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Journal of the London Mathematical Society 1986 s2-34(2):335-348; doi:10.1112/jlms/s2-34.2.335
© 1986 by London Mathematical Society
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© Oxford University Press

A Short Proof Concerning the Invariant Subspace Problem

C. J. Read

Department of Pure Mathematics and Mathematical Statistics 16 Mill Lane, Cambridge CB2 1SB

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 l1, 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 l1, 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.


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