arXiv:1211.4854v1 [math.FA] 20 Nov 2012 NARROW AND ℓ 2 -STRICTLY SINGULAR OPERATORS FROM L p V. MYKHAYLYUK, M. POPOV, B. RANDRIANANTOANINA, AND G. SCHECHTMAN Dedicated to the memory of Joram Lindenstrauss Abstract. 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 uncon- ditional 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   L 1 (A) is not an isomorphic embedding, then T is narrow. (Here L1(A)= {x ∈ L1 : supp x ⊆ 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. 1. Introduction In this paper we study narrow operators on the real spaces L p , for 1 ≤ p< ∞ (by L p we mean the L p [0, 1] with the Lebesgue measure µ). Let X be a Banach space. We say that a linear operator T : L p → X is narrow if for every ε> 0 and every measurable set A ⊆ [0, 1] there exists x ∈ L p with x 2 = 1 A ,  [0,1] x dµ = 0, so that ‖Tx‖ <ε. By Σ we denote the σ-algebra of Lebesgue measurable subsets of [0, 1], and set Σ + = {A ∈ Σ: µ(A) > 0}. As is easy to see, every compact operator T : L p → X is narrow. However in general the class of narrow operators is much larger than that of compact 2010 Mathematics Subject Classification. Primary 47B07; secondary 47B38, 46B03, 46E30. Key words and phrases. Narrow operator, ℓ2-strictly singular operator, Lp(µ)-spaces. G.S. was supported by the Israel Science Foundation and by the U.S.-Israel Binational Science Foundation. 1