Strictly Singular and Cosingular Multiplications

Abstract

AbstractLet L(X) be the space of bounded linear operators on the Banach space X. We study the strict singularity and cosingularity of the two-sidedmultiplication operators S ↦ ASB on L(X), where A, B ∈ L(X) are fixed bounded operators and X is a classical Banach space. Let 1 < p < ∞and p ≠ 2. Our main result establishes that the multiplication S ↦ ASB is strictly singular on L(Lp(0, 1)) if and only if the non-zero operators A, B ∈ L(Lp(0, 1)) are strictly singular. We also discuss the case where X is a L1- or a L∞-space, as well as several other relevant examples.