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