Beurling type invariant subspaces on Hardy and Bergman spaces of the unit ball or polydisk

McCullough and Trent generalize Beurling-Lax-Halmos invariant subspace theorem for the shift on Hardy space of the unit disk to the multi-shift on Drury-Arveson space of the unit ball by representing an invariant subspace of the multi-shift as the range of a multiplication operator that is a partial isometry. By using their method, we obtain similar representations for a class of invariant subspaces of the multi-shifts on Hardy and Bergman spaces of the unit ball or polydisk. Our results are surprisingly general and include several important classes of invariant subspaces on the unit ball or polydisk.