Narrow operators and the Daugavet property for ultraproducts

Dmitriy Bilik; Vladimir Kadets; Roman Shvidkoy; Dirk Werner

doi:10.1007/s11117-003-9339-9

paper

Narrow operators and the Daugavet property for ultraproducts

arXiv:math/0107132 · doi:10.1007/s11117-003-9339-9

Abstract

We show that if $T$ is a narrow operator on $X = X_{1} \oplus_{1} X_{2}$ or $X = X_{1} \oplus_{\infty} X_{2}$ , then the restrictions to $X_{1}$ and $X_{2}$ are narrow and conversely. We also characterise by a version of the Daugavet property for positive operators on Banach lattices which unconditional sums of Banach spaces inherit the Daugavet property, and we study the Daugavet property for ultraproducts.

15 pages