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On one-sided ideals of rings of continuous linear operators

✍ Scribed by Mehdi Radjabalipour; Bamdad R. Yahaghi

Publisher
Elsevier Science
Year
2008
Tongue
English
Weight
119 KB
Volume
429
Category
Article
ISSN
0024-3795

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✦ Synopsis

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 one-sided ideals of the ring L c (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 L c (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.)

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