The invariant subspace problem on a class of nonreflexive Banach spaces, 1

Abstract

In [1] we gave a fairly short proof that there is an operator on the space l1, without nontrivial invariant subspaces, and we conjectured that the same might be true of any space l1 ⊕ W where W is a separable Banach space. This conjecture turns out to be true, and by proving it here we give the first example of a reasonably large class of Banach spaces for which the solution to the invariant subspace problem is known. This continues the sequence of counter-examples which began on an unknown Banach space (Enflo [2], Read [4], Beauzamy [3], (simplification of [2])), proceeded to the space l1 (Read [5,1]) and here continues with the case of any separable Banach space containing l1 as a complemented subspace. No counter-example is known to the author for a Banach space which does not contain l1.