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(a) C.J. Read showed that if X = l1 then there exists a bounded operator on l1 having no nontrivial invariant subspace.
(cf. [C.J. Read, A solution to the invariant subspace problem, Bull. London Math. Soc., 16(1984), 337-401.]
(b) If X = Cn (n > 1), T Î B(Cn)-CI is an arbitrary operator and a Î C is an eigenvalue of T, then M = Ker(T - aI) is a nontrivial subspace of X and TM Í M.( I is the identity operator on Cn)