Richness of invariant subspace lattices for a class of operators

In 1994, H. Mohebi and M. Radjabalipour proved that every operator in a certain class of operators on reflexive Banach spaces has infinitely many invariant subspaces. In this paper, we prove that the invariant subspace lattice for every operator in the class of operators on (general) Banach spaces is rich, and we give an example of an operator $T$ that has infinitely many invariant subspaces, while the invariant subspace lattice $\operatorname{Lat} (T)$ for $T$ is not rich. Here we call an invariant lattice subspace $\operatorname{Lat} (T)$ for the operator $T$ rich if there exists an infinite dimensional Banach space $E$ such that $\operatorname{Lat} (T)$ contains a sublattice that is order isomorphic to the lattice $\operatorname{Lat}(E)$ of all closed liner subspaces of $E$. Finally we show that the invariant subspace lattice $\operatorname{Lat} (T)$ for a bounded linear operator $T$ on a reflexive Banach space $X$ is reflexive-rich if and only if the invariant subspace lattice $\operatorname{Lat} (T^{*})$ for $T^{*}$ is reflexive-rich.