The Invariant Subspace Problem on Some Banach Spaces with Separable Dual

Operators without non-trivial invariant subspaces (so called ‘cyclic operators’) are now known to exist on a number of Banach spaces, but all such Banach spaces are non-reflexive and contain copies of the sequence space l1 In this paper, we find cyclic operators on some new Banach spaces which do not contain l1. The Banach spaces involved are not reflexive, but they include the space co, which has separable dual, and a space j∞ (the l2 direct sum of countably many copies of the James space J) which has a separable bidual (indeed, all the spaces J ∞ * , J ∞ ** , J ∞ *** and so on, are separable). This seems to be a best possible result, short of finding a solution to the invariant subspace problem on a reflexive Banach space.