S. ter Horst

Coupling and Relaxed Commutant Lifting

Abstract.A Redheffer type description of the set of all contractive solutions to the relaxed commutant lifting problem is given. The description involves a set of Schur class functions which is obtained by combining the method of isometric coupling with results on isometric realizations. For a number of special cases, including the case of the classical commutant lifting theorem, the description yields a proper parameterization of the set of all contractive solutions, but examples show that, in general, the Schur class function determining the contractive lifting does not have to be unique. Also some sufficient conditions are given guaranteeing that the corresponding relaxed commutant lifting problem has only one solution.

Relaxed commutant lifting: an equivalent version and a new application

This paper presents a few additions to commutant lifting theory. An operator interpolation problem is introduced and shown to be equivalent to the relaxed commutant lifting problem. Using this connection a description of all solutions of the former problem is given. Also a new application, involving bounded operators induced by H 2 operator-valued functions, is presented.

Redheffer Representations and Relaxed Commutant Lifting

It is well known that the solutions of a (relaxed) commutant lifting problem can be described via a linear fractional representation of the Redheffer type. The coefficients of such Redheffer representations are analytic operator-valued functions defined on the unit disc

Equivalence after extension for compact operators on Banach spaces

{\mathbb {D}}

Standard versus strict Bounded Real Lemma with infinite-dimensional state space II: The storage function approach

of the complex plane. In this paper we consider the converse question. Given a Redheffer representation, necessary and sufficient conditions on the coefficients are obtained guaranteeing the representation to appear in the description of the solutions to some relaxed commutant lifting problem. In addition, a result concerning a form of non-uniqueness appearing in the Redheffer representations under consideration and an harmonic maximal principle, generalizing a result of A. Biswas, are proved. The latter two results can be stated both on the relaxed commutant lifting as well as on the Redheffer representation level.

The Bézout Equation on the Right Half-plane in a Wiener Space Setting

For the strictly positive case (the suboptimal case), given stable rational matrix functions G and K, the set of all H ∞ solutions X to the Leech problem associated with G and K, that is, G(z)X(z) = K(z) and \( {\mathrm {sup}}_{{\mid z \mid}{\leq}{1}} {\parallel {X}{(z)} \parallel} \ {\leq} {1} \), is presented as the range of a linear fractional representation of which the coefficients are presented in state space form. The matrices involved in the realizations are computed from state space realizations of the data functions G and K. On the one hand the results are based on the commutant lifting theorem and on the other hand on stabilizing solutions of algebraic Riccati equations related to spectral factorizations.

Graphs with sparsity order at most two: the complex case

In recent years the coincidence of the operator relations equivalence after extension and Schur coupling was settled for the Hilbert space case, by showing that equivalence after extension implies equivalence after one-sided extension. In the present paper we investigate consequences of equivalence after extension for compact Banach space operators. We show that generating the same operator ideal is necessary but not sufficient for two compact operators to be equivalent after extension. In analogy with the necessary and sufficient conditions for compact Hilbert space operators to be equivalent after extension, in terms of their singular values, we prove, under certain additional conditions, the necessity of a similar relationship between the s-numbers of two compact Banach space operators that are equivalent after extension, for arbitrary s-functions. We investigate equivalence after extension for operators on lp-spaces. We show that two operators that act on different lp-spaces cannot be equivalent after one-sided extension. Such operators can still be equivalent after extension, for instance all invertible operators are equivalent after extension; however, if one of the two operators is compact, then they cannot be equivalent after extension. This contrasts the Hilbert space case where equivalence after one-sided extension and equivalence after extension are, in fact, identical relations. Finally, for general Banach spaces X and Y, we investigate consequences of an operator on X being equivalent after extension to a compact operator on Y. We show that, in this case, a closed finite codimensional subspace of Y must embed into X, and that certain general Banach space properties must transfer from X to Y. We also show that no operator on X can be equivalent after extension to an operator on Y, if X and Y are essentially incomparable Banach spaces.

The discrete twofold Ellis-Gohberg inverse problem

The main results presented in this paper provide a complete and explicit description of all solutions to the left tangential operator Nevanlinna- Pick interpolation problem assuming the associated Pick operator is strictly positive. The complexity of the solutions is similar to that found in descriptions of the sub-optimal Nehari problem and variation on the Nevanlinna-Pick interpolation problem in the Wiener class that have been obtained through the band method. The main techniques used to derive the formulas are based on the theory of co-isometric realizations, and use the Douglas factorization lemma and state space calculations. A new feature is that we do not assume an additional stability assumption on our data, which allows us to view the Leech problem and a large class of commutant lifting problems as special cases. Although the paper has partly the character of a survey article, all results are proved in detail and some background material has been added to make the paper accessible to a large audience including engineers.