Unitary colligations, reproducing kernel Hilbert spaces, and Nevanlinna-Pick interpolation in several variables
Some progress of late in \(d\)-variable systems and interpolation theories was initiated by the remarkable paper of \textit{J. Agler} [Oper. Theory, Adv. Appl. 48, 47-66 (1990; Zbl 0733.32002)] where the generalized Schur class \({\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) was introduced and described in different terms. This class consists of all holomorphic functions \(W(z)=\sum_{t\in {\mathbb Z}^d_+}\widehat{W}_tz^t\) on the polydisk \({\mathbb D}^d\) (here \({\mathbb Z}^d_+=\{ t\in {\mathbb Z}^d| t_k\geq 0, k=1,\ldots ,d\}\), \(z^t={z_1}^{t_1}\cdots {z_d}^{t_d}\)), with values in \({\mathcal L}({\mathcal E}, {\mathcal E}_*)\) for Hilbert spaces \({\mathcal E}\) and \({\mathcal E}_*\), such that for any \(d\)-tuple \(T=(T_1,\ldots ,T_d)\) of commuting contractions on some Hilbert space \({\mathcal X}\) and any \(0<r<1\) the inequality \(| | W(rT)| | \leq 1\) holds, where \(W(rT)=W(rT_1,\ldots , rT_d)=\sum_{t\in{\mathbb Z}^d_+}\widehat{W}_t\otimes {(rT)}^t\in {\mathcal L}({\mathcal E}\otimes {\mathcal X}, {\mathcal E}_*\otimes {\mathcal X})\). J. Agler proved that \(W(z)\in {\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) if and only if there exist auxiliary Hilbert spaces \({\mathcal M}_k\) and holomorphic functions \(H_k(z)\) on \({\mathbb D}^d\), with values in \({\mathcal L}({\mathcal M}_k, {\mathcal E}_*)\), \(k=1,\ldots ,d\), such that (*) \(I- W(z)W(w)^*=\sum_{k=1}^d(1-z_k\bar{w}_k)H_k(z)H_k(w)^*\) \((z, w \in {\mathbb D}^d)\), and that \(W(z)\in {\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) if and only if there exists a unitary colligation \(U=({\mathcal H}={\mathcal H}_1\oplus\cdots\oplus {\mathcal H}_d, {\mathcal E}, {\mathcal E}_*; A, B, C, D)\) such that (**) \(W(z)=D+CZ(z){(I-AZ(z))}^{-1}B\) \((z\in {\mathbb D}^d)\), where \(Z(z)=\sum_{k=1}^dz_kP_k\) and \(P_k: {\mathcal H}\to {\mathcal H}_k\) is the orthogonal projection. Besides, Agler established a criterion for the solvability of the \(d\)-variable Nevanlinna-Pick problem, that is \(W(z^i)=w_i\) \((z^i\in {\mathbb D}^d, w_i\in {\mathbb C})\), \(i=1,\ldots ,n\), in the class \({\mathcal S}_d={\mathcal S}_d({\mathbb C}, {\mathbb C})\) in terms of relation (*) valid in the interpolation nodes. The paper under review developes Agler's ideas, for that the emphasis is done on systems and interpolation theoretical aspects. In particular, (**) means that \(W(z)\) is the transfer function of some \(d\)-dimensional linear input-state-output system of the Roesser type [see e.g. \textit{T. Kaczorek}, ``Two-dimensional linear systems'', Lect. Notes Control Inform. Sci. 68 (1985; Zbl 0593.93031], with the unitarity condition imposed. Such a system (unitary realization of \(W(z)\)) is essentially not unique for \(d>1\). The authors show, however, that if the operator-valued functions \(H_k(z)\), \(k=1,\ldots ,d\), from (*) are given then, together with \(W(z)\), they determine an essentially unique unitary realization of \(W(z)\). The functional model of such a realization is constructed, and also the functional model determined by \(W(z)\) and given operator-valued functions \(H_{*k}(z)\), \(k=1,\ldots ,d\), from the equality dual to (*), as well as the functional model corresponding to the ``two-sided'' equality analogous to (*) (which is also established in this paper), are constructed. The main instrument in these constructions is the reproducing kernel Hilbert spaces technique. The authors introduce the notions of \(d\)-variable left, right, and two-sided tangential Nevanlinna-Pick problems in the class \({\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\) and give criteria for their solvability, which generalize the Agler's criterion to the case of operator-valued functions. Using the result of D. Z. Arov and L. Z. Grossman on the parametrization of unitary extensions of a given partially defined isometry [see \textit{D. Z. Arov} and \textit{L. Z. Grossman}, Math. Nachr. 157, 105-123 (1992; Zbl 0777.47007)], they give the parametrization of all solutions for these problems where the parameter is an arbitrary function from \({\mathcal S}_d({\mathcal L}, {\mathcal N})\) for some definite Hilbert spaces \({\mathcal L}, {\mathcal N}\). The \(d\)-variable version of the Toeplitz corona theorem [see \textit{M. Rosenblum}, Integral Equations Oper. Theory 3, 125-137 (1980; Zbl 0452.46032) and \textit{J. W. Helton}, J. Oper. Theory 15, 359---375 (1986; Zbl 0605.46043)] is also obtained in the paper under review. Note that some results of this paper were obtained simultaneously [\textit{J. Agler} and \textit{J. McCarthy}, J. Reine Angew. Math. 506, 191-204 (1999)]. Let us note also that another type of multidimensional dissipative (contractive) and conservative (unitary) linear input-state-output systems, that is also intimately connected with the class \({\mathcal S}_d({\mathcal E}, {\mathcal E}_*)\), was introduced and investigated by the reviewer in two papers that are still undergoing the publishing process, however available in Web e-print archive [func-an/9804130, func-an/9810119].
linear input-state-output system of the Roesser type
scientific article; zbMATH DE number 1214062
scientific article