On one-sided ideals of rings of continuous linear operators

Abstract

Let X be a real or complex locally convex vector space and Lc(X) denote the ring (in fact the algebra) of continuous linear operators on X. In this note, we characterize certain one-sided ideals of the ring Lc(X) in terms of their rank-one idempotents. We use our main result to show that a one-sided ideal of the ring of continuous linear operators on a real or complex locally convex space is triangularizable if and only if the one-sided ideal is generated by a rank-one idempotent if and only if rank(AB-BA)⩽1 for all A,B in the one-sided ideal. Also, a description of irreducible one-sided ideals of the ring Lc(X) in terms of their images or coimages will be given. (The counterparts of some of these results hold true for one-sided ideals of the ring of all right (resp. left) linear transformations on a right (resp. left) vector space over a general division ring.)