Invariant subspaces having two side frames over the bidisk

An invariant subspace M   of the Hardy space over the bidisk is said to have two side frames if M⊖zMM⊖zM and M⊖wMM⊖wM contain nonzero TwTw and TzTz invariant subspaces, respectively. If one frame is in H2(z)H2(z) and the other is in H2(w)H2(w), M is said to have two side rigid frames. We shall show an example of an invariant subspace having two side frames which is not unitarily equivalent to any one having two side rigid frames. We also give some sufficient conditions on M for M to be unitarily equivalent to a rigid one.