Lifting Non-topological Divisors of Zero modulo the Compact Operators

Abstract

This paper provides the first examples of non-semiFredholm operators S on a Banach space such that the left or right multiplication operators R ↦ SR or R ↦ RS define linear embeddings of the corresponding Calkin algebra into itself. For instance, if S is a bounded linear operator on C(0, 1) with closed range such that Ker S ∼ l1, then there is a constant c > 0 with dist (SR, K(C(0, 1))) ≥ c dist(R, K(C(0, 1))) for all bounded operators R ∈ L(C(0, 1)). Here K(C(0, 1)) stands for the compact operators on C(0, 1). Moreover, if S: L1 → L1 has closed range and L1/Im S contains no copies of l1, then there is a constant c > 0 such that dist(RS, W(L1)) ≥ c dist(R, W(L1)) for all R ∈ L(L1). Here W(L1) denotes the weakly compact operators on L1.