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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.\n  (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 $1<p<2$. 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