# Journal of Operator Theory

Volume 43, Issue 2, Spring 2000 pp. 263-294.

Optimal time-variant systems and factorization of operators. II: Factorization**Authors**: D.Z. Arov (1), M.A. Kaashoek (2), and D.R. Pik (3)

**Author institution:**(1) Department of Mathematics, South-Ukrainian Pedagogical University, 270020 Odessa, Ukraine

(2) Faculteit der Wiskunde en Informatica, De Boelelaan 1081 a, 1081 HV Amsterdam, The Netherlands

(3) Faculteit der Wiskunde en Informatica, De Boelelaan 1081 a, 1081 HV Amsterdam, The Netherlands

**Summary:**For a block lower triangular contraction $T$ the maximal block lower triangular outer solutions $F$ and $G$ of the operator i nequalities $I - T^\ast T \geq F^\ast F$, and $I - T T^\ast \geq G G^\ast$ are identified in terms of optimal and star-optimal time-variant realizations of $T$, respectively. Special attention is given to the case when the inequality $I - T^\ast T \geq F^\ast F$ is satisfied for $F = 0$ only, and to the case when equality can be obtained. As a byproduct a characterization is derived of optimality of a time-variant sys tem in terms of the input coefficients of the systems only. The existence of maximal block lower triangular solutions $F$ of the operator inequality $I - T^\ast T \geq F^\ast F$ is also used to derive an optimal realization of $T$.

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