Bounded operators without invariant subspaces on certain Banach spaces

This thesis presents the author's research result on the “invariant subspace problem” in the negative direction. Let X be any nonreflexive Banach space with a basis and let lp(X) (1 ≤ p < ∞) be the l_p-direct sum of X. A family of bounded operators are constructed on the class of separable Banach spaces that contain lp( X) as a complemented subspace. It is then proved that the operators constructed have no non-trivial closed invariant subspaces. Both the construction and the proof given remain valid if lp( X) is replaced by c0(X)—the c₀-direct sum of X. These two classes of Banach spaces contain most of the Banach spaces that are currently known to the author to have bounded operators without invariant subspaces.