On Enflo and narrow operators acting on $L_p$

Details On Enflo and narrow operators acting on $L_p$ On Enflo and narrow operators acting on $L_p$

2012-01-19

arxiv.org/abs/1201.4041v3

Mathematics

Functional Analysis

This paper has been withdrawn, since a new better proof of one of the main results has been communicated to us. We will edit t

Scientific paper

The first part of the paper is inspired by a theorem of H. Rosenthal, that if an operator 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 the operator is narrow. (Here $L_1(A) = \bigl\{x \in L_1: \,\, {\rm supp} \, x \subseteq A \bigr\}$.) This leads to a natural question of finding mildest possible assumptions for operators on a given space $X$, which will imply that the operator is narrow. We find a partial answer to this question for operators on $L_p(0,1)$ with $1

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