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# Journal of Operator Theory

Volume 55, Issue 2, Spring 2006  pp. 393-438.

The Kalman-Yakubovich-Popov inequality for discrete time systems of infinite dimension

Authors D.Z. Arov (1) and M.A. Kaashoek (2) and D. Pik (3)
Author institution: (1) Physical-Mathematical Institute, Dept. of Mathematical Analysis, South-Ukrainian Pedagogical University, 65020 Odessa, Ukraine
(2) Faculty of Sciences, Dept. of Mathematics, Vrije Universiteit, De Boelelaan 1081 a, 1081 HV Amsterdam, The Netherlands
(3) Faculty of Sciences, Mathematical Institute, Universiteit Leiden, Niels Bohrweg 1, 2333 CA Leiden, The Netherlands

Summary:  Infinite dimensional discrete time dissipative scattering systems are introduced in terms of generalized (possibly unbounded) solutions of the Kalman-Yakubovich-Popov inequality (KYP-inequality). It is shown that for a minimal system the KYP-inequality has a generalized solution if and only if the transfer function of the system coincides with a Schur class function $\theta$ in a neighborhood of zero. The set of solutions of the KYP-inequality, its order structure, and the corresponding contractive systems are studied in terms of $\theta$. Also using the KYP-inequality a number of stability theorems are derived.

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