The Daugavet equation for polynomials

Abstract

In this paper 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 equation max |ω|=1 ‖Id + ω P‖ = 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. The 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 currently 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 L1(μ), μ atomless, satisfy Daugavet equation (see [25] for an elementary approach). We refer the reader to the books [1, 2] and the papers [20, 26] for more information and background. It is also a remarkable result given in 1970 by J. Duncan et al. [16] that, for every compact Hausdorff space K and every bounded linear operator T on C(K), the equality max ω∈T ‖Id + ωT‖ = 1 + ‖T‖ holds, where we use T to denote the unit sphere of the base field. The above norm equality is now known as the alternative Daugavet equation [22], and it is satisfied by all bounded linear operators on C(K) and L1(μ), K and μ arbitrary. We refer the reader to [16, 21, 22] and references there in for background. The aim of this paper is to study the Daugavet equation and the alternative Daugavet equation for polynomials in Banach spaces. There is a concept, the numerical range of an operator (see below for the definition), intimately related to the Daugavet and alternative Daugavet equations. The definition of numerical range for bounded linear operators on Banach spaces was given in 1962 by F. Bauer [5] (see [9, 10] for background) extending the 1918 classical definition of numerical range (or field of values) of a matrix given by O. Toeplitz [24]. In 1968, the concept of numerical range was extended to arbitrary continuous functions from the unit sphere of a real or complex Banach space into the space by F. Bonsall, B. Cain, and H. Schneider [7]. In the seventies, Date: March 24th, 2006; revised September 13th, 2006. 2000 Mathematics Subject Classification. Primary 46G25; Secondary 46B20, 47A12.