A Beurling-Lax type theorem in the unit ball (Q1600113)

Let \(\mathcal F,G\) be two Pontryagin spaces with the same negative index, and let \({\mathbf L}(\mathcal F,G)\) be the set of all bounded linear operators from \(\mathcal F\) into \(\mathcal G\). The Schur multipliers are the set of \({\mathbf L}(\mathcal F,G)\)-valued functions \(S\), analytic in an open subset \(\Omega(S)\) in the unit ball of \({\mathbb{C}}^N\), such that the kernel \(K_S(z,w)=(1-\langle z,w\rangle)^{-1}(I_{\mathcal G}-S(z)S(w)^*)\) is positive for all \(z,w\in\Omega(S)\). The authors characterize the spaces with reproducing kernel \(K_S\) (which is their version of the Beurling-Lax theorem) and associate to each Schur multiplier a coisometric realization.