A pseudo-Daugavet property for narrow projections in Lorentz spaces

Abstract

Let $X$ be a rearrangement-invariant space. An operator $T:X\to X$ is called narrow if for each measurable set $A$ and each $\epsilon >0$ there exists $x\in X$ with $x^2={\chi }_A,\int xd\mu =0$ and $\Vert Tx\Vert <\epsilon$. 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 $k_X>1$ so that for every narrow projection $P$ on $L_{w,p}$ $\Vert Id-P\Vert \ge k_X.$ 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, which gives a negative answer to a question of A.Plichko and M.Popov.

24 pages