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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)
Lutz Weis and Dirk Werner
Abstract. 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.
1. Introduction
An operator T : X → X on a Banach space is said to satisfy the Daugavet
equation if
‖Id+ T‖ = 1 + ‖T‖; (1.1)
this terminology is derived from Daugavet’s theorem that a compact oper-
ator 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 L1[0, 1] is due to Plichko
and Popov [7, Th. 9.2 and Th. 9.8] who prove that an operator on L1[0, 1]
which does not fix a copy of L1[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 ? X isomorphic to E such that T |F is an (into-) isomorphism.) In this
paper we shall establish the corresponding result for operators on spaces of
continuous functions.
Theorem 1.1 Let (S, d) be a compact metric space without isolated points.
If T : C(S) → C(S) does not fix a copy of C(S), then T satisfies the Daugavet
equation.
2 Lutz Weis and Dirk Werner
In the next section we shall provide a direct proof of this theorem which is
measure theoretic in spirit, and in Section 3 we shall give another argument
which relies on a deep result due to Rosenthal [8]. Though this proof is much
shorter it is less revealing than the one in Section 2; therefore we found it
worthwhile to present both arguments.
Either of our approaches depends on the analysis of the representing
kernel of an operator T : C(S) → C(S); this is the family of Borel measu
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