The Daugavet equation for polynomials

Abstract

Abstract. We study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality Id + P = 1 + P is satisfied for all weakly compact polynomials P : X → X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equationId + ωP = 1 + P for polynomials P : X → X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. This result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-homogeneous polynomials.In 1963, I. K. Daugavet [13] showed that every compact linear operator T on C[0, 1] satisfies Id + T = 1 + T , a norm equality which has become known as the Daugavet equation. Over the years, the validity of the above equality has been established for many classes of operators on many Banach spaces. 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]

Polynomial Numerical Index for Some Complex Vector-Valued Function Spaces

“…That is, for k ∈ N, we define n (k) (X) = inf{v(P ) : P ∈ P( k X; X), P = 1}, where P( k X; X) is the space of all continuous k-homogeneous polynomials from X into X, and call it the polynomial numerical index of order k of X. For more information and background, we refer the reader to the recent papers [17] (a survey) and to [7,9,19,20] and references therein.…”

“…We refer the reader to the books [1,2] and the papers [18,28] for more information and background. Very recently [9], the study of the Daugavet equation has been extended to polynomials and, more generally, to bounded functions from the unit sphere of a Banach space into the space. Let us recall the relevant definitions.…”

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

“…The main examples of Banach spaces having the polynomial Daugavet property are: C b (Ω, E) when the completely regular space Ω is perfect (E is any Banach space) and its finite-codimensional subspaces, L ∞ (µ, E) when the measure µ is atomless, and C w (K, E), C w * (K, E * ) when the compact space K is perfect. We refer the reader to [3,4] for more information and background. Let us remark that the polynomial Daugavet property may be characterized in terms of scalar polynomials.…”

“…In 2007 the study of the Daugavet equation was extended to polynomials [3] and, moreover, to bounded functions from the unit ball of a Banach space into the space. Let us recall the relevant definitions.…”

Equação de Daugavet para polinômios

“…Atualmente tais equações têm sido amplamente estudadas. Além disto, em 2007 Y. S. Choi et al [9] ampliaram o seu estudo para funções mais gerais, particularmente para polinômios definidos em espaços de Banach da seguinte maneira. Dado um espaço de Banach X, um polinômio P ∈ P(X, X) satisfaz a equação de Daugavet se…”