A property of weakly compact operators on 𝐶[0,1]

Daugavet’s equation and orthomorphisms

“…In particular, Theorem 3.2 corrects a statement of Chauveheid [5] and Holub [9] concerning L°°(p), and Corollary 3.3 applies to absolute kernel operators on L (p.) or L°°(p) ; see [19,Chapter 13].…”

“…This result was subsequently extended into various directions [1, 4, 5, 7-13, 17, 18]; in particular, it follows from results of Foias and Singer [8] and Holub [9,10] that Daugavet's equation holds for every weakly compact operator on C[0,1] or L [0, 1], and that every bounded operator on these spaces satisfies max{||/ + r||, || J -T\\} = 1 + ||r||. It is remarkable that, with the exception of the results due to Foias and Singer [8] and Krasnoselskii [12], all known results on Daugavet's equation concern linear operators on a Banach lattice, whereas Banach lattice methods have only been used by Lozanovskii [13], Synnatzschke [17,18], and Abramovich [1].…”

“…D Theorem 2.2 is, up to an application of the representation theorems for ALand AM-spaces, essentially due to Holub [9,10]; see also Abramovich [1] for a short proof of Holub's result. …”

A generalization of a theorem of J. Holub

“…In a recent work [4] In other words either T or -T or both satisfy the so-called Daugavet equation. We refer to [2,6] for a history on the Daugavet equation and to [2,4,5] for some related results.…”

“…In other words either T or -T or both satisfy the so-called Daugavet equation. We refer to [2,6] for a history on the Daugavet equation and to [2,4,5] for some related results. The aim of this article is to present a very simple proof of this (or rather, of a slightly more general) result, which allows one to understand its geometrical nature.…”