Kreĭn space representation and Lorentz groups of analytic Hilbert modules (Q1635855)

A subspace \(M\) of the Hardy space \(H^{2}(\mathbb D^{2})\) is called submodule if it is invariant under multiplications by the both coordinate functions on \(\mathbb D^{2}\). An associated integral operator \(C_{M}\) (defect operator) in \(H^{2}(\mathbb D^{2})\) is defined by the formula \((C_{M}f)(z)=\int_{\mathbb T^{2}}G^{M}(z,\lambda)f(\lambda)\,dm(\lambda)\), \(z\in\mathbb D^{2}\), where \(G^{M}(z,\lambda)\) is the quotient of the reproducing kernel of \(M\) over the Szegő kernel for \(H^{2}(\mathbb D^{2})\). There is constructed the Kreĭn space connected with this operator. For the submodule \(M\) with finite rank operator \(C_{M}\), there is obtained a bidisk version of Beurling's theorem. There is considered the group \(\mathcal G(M)\) of invertible operators \(T\) on \(M\) such that \(T^{\ast}C_{M}T=C_{M}\). It is shown that, in the case of finite rank defect operators, two submodules are congruent if and only if these associated groups are isomorphic. There is obtained a subgroup of \(\mathcal G(M)\) being invariant under unitary equivalence of submodules.