Commutators on

Let T be a bounded linear operator on X = ( ∑ ℓ q ) ℓ 1 with 1 ⩽ q < ∞ . T is said to be X-strictly singular if the restriction of T on any subspace of X that is isomorphic to X is not an isomorphism. It is shown that the unique proper maximal ideal in L ( X ) is the set of all X-strictly singular operators. With some more efforts, we prove that T is a commutator in L ( X ) if and only if for all non-zero λ ∈ C , the operator T − λ I is not X-strictly singular.