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Let X be a rearrangement-invariant space. An operator T : X → X is called narrow if for each measurable set A and each " > 0 there exists x ∈ X with x 2 = A, R xdµ = 0 and k Txk 2, then there exists a constant kX > 1 so that for every narrow projection P on Lw,p k Id − P k ≥ kX. This generalizes earlier results on Lp 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.

A pseudo-Daugavet property for narrow projections in Lorentz spaces