Parameterization of all solutions of the three chains completion problem

The family of all solutions of the three chains completion problem with a prescribed tolerance is described explicitly. It is shown that this family can be parameterized by a natural set of contractive upper triangular operators. As an application all solutions to a suboptimal nonstationary Nehari problem are described.     

A harmonic-type maximal principle in the three chains completion problem

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all contractive interpolants in the three chains completion problem (see [4]), which is analogous to the maximal principle proven in [2] in case of the Schur parametrization of all contractive intertwining liftings in the commutant lifting theorem.     

A band method approach to a positive expansion problem in a unitary dilation setting

In this paper a positive commuting expansion problem is presented and solved. The problem is set up in the framework of a minimal unitary dilation of a contraction acting on a Hilbert space and includes the Carathéodory and other classical interpolation problems. By combining the geometry of the minimal unitary dilation with state space techniques from system theory, a special solution is constructed. Next using the band method approach and spectral factorizations of this special solution a linear fractional parameterization of all solutions is obtained. Explicit state space formulas and applications to some classical interpolation problems are given.     

A time-variant version of the commutant lifting theorem and nonstationary interpolation problems

This paper contains a generalization of the commutant lifting theorem to a time-variant setting. The main result, which is called the three chains completion theorem, is used to solve various nonstationary norm constrained interpolation problems.     

Discrete time-variant interpolation as classical interpolation with an operator argument

Necessary and sufficient conditions are derived for the existence of solutions to discrete time-variant interpolation problems of Nevanlinna-Pick and Nudelman type. The proofs are based on a reduction scheme which allows one to treat these time-variant interpolation problems as classical interpolation problems for operator-valued functions with operator arguments. The latter ones are solved by using the commutant lifting theorem.     

Nudelman interpolation and the band method

In previous work the authors developed a new addition of the band method based on a Grassmannian approach for solving a completion/extension problem in a general, abstract framework. This addition allows one to obtain a linear fractional representation of all solutions of the abstract completion problem from special extensions which are not necessarily band extensions (for the positive case) or triangular extensions (for the contractive case). In this work we extend this framework to a somewhat more general setting and show how one can obtain formulas for the required special extensions from solutions of a system of linear equations. As an application we show how the formalism can be applied to the bitangential Nevanlinna-Pick interpolation problem, a case which, up to now, was not amenable to the band method.The first author was partially supported by National Science Foundation grant DMS-9500912.     

On the Nevanlinna-Pick interpolation problem for generalized stieltjes functions

The solutions of the Nevanlinna-Pick interpolation problem for generalized Stieltjes matrix functions are parametrized via a fractional linear transformation over a subset of the class of classical Stieltjes functions. The fractional linear transformation of some of these functions may have a pole in one or more of the interpolation points, hence not all Stieltjes functions can serve as a parameter. The set of excluded parameters is characterized in terms of the two related Pick matrices.Dedicated to the memory of M. G. Kreîn     

The band method and generalized Carathéodory-Toeplitz interpolation at operator points

A generalization of the band method is presented. In the new set up there are two semi-band structures, a right one and a left one, and the two are coupled. Moreover the role of the band extension in the classical version is taken over by a special positive real part extension. It is shown that with minor modifications the main results of the band method extend to this more general set up. As an application a generalization of the Carathéodory-Teoplitz interpolation problem for operator functions with operator arguments is solved.     

Relaxation of metric constrained interpolation and a new lifting theorem

In this paper a new lifting interpolation problem is introduced and an explicit solution is given. The result includes the commutant lifting theorem as well as its generalizations in [27] and [2]. The main theorem yields explicit solutions to new natural variants of most of the metric constrained interpolation problems treated in [9]. It is also shown that via an infinite dimensional enlargement of the underlying geometric structure a solution of the new lifting problem can be obtained from the commutant lifting theorem. However, the new setup presented obtained from the commutant lifting theorem. However, the new setup presented in this paper appears to be better suited to deal with interpolations problems from systems and control theory than the commutant lifting theorem.Dedicated to Israel Gohberg, as a token of admiration for his inspiring work in analysis and operator theory, with its far reaching applications, in friendship and with great affection.     

On maximum entropy interpolants and maximum determinant completions of associated Pick matrices

In this paper we shall show that the value of the maximum -entropy solution of an interpolation problem of the Nevanlinna-Pick type at the point maximizes the determinant of the solution of an associated matrix completion problem. This serves to show that the solutions of two distinct extremal problems coincide.Harry Dym wishes to express his thanks to Renee' and Jay Weiss for endowing the chair which supports his research.     

A function-theoretic approach to a parametrization of the set of solutions of a completely indeterminate matrical Nehari problem

In this paper, a function-theoretic approach to the completely indeterminate matricial Nehari Problem is given. The uniqueA-normalized resolvent matrixU is constructed by a limit procedure. Moreover, some limit relations for the Potapov-Ginzburg transform ofU are obtained.     

The quadratic moment problem for the unit circle and unit disk

For the quadratic complex moment problem , we obtain necessary and sufficient conditions for the existence of representing measures supported in the unit circleT or in the closed unit disk . We explicitly construct all finitely atomic representing measures supported inT or which have the fewest atoms possible. For the quadratic -moment problem in which the moment matrixM(1) is positive and invertible, there exists an ellipse D such that the minimal (3-atomic) representing measures are supported in the complement of the interior region of . Finally, we apply these results to obtain information on the location of the zeros of certain cubic polynomials.Dedicated to the memory of Velaho D. Bowman-FialkowBoth authors were partially supported by research grants from the National Science Foundation. The second-named author was also partially supported by an award from the State of New York/UUP Professional Development and Quality of Working Life Committee.     

Tangential interpolation with symmetries and two-point interpolation problem for matrix-valued H2-functions

We describe all solutions of the two-sided tangential interpolation problem in the class of matrix-valued Hardy functions when symmetries are added: these symmetries are defined in terms of involutions ofH ₂. The obtained results are applied to a one-sided two-points tangential interpolation for matrix functions.The research of this author is partially supported by the NSF Grant DMS 9500924 and by the Binational United States-Israel Foundation Grant 9400271.     

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.     

On three Krein extension problems and some generalizations

Three basic extension problems which were initiated by M. G. Krein are discussed and further developed. Connections with interpolation problems in the Carathéodory class are explained. Some tangential and bitangential versions are considered. Full characterizations of the classes of resolvent matrices for these problems are given and formulas for the resolvent matrices of left tangential problems are obtained using reproducing kernel Hilbert space methods.Dedicated to the memory of M. G. Krein, a beacon for us both.The authors wish to acknowledge the partial support of the Israel-Ukraine Exchange Program. D. Z. Arov also wishes to thank the Weizmann Institute of Science for partial support and hospitality; H. Dym wishes to thank Renee and Jay Weiss for endowing the chair which supports his research.