A Short Proof Concerning the Invariant Subspace Problem

Counterexamples for the invariant subspace problem on a general Banach space exist due to Enflo [ 1 ], Read [ 3 ], Beauzamy [ 3 ] (simplification of [ 1 ]). On the space l 1 , there is a counterexample due to Read [ 4 ], which is rather long since it uses all of [ 3 ]. Here we present a short, direct proof that there is an invariant subspace free operator T on l 1 , and we also establish the following facts about our operator. First, it is a perturbation of a weighted shift operator by a nuclear operator. Secondly, its spectrum is identical with its approximate point spectrum, which is the unit ball of C. Thirdly, we can arrange either that no positive power of T has invariant subspaces, or alternately that every positive power T ∗ except T itself has non‐trivial invariant subspaces.