Narrow operators and the Daugavet property for ultraproducts

Dmitriy Bilik; Vladimir Kadets; Roman Shvidkoy; Dirk Werner

doi:10.1007/s11117-003-9339-9

Narrow operators and the Daugavet property for ultraproducts

Dmitriy Bilik, Vladimir Kadets, Roman Shvidkoy, Dirk Werner

Research output: Contribution to journal › Article › peer-review

19 Scopus citations

Abstract

We show that if T is a narrow operator (for the definition see below) on X = X₁ ⊕₁ X₂ or X = X₁ ⊕∞ X₂ 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.

Original language	English (US)
Pages (from-to)	45-62
Number of pages	18
Journal	Positivity
Volume	9
Issue number	1
DOIs	https://doi.org/10.1007/s11117-003-9339-9
State	Published - Mar 2005
Externally published	Yes

Bibliographical note

Funding Information:
The work of V.K. was supported by a grant from the Alexander-von-Humboldt Stiftung.

Keywords

Daugavet property
Narrow operator
Strong Daugavet operator
Ultraproducts of Banach spaces

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AU - Bilik, Dmitriy

AU - Kadets, Vladimir

AU - Shvidkoy, Roman

AU - Werner, Dirk

N1 - Funding Information: The work of V.K. was supported by a grant from the Alexander-von-Humboldt Stiftung.

PY - 2005/3

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N2 - We show that if T is a narrow operator (for the definition see below) on X = X1 ⊕1 X2 or X = X1 ⊕∞ X2 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.

AB - We show that if T is a narrow operator (for the definition see below) on X = X1 ⊕1 X2 or X = X1 ⊕∞ X2 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.

KW - Daugavet property

KW - Narrow operator

KW - Strong Daugavet operator

KW - Ultraproducts of Banach spaces

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Narrow operators and the Daugavet property for ultraproducts

Abstract

Bibliographical note

Keywords

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