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Realizations of Herglotz-Nevanlinna functions via \(F\)-systems. (English) Zbl 1057.47012

Albeverio, S. (ed.) et al., Operator methods in ordinary and partial differential equations. Proceedings of the S. Kovalevsky symposium, Stockholm, Sweden, June 2000. Basel: Birkhäuser (ISBN 3-7643-6790-3/hbk). Oper. Theory, Adv. Appl. 132, 183-198 (2002).
The class of Herglotz-Nevanlinna functions of the special form \(V(z)=Q+Lz+\int_{-\infty}^{+\infty}\frac{d\Sigma}{t-z},\) with square matrices \(Q=Q^*\) and \(L\geq 0\), and non-decreasing matrix-valued function \(\Sigma(t)\) such that \(\int_{-\infty}^{+\infty}(d\Sigma(t)x,x)<+\infty\) for all vectors \(x\) from a finite-dimensional space \(\mathfrak{C}\), is considered. Realizations of the form \(V(z)=K^*(D-zF)^{-1}K,\;z\in\mathbb{C}\setminus\mathbb{R}\), are constructed, where \(K\) is an operator from \(\mathfrak{C}\) into an auxiliary Hilbert space \(\mathfrak{H}\), the operator \(F\) is an orthogonal projection in \(\mathfrak{H}\), and there exists a (possibly, unbounded) operator \(M\) in \(\mathfrak{H}\) such that \(\text{ Re}\, M=D,\;\text{ Im}\,M=KJK^*\), with an operator \(J=J^*=J^{-1}\) on \(\mathfrak{C}\). The aggregate \[ \Theta_F=\left(\begin{matrix} M & F & K & J \\ \mathfrak{H} & & & \mathfrak{C} \end{matrix}\right) \] is called in the paper an \(F\)-system. For the case when the operator \(M\) is bounded and \(F=I_{\mathfrak{H}}\), it coincides with the Brodskiĭ-Livšic operator system \[ \Theta=\left(\begin{matrix} M & K & J \\ \mathfrak{H} & & \mathfrak{C} \end{matrix}\right). \] Thus, the realization results of the authors generalize certain results of M. S. Brodskiĭ [“Triangular and Jordan representations of linear operators”, (Translations of Mathematical Monographs 32, AMS, Providence) (1971; Zbl 0214.38901)] and M. S. Livšic [“Operators, oscillations, waves (open systems)” (Translations of Mathematical Monographs 34, AMS, Providence) (1973; Zbl 0254.47001)]. In the paper under review, the results are given for the case \(J=I_{\mathfrak{C}}\) (the general case of \(J=J^*=J^{-1}\) is more involved and needs a further investigation).
Note of the reviewer: the authors of reference [2], i.e., [IEOT, 24 (1996), 1–45] are D. Z. Arov and M. A. Nudel’man (not A. A. Nudel’man!).
For the entire collection see [Zbl 1005.00033].

MSC:

47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
30D50 Blaschke products, etc. (MSC2000)
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
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