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Winter 2002 A pseudo-Daugavet property for narrow projections in Lorentz spaces
Mikhail M. Popov, Beata Randrianantoanina
Illinois J. Math. 46(4): 1313-1338 (Winter 2002). DOI: 10.1215/ijm/1258138482


Let $X$ be a rearrangement-invariant space. An operator $T: X\to X$ is called narrow if for each measurable set $A$ and each $\varepsilon > 0$ there exists $x \in X$ with $x^2= \chi_A,\ \int x d \mu = 0$ and $\| Tx \| < \varepsilon$. In particular, all compact operators are narrow. We prove that if $X$ is a Lorentz function space $L_{w,p}$ on [0,1] with $p > 2$, then there exists a constant $k_X > 1$ such that for every narrow projection $P$ on $L_{w,p}$ $\| \operatorname{Id} - P \| \geq k_X. $ This generalizes earlier results on $L_p$ and partially answers a question of E. M. Semenov. Moreover, we prove that every rearrangement-invariant function space $X$ with an absolutely continuous norm contains a complemented subspace isomorphic to $X$ which is the range of a narrow projection and a non-narrow projection. This gives a negative answer to a question of A. Plichko and M. Popov.


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Mikhail M. Popov. Beata Randrianantoanina. "A pseudo-Daugavet property for narrow projections in Lorentz spaces." Illinois J. Math. 46 (4) 1313 - 1338, Winter 2002.


Published: Winter 2002
First available in Project Euclid: 13 November 2009

zbMATH: 1030.46027
MathSciNet: MR1988266
Digital Object Identifier: 10.1215/ijm/1258138482

Primary: 46E30
Secondary: 46B20

Rights: Copyright © 2002 University of Illinois at Urbana-Champaign


Vol.46 • No. 4 • Winter 2002
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