Daugavet’s equation and orthomorphisms

Schmidt, Klaus D.

doi:10.1090/S0002-9939-1990-1002165-3

Daugavet’s equation and orthomorphisms
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by Klaus D. Schmidt

Proc. Amer. Math. Soc. 108 (1990), 905-911

DOI: https://doi.org/10.1090/S0002-9939-1990-1002165-3

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Abstract:

The main result of this paper asserts that every Dunford-Pettis operator on an AL-space having no discrete elements satisfies Daugavet’s equation $\left \| {I + T} \right \| = 1 + \left \| T \right \|$; this extends a recent result of Holub on weakly compact operators. The proof is based on properties of orthomorphisms on a Banach lattice which also yield a short proof of another result of Holub concerning Daugavet’s equation for bounded operators on an arbitrary AL- or AM-space.

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