This is an announcement for the paper "On Enflo and narrow operators acting on $L_p$" by V. Mykhaylyuk, M. Popov and B. Randrianantoanina.

Abstract: The paper is devoted to proofs of the following three results. Theorem A. For $1 < p < 2$ every non-Enflo operator $T$ on $L_p$ is narrow. Theorem B. For $1 < p < 2$ every operator $T$ on $L_p$ which is unbounded from below on $L_p(A)$, $A \subseteq [0,1]$, by means of function having a ``gentle'' growth, is narrow. Theorem C. For $2 < p, r < \infty$ every operator $T: L_p\rightarrow\ell_r$ is narrow. Theorem A was mentioned by Bourgain in 1981, as a result that can be deduced from the proof of a related result in Johnson-Maurey-Schechtman-Tzafriri's book, but the proof from there needed several modifications. Theorems B and C are new results. We also discuss related open problems.

Archive classification: math.FA

Mathematics Subject Classification: Primary 47B07, secondary 47B38, 46B03

Submitted from: randrib@muohio.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1201.4041

or

http://arXiv.org/abs/1201.4041