All solutions to an operator Nevanlinna-Pick interpolation problem (Q1728891)

In this survey paper, a characterization of all solutions of the left tangential operator valued Nevanlinna-Pick (LTONP) problem is reviewed and derived using tools from mathematical systems theory: co-isometric realizations, the Douglas factorization, and state space realizations. The problem is formulated as follows: Given Hilbert spaces \(\mathcal{Z}, \mathcal{X},\mathcal{Y}\), and a data set, which is an operator triple \((W,\widetilde{W},Z)\) defined by \[ Z:\mathcal{Z}\to\mathcal{Z},\ W:\ell_+^2(\mathcal{Y})\to\mathcal{Z},\ \widetilde{W}: \ell_+^2(\mathcal{U})\to\mathcal{Z} \] such that \(ZW=WS_{\mathcal{Y}}\) and \(Z\widetilde{W}=\widetilde{W}S_{\mathcal{U}}\) where \(S_{\mathcal{Y}}\) and \(S_{\mathcal{U}}\) are forward shift operators in the respective \(\ell^2_+\) spaces of unilateral square summable sequences. \(F\) is a solution if \(F\in\mathcal{S}(\mathcal{U},\mathcal{Y})\) and \(WT_F=\widetilde{W}\) where \(T_F\) is the Toeplitz operator with symbol \(F\) and \(\mathcal{S}(\mathcal{U},\mathcal{Y})\) is the Schur class of operators from \(\mathcal{U}\) to \(\mathcal{Y}\), i.e., \(\|F\|_\infty\le1\) in the unit disk \(\mathbb{D}\). A~key observation is that \(W\), \(\widetilde{W}\) and even \(WT_F\) can be considered as controllability operators. First the problem is considered where the spectral radius of \(Z\) need not be bounded by 1 as in [\textit{C. Foias} et al., Metric constrained interpolation, commutant lifting and systems. Basel: Birkhäuser (1998; Zbl 0923.47009)], but in the main theorems that follows, we do have \(\|Z\|\le1\). If the Pick operator \(\Lambda=WW^*-\widetilde{W}\widetilde{W}^*\ge0\), then \(\|Z^*\|\le1\) and there exists some Hilbert space \(\mathcal{E}\) such that all solutions are given by a linear fractional transform \[ F(\lambda)=\left(\Upsilon_{11}(\lambda)X(\lambda)+\Upsilon_{12}(\lambda)\right) \left(\Upsilon_{21}(\lambda)X(\lambda)+\Upsilon_{22}(\lambda)\right)^{-1},~~\lambda\in\mathbb{D}, \] with arbitrary \(X\in\mathcal{S}(\mathcal{U},\mathcal{E})\). Explicit expressions are given for the \(\Upsilon_{ij}\) as functions of an admissible pair of complementary operators \(C:\mathcal{Z}\to\mathcal{E}\) and \(D:\mathcal{Y}\to\mathcal{E}\). The latter can be expressed in terms of an inner function \(\Theta\in\mathcal{S}(\mathcal{E},\mathcal{U})\). If \(\Lambda\) and \(P=WW^*\) are strictly positive, then \(A=W^*P^{-1}W:\ell_+^2(\mathcal{U})\to\ell_+^2(\mathcal{U})\) is a strict contraction and that can somewhat simplify the expressions for \(\Upsilon_{ij}\) showing that they are \(H^2\) functions that are natural generalizations of corresponding functions for the Nehari problem. Moreover, the \(\Upsilon_{i,j}\) form a \(2\times2\) \(J\)-contactive operator on \(\mathcal{E}\oplus\mathcal{U}\). The connection between \(F\) and \(X\) is one-to-one. The central solution, corresponding to \(X=0\), is the unique solution maximizing entropy. Finally, since no stability of the data is assumed, it is possible to consider several forms of commutant lifting and the Leech problem [\textit{R. B. Leech}, Integral Equations Oper. Theory 78, No. 1, 71--73 (2014; Zbl 1304.47022)] as special cases. Even though this is a survey paper, the proofs are all included, if not in the main text, then in several appendices. For the entire collection see [Zbl 1392.45001].