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All solutions to an operator Nevanlinna-Pick interpolation problem. (English) Zbl 1470.47017

Duduchava, Roland (ed.) et al., Operator theory in different settings and related applications, 26th international workshop on operator theory and its applications, IWOTA, Tbilisi, Georgia, July 6–10, 2015. Cham: Birkhäuser. Oper. Theory: Adv. Appl. 262, 139-220 (2018).
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 [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 [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].

MSC:

47A57 Linear operator methods in interpolation, moment and extension problems
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
47A62 Equations involving linear operators, with operator unknowns
28D20 Entropy and other invariants
47-02 Research exposition (monographs, survey articles) pertaining to operator theory
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