On Narrow Operators from $$L_p$$ into Operator Ideals

Abstract It is well known that every $$l_2$$ l 2 -strictly singular operator from $$L_p$$ L p , $$1<p<\infty $$ 1 < ∞ to any Banach space X with an unconditional basis narrow. In this article, we extend result the setting of spaces without basis. We show if $$1 \le p,r <\infty ≤ , r then $$\ell _2$$ ℓ T $$L_p into Schatten–von Neumann r -class $$C_r$$ C This a noncommutative complement results in Mykhaylyuk et al. (in Israel J Math 203:81–108, 2014).