Solution of the reconstruction-of-the-measure problem for canonical invariant subspaces (Q2143359)

Let \(\mathcal{H}\) be a complex Hilbert space and let \(\mathcal{B}(\mathcal{H})\) denote the algebra of bounded linear operators on \(\mathcal{H}\). We say that \(T \in \mathcal{B}(\mathcal{H})\) is normal if \(T^*T = TT^*\), subnormal if \(T = N|\mathcal{H}\), where \(N\) is normal and \(N(\mathcal{H}) \subseteq \mathcal{H}\), and hyponormal if \(T^*T \geq TT^*\). These notions extend to \(n\)-tuples of Hilbert space operators \(\mathbf{T} \equiv (T_1,\dots,T_n)\); for instance, \(\mathbf{T}\) is normal if \(T_i\) is normal and \(T_iT_j = T_jT_i\) for all \(i,j = 1,\dots,n\), and subnormal if \(\mathbf{T} = \mathbf{N}|\mathcal{H}\), where \(\mathbf{N}\) is normal on a larger Hilbert space \(\mathcal{K}\) and \(N_i{\mathcal{H}} \subseteq \mathcal{H}\) for all \(i=1,\dots,n\). In this paper, the authors focus on the case of \(\mathcal{H} = \ell^2({\mathbb{Z}_+}^2)\) and \(\mathbf{T}\) a 2-variable weighted shift \(W_{(\alpha, \beta)} \equiv (T_1, T_2)\). As is well known, the subnormality of such pairs is characterized by the existence of a representing measure (known as the Berger measure of \(W_{(\alpha, \beta)}\)) for the family of moments of the pair of double-indexed sequences \((\alpha, \beta)\). They study the Reconstruction-of-the-Measure Problem (ROMP) for commuting 2-variable weighted shifts \(W_{(\alpha, \beta)}\), when the initial data are given as the Berger measure of the restriction of \(W_{(\alpha, \beta)}\) to a canonical invariant subspace, together with the marginal measures for the 0th row and 0th column in the weight diagram for \(W_{(\alpha, \beta)}\). They prove that the natural necessary conditions are indeed sufficient. When the initial data correspond to a soluble problem, they give a concrete formula for the Berger measure of \(W_{(\alpha, \beta)}\). Their strategy is to build on previous results for back-step extensions and one-step extensions. The main result (Theorem 6.5) allows us to solve ROMP for two-step extensions. This, in turn, leads to a solution of ROMP for arbitrary canonical invariant subspaces of \(\ell^2({\mathbb{Z}_+}^2)\).