Quasiaffine orbits of invariant subspaces for uniform Jordan operators

We consider the problem of classification of invariant subspaces for the class of uniform Jordan operators. We show that given two invariant subspaces M1M1 and M2M2 of a uniform Jordan operator T=S(θ)⊕S(θ)⊕⋯T=S(θ)⊕S(θ)⊕⋯, the subspace M2M2 belongs to the quasiaffine orbit of M1M1 if and only if the restrictions T|M1T|M1 and T|M2T|M2 are quasisimilar and the compression TM2⊥ can be injected in the compression TM1⊥. Our result refines previous work on the subject by Bercovici and Smotzer.