The Daugavet property is equivalent to the polynomial Daugavet property

2025/07/08 by Sheldon Dantas, Dantas, Sheldon, Miguel Martı́n +3
Engineering · #Control Systems and Identification #FOS: Mathematics #Functional Analysis (math.FA)

paper · pdf · doi:10.48550/arxiv.2507.06143

openalex publication_date 2025/07/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

In this note, we prove that the Daugavet property implies the polynomial Daugavet property, solving a longstanding open problem in the field. Our approach is based on showing that a geometric characterization of the Daugavet property due to Shvidkoy, originally formulated in terms of the weak topology, remains valid when the weak topology is replaced by the weak polynomial topology. Using similar techniques, we further establish that every linear Daugavet center is also a polynomial Daugavet center, and that the weak operator Daugavet property implies its polynomial counterpart. As an application of the latter result, we present new examples of Banach spaces whose N-fold symmetric tensor products satisfy the Daugavet property.

Citations

• Weak operator Daugavet property and weakly open sets in tensor product spaces
• Daugavet property of Banach algebras of holomorphic functions and norm-attaining holomorphic functions
• L-orthogonal elements and L-orthogonal sequences
• Daugavet points in projective tensor products
• Daugavet property in projective symmetric tensor products of Banach spaces
• L-orthogonality, octahedrality and Daugavet property in Banach spaces
• Daugavet property in tensor product spaces
• A characterisation of the Daugavet property in spaces of Lipschitz\n functions
• Strong diameter two property and convex combination of slices reaching the unit sphere
• Daugavet centers
• The Daugavet property for spaces of Lipschitz functions
• The Daugavet property of C^*-algebras, JB^*-triples, and of their isometric preduals
• Banach spaces with the Daugavet property

Related

• Daugavet property in projective symmetric tensor products of Banach spaces
• Daugavet property in tensor product spaces
• Remarks on the Daugavet Property for Complex Banach Spaces
• Daugavet points in projective tensor products
• The Daugavet property of C^*-algebras, JB^*-triples, and of their isometric preduals