This is an announcement for the paper "Narrow and $\ell_2$-strictly singular operators from $L_p$" by V. Mykhaylyuk, M. Popov, B. Randrianantoanina, and G. Schechtman.
Abstract: In the first part of the paper we prove that for $2 < p, r < \infty$ every operator $T: L_p \to \ell_r$ is narrow. This completes the list of sequence and function Lebesgue spaces $X$ with the property that every operator $T:L_p \to X$ is narrow. Next, using similar methods we prove that every $\ell_2$-strictly singular operator from $L_p$, $1<p<\infty$, 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 $L_1[0,1]$ satisfies the assumption that for each measurable set $A \subseteq [0,1]$ the restriction $T \bigl|_{L_1(A)}$ is not an isomorphic embedding, then $T$ is narrow. (Here $L_1(A) = {x \in L_1: {\rm supp} , x \subseteq A}$.) Inspired by this result, in the last part of the paper, we find a sufficient condition, of a different flavor than being $\ell_2$-strictly singular, for operators on $L_p[0,1]$, $1<p<2$, 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 $L_p$ which, for every $A\subseteq[0,1]$, sends a function of ``gentle" growth supported on $A$ to a function of arbitrarily small norm is narrow.
Archive classification: math.FA
Mathematics Subject Classification: Primary 47B07, secondary 47B38, 46B03, 46E30
Remarks: Dedicated to the memory of Joram Lindenstrauss
Submitted from: randrib@muohio.edu
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/1211.4854
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