NARROW AND l2-STRICTLY SINGULAR OPERATORS FROM Lp

In the first part of the paper we prove that for 2 < p, r < ∞ every operator T : Lp → lr is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : Lp → X is narrow. Next, using similar methods we prove that every l2-strictly singular operator from Lp, 1 < p < ∞, to any Banach space with an unconditional basis, is narrow, which partially answers a question of Plichko and Popov posed in 1990. A theorem of H. P. Rosenthal asserts that if an operator T on L1[0, 1] satisfies the assumption that for each measurable set A ⊆ [0, 1] the restriction T ∣∣ L1(A) is not an isomorphic embedding, then T is narrow. (Here L1(A) = {x ∈ L1 : suppx ⊆ A}.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being l2-strictly singular, for operators on Lp[0, 1], 1 < p < 2, to be narrow. We define a notion of a “gentle” growth of a function and we prove that for 1 < p < 2 every operator T on Lp which, for every A ⊆ [0, 1], sends a function of “gentle” growth supported on A to a function of arbitrarily small norm is narrow.