Representable spaces have the polynomial Daugavet property. (English) Zbl 1357.46012
J. B. Guerrero and A. Rodríguez-Palacios [J. Funct. Anal. 254, No. 8, 2294–2302 (2008; Zbl 1149.46012)] introduced the class of representable Banach spaces and demonstrated that every representable space \(X\) possesses the Daugavet property, i.e., every weakly compact linear operator \(T: X \to X\) satisfies the Daugavet equation \(\|\mathrm{Id} + T\| = 1 + \|T\|\).
In the article under review, the authors prove that, in fact, every representable space \(X\) has the polynomial Daugavet property, that is, \(\|\mathrm{Id} + \Phi\| = 1 + \|\Phi\|\) for every weakly compact polynomial \(\Phi: B_X \to X\), where the norm of a map \(F: B_X \to X\) is defined as \(\|F\|=\sup\{\|F(x)\| : x \in B_X\}\).
Reviewer’s remark. To the best of my knowledge, the general question whether every space with the Daugavet property also has the polynomial Daugavet property (Problem 1.2, [M. Martín et al., Arch. Math. 94, No. 4, 383–389 (2010; Zbl 1201.46012)]) remains open.
In the article under review, the authors prove that, in fact, every representable space \(X\) has the polynomial Daugavet property, that is, \(\|\mathrm{Id} + \Phi\| = 1 + \|\Phi\|\) for every weakly compact polynomial \(\Phi: B_X \to X\), where the norm of a map \(F: B_X \to X\) is defined as \(\|F\|=\sup\{\|F(x)\| : x \in B_X\}\).
Reviewer’s remark. To the best of my knowledge, the general question whether every space with the Daugavet property also has the polynomial Daugavet property (Problem 1.2, [M. Martín et al., Arch. Math. 94, No. 4, 383–389 (2010; Zbl 1201.46012)]) remains open.
Reviewer: Vladimir Kadets (Kharkiv)
References:
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