On Narrow Operators from L_p into Operator Ideals

Abstract

It is well known that every l_2-strictly singular operator from L_p, 1<p<infty to any Banach space X with an unconditional basis is narrow. In this article, we extend this result to the setting of Banach spaces without an unconditional basis. We show that if 1 le p,r <infty , then every ell _2-strictly singular operator T from L_p into the Schatten–von Neumann r-class C_r is narrow. This is a noncommutative complement to results in Mykhaylyuk et al. (in Israel J Math 203:81–108, 2014).