lutz Weis

Dirk Werner

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The Daugavet equation for operators not fixing a copy of C(S)

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.241.4481

The Daugavet equation for operators not fixing a copy of $C(S)$

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The Daugavet equation for operators not fixing a copy of $C(S)$