Abstract
In the first part of the paper we prove that for 2 < p, r < ∞ every operator T: L p → ℓ r is narrow. This completes the list of sequence and function Lebesgue spaces X with the property that every operator T : L p → X is narrow.
Next, using similar methods we prove that every ℓ2-strictly singular operator from L p , 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 from L 1[0, 1] to itself 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 L 1(A) = {x ∈ L 1 : 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 from L p [0, 1] to itself, 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 from L p to itself which, for every A ⊆ [0, 1], sends a function of “gentle” growth supported on A to a function of arbitrarily small norm is narrow.
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Dedicated to the memory of Joram Lindenstrauss
G. S. supported by the Israel Science Foundation and by the U.S.-Israel Binational Science Foundation.
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Mykhaylyuk, V., Popov, M., Randrianantoanina, B. et al. Narrow and ℓ2-strictly singular operators from L p . Isr. J. Math. 203, 81–108 (2014). https://doi.org/10.1007/s11856-014-0012-8
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DOI: https://doi.org/10.1007/s11856-014-0012-8