[Paper Review] An elementary approach to the Daugavet equation

Dirk Werner|arXiv (Cornell University)|Dec 12, 1994

Advanced Banach Space Theory9 references22 citations

TL;DR

This paper presents an elementary, unified approach to the Daugavet equation for operators on C(S)-spaces using stochastic kernels. By representing operators via their representing measures (μ_s), the authors derive a necessary and sufficient condition for the Daugavet equation ‖I+T‖=1+‖T‖, which simplifies proofs for weakly compact and c₀-factorable operators on compact Hausdorff spaces without isolated points.

ABSTRACT

Let $T\dopu C(S) o C(S)$ be a bounded linear operator. We present a necessary and sufficient condition for the so-called Daugavet equation $$ \|\Id+T\| = 1+\|T\| $$ to hold, and we apply it to weakly compact operators and to operators factoring through $c_{0}$. Thus we obtain very simple proofs of results by Foias, Singer, Pelczynski, Holub and others.

Motivation & Objective

• To provide a unified, elementary framework for proving the Daugavet equation across various classes of operators on C(S)-spaces.
• To generalize and simplify existing proofs of the Daugavet equation for weakly compact and c₀-factorable operators.
• To establish a necessary and sufficient condition for the Daugavet equation using the representing kernel (μ_s) of an operator T on C(S).
• To extend results to (AL)- and (AM)-spaces via duality and adjoint operators.
• To handle the complex case by adapting the main condition to complex scalars using modulus conditions.

Proposed method

• Represent a bounded linear operator T: C(S) → C(S) via its stochastic kernel (μ_s)_{s∈S}, where μ_s = T*δ_s.
• Use the identity ‖T‖ = sup_s ‖μ_s‖ to express operator norms in terms of the total variation of μ_s.
• Prove that max{‖I±T‖} = 1 + ‖T‖ holds if and only if sup_s [1 + |μ_s({s})| + |μ_s|(S\{s})] = 1 + ‖T‖.
• Establish condition (∗) as necessary and sufficient: sup_{s∈U} μ_s({s}) ≥ 0 for all nonvoid open U ⊂ S.
• Apply Baire category theorem to show that the set {t ∈ S : μ_s({t}) = 0 for all s} is dense in S when S has no isolated points.
• Use duality to extend results from C(S) to (AL)- and (AM)-spaces via adjoint operators and representation theorems.

Experimental results

Research questions

• RQ1 What is a necessary and sufficient condition for the Daugavet equation ‖I+T‖ = 1 + ‖T‖ to hold for operators on C(S)?
• RQ2 How can the stochastic kernel representation (μ_s) be used to simplify proofs of the Daugavet equation for weakly compact operators?
• RQ3 Under what conditions on the representing measures μ_s does the Daugavet equation hold for operators factoring through c₀?
• RQ4 Can the results be extended to complex Banach spaces, and how must the condition be modified?
• RQ5 What role does the absence of isolated points in S play in ensuring the Daugavet equation for weakly compact operators?

Key findings

• The Daugavet equation holds for T: C(S) → C(S) if and only if sup_{s∈U} μ_s({s}) ≥ 0 for every nonvoid open set U ⊂ S.
• For weakly compact operators on C(S) with S compact Hausdorff and without isolated points, the Daugavet equation holds due to the continuity of s ↦ μ_s in the weak topology.
• Operators factoring through c₀ on C(S) with S without isolated points satisfy the Daugavet equation, as shown by the Baire category argument on the set {t ∈ S : μ_s({t}) = 0 ∀s}.
• The result extends to (AL)- and (AM)-spaces via duality: if E is an (AL)- or (AM)-space, then max{‖I±T‖} = 1 + ‖T‖ for any bounded operator T on E.
• The condition for the Daugavet equation in the complex case is sup_{s∈U} (|1 + μ_s({s})| − (1 + |μ_s({s})|)) ≥ 0 for all nonvoid open U ⊂ S.
• The proof shows that positive operators on C(S), (AL)-, and (AM)-spaces satisfy the Daugavet equation, as μ_s({s}) ≥ 0 implies the condition (∗) holds.