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• W1901450612 abstract "Let $X$ be a rearrangement-invariant space. An operator $T: Xto X$ 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." @default.
• W1901450612 created "2016-06-24" @default.
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• W1901450612 date "2002-10-01" @default.
• W1901450612 modified "2023-09-26" @default.
• W1901450612 title "A pseudo-Daugavet property for narrow projections in Lorentz spaces" @default.
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• W1901450612 doi "https://doi.org/10.1215/ijm/1258138482" @default.
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