LAUSR

Let M be an invariant subspace of $$H^2$$H2 over the bi-disk and $$N=H^2\ominus M$$N=H2⊖M. Let $$S_{z,N},S_{w,N}$$Sz,N,Sw,N be the compression of the multiplication operators $$T_z,T_w$$Tz,Tw on $$H^2$$H2 onto N. For a two-variable inner function $$\theta $$θ, let $$M_\theta = \theta M$$Mθ=θM and $$N_\theta =H^2\ominus M_\theta $$Nθ=H2⊖Mθ. We shall study the relationship of the ranks of the cross-commutators $$[S_{z,N},S^*_{w,N}]$$[Sz,N,Sw,N∗] and $$[S_{z,N_\theta },S^*_{w,N_\theta }]$$[Sz,Nθ,Sw,Nθ∗]. We also characterize M such that rank $$[S_{z,N},S^*_{w,N}]$$[Sz,N,Sw,N∗]$$\not =$$≠ rank $$[S_{z,N_\theta },S^*_{w,N_\theta }]$$[Sz,Nθ,Sw,Nθ∗] for any non-constant inner function $$\theta $$θ.