A general approach to Read's type constructions of operators without non-trivial invariant closed subspaces

We present a general method for constructing operators without non-trivial invariant closed subsets on a large class of non-reflexive Banach spaces. In particular, our approach unifies and generalizes several constructions due to Read of operators without non-trivial invariant subspaces on the spaces ${\mathrm{\ell }}_1$, $c_0$ or ${\oplus }_{{\mathrm{\ell }}_2}J$, and without non-trivial invariant subsets on ${\mathrm{\ell }}_1$. We also investigate how far our methods can be extended to the Hilbertian setting, and construct an operator on a quasireflexive dual Banach space which has no non-trivial $w^{\ast }$-closed invariant subspace.