(a) A Banach space X and an operator T ∈ B(X) having no nontrivial invariant subspace.

(b) A Banach space X and an operator T ∈ B(X) having a nontrivial invariant subspace.

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(a) C.J. Read showed that if X = l^{1} then there exists a bounded operator on l^{1} 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 = C^{n}(n > 1), T ∈ B(C^{n}) − CI is an arbitrary operator and α ∈ C is an eigenvalue of T, then M = Ker(T − αI) is a nontrivial subspace of X and T M ⊆ M.( I is the identity operator on C^{n})