Narrow operators and the Daugavet property for ultraproducts

• DOI: 10.1007/s11117-003-9339-9
• Publication Date: 2005-04-20T10:52:37Z
• Authors: Dmitriy Bilik; Vladimir Kadets; Roman Shvidkoy; Dirk Werner
• Keywords: Ultraproduct; Operator (biology)

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.<br/>15 pages<br/>