A harmonic-type maximal principle in commutant lifting

In this note, we prove a harmonic-type maximal principle for the Schur parametrization of all intertwining liftings of an intertwining contraction in the commutant lifting theorem.     

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.     

The maximum principle for the three chains completion problem

A time-variant version of the maximum principle for the central solution in the commutant lifting theorem is given. The main result is illustrated on the Parrott completion problem.     

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.     

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.     

Commutant lifting theorem and interpolation in discrete nest algebras

Ball in [Ba] showed that the commutant lifting theorem for the nest algebras due to Paulsen and Power gives a unified approach to a wide class of interpolation problems for nest algebras. By restricting our attention to the case when nest algebras associated with the problems are discrete we derive a variant of the commutant lifting theorem which avoids language of representation theory and which is sufficient to treat an analog of the generalized Schur-Nevannlinna-Pick (SNP) problem in the setting of upper triangular operators.     

Commutant lifting and factorization of reproducing kernels

A general version of the commutant lifting theorem for operators between different spaces is proved. It includes as special cases the lifting theorems of Ball-Trent-Vinnikov and Volberg-Treil. A multivariable variant of the Volberg-Treil theorem is obtained as a corollary. A certain factorization property of reproducing kernels is shown to be a sufficient condition for the lifting. Another factorization property is shown to be a necessary condition.     

The weighted commutant lifting theorem in the coupling approach

A simple coupling argument is seen to provide an alternate proof of the weighted commutant lifting theorem of Biswas, Foias and Frazho (which includes, as a particular case, the abstract Nehari theorem of Treil and Volberg).     

Commutant lifting and interpolation: The time-varying case

We show how the commutant lifting theorem for nest algebras due to Paulsen and Power can be used to give a unified framework for the treatment of a variety of interpolation problems for nest algebras which have been considered recently in the literature. Applications include the treatment of robust control for time-varying systems.Partially supported by NSF grant DMS-9500912     

Dilation of generalized Toeplitz kernels on ordered groups

We introduce the concept of Toeplitz-Kre?n-Cotlar triplet on ordered groups and we prove a general dilation result and a general representation result for the positive definite case.This general result includes and extends previous generalizations of the Kre?n extension theorem, the Sz.-Nagy and Foias commutant lifting theorem and the generalized Herglotz-Bochner-Weil theorem.     

Complete Nevanlinna-Pick Property of Dirichlet-Type Spaces

We prove that all Dirichlet-type spaces of functions analytic in the unit disk whose derivatives are square area integrable with superharmonic weights have complete Nevanlinna-Pick reproducing kernels. As a corollary, we obtain a commutant lifting theorem for cyclic analytic two-isometries.     

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.     

On an intertwining lifting theorem for certain reproducing kernel Hilbert spaces

We provide an alternate approach to an intertwining lifting theorem obtained by Ball, Trent and Vinnikov. The results are an exact analogue of the classical Sz-Nagy-Foias theorem in the case of multipliers on a class of reproducing kernel spaces, which satisfy the Nevanlinna-Pick property.     

The intertwining lifting theorem for ordered groups

Sz.-Nagy's dilation theorem for a contraction on a Hilbert space has been extended by Mlak to the case of a contraction semigroup whose indices are the positive elements of a totally ordered discrete group. We generalize to this case the intertwining lifting theorem of Sz.-Nagy and Foias. Some previous interpolation results on ordered groups are obtained as consequences.     

Contractive liftings and the commutatorDilatations contractive et le commutateur

This paper presents the solution to a problem proposed by B.Sz.-Nagy about extending the commutant lifting theorem to the case when the underlying operators do not intertwine. The main theorem establishes minimal norm liftings of certain commutators. The proof is constructive. To cite this article: C. Foias et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 431–436.     

Two classical theorems of function model theory in coordinate-free presentation

This paper is devoted to the presentation, within the framework of a coordinate-free model, of two known Sz. Nagy—Foia theorems: The first one deals with the correspondence between the invariant subspaces of a contraction T and the regular factorizations of its characteristic function _T, while the second one is the commutant lifting theorem. The proofs are based on a coordinate-free approach to the model. In the first theorem an essential point is the singling out of the role of functional imbeddings and the formulation of a criterion for the existence of an invariant subspace in terms of a functional imbedding of a special form. As far as the commutant lifting theorem is concerned, our approach enables us to give a parametrization of the lifted operators with the aid of one free parameter instead of two dependent ones, as done by Sz.-Nagy and Foia.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 178, pp. 5–22, 1989.     

Projective modules and Hilbert spaces with a Nevanlinna-Pick kernel

In this paper we solve a mapping problem for a particular class of Hilbert modules over an algebra multipliers of a diagonal Nevanlinna-Pick (NP) kernel. In this case, the regular representation provides a multiplier norm which induces the topology on the algebra. In particular, we show that, in an appropriate category, a certain class of Hilbert modules are projective. In addition, we establish a commutant lifting theorem for diagonal NP kernels.