Quotients of Banach Spaces with the Daugavet Property

Abstract

We consider a general concept of Daugavet property with respect to a norming subspace. This concept covers both the usual Daugavet property and its weak * analogue. We introduce and study analogues for narrow operators and rich subspaces in this general setting and apply the results to show that a quotient of L1[0, 1] over an ℓ1subspace can fail the Daugavet property. The latter answers a question posed to us by A. Pe lczyński in the negative.

The Daugavet equation for polynomials

“…For instance, weakly compact linear operators on C(K), K perfect, and L 1 (µ), µ atomless, satisfy the Daugavet equation (see [25] for an elementary approach). We refer the reader to the books [1, 2] and papers [20,26] …”

The Daugavet property in the Musielak–Orlicz spaces

“…Since µ is σ-finite, there is an ascending sequence of measurable sets with finite and positive measure (Ω n ) ∞ n=1 such that Ω = ∪ ∞ n=1 Ω n . Let T n = {t ∈ Ω n : M (t, N ′ (t, (1 + ǫ)|g|(t))) n}, n ∈ N. Clearly (T n ) ∞ n=1 is an ascending sequence of measurable sets of finite measure satisfying (18) sup t∈Tn M (t, N ′ (t, (1 + ǫ)|g|(t))) < ∞, n ∈ N.…”

The polynomial Daugavet property for atomless L 1(μ)-spaces

“…This is the case, among others, of the spaces C(K, E) when the compact space K is perfect (E is any Banach space), L 1 (µ, E) and L ∞ (µ, E) when the measure µ is atomless, the disk algebra A(D) and the algebra of bounded analytic functions H ∞ . We refer the reader to [1,7,8] for more information and background.…”