The Commutant Lifting Approach to Interpolation Problems

Foiaş, Ciprian; Frazho, A. E.

doi:10.1007/978-3-0348-7712-1

Cited by 473 publications

(318 citation statements)

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“…For an elaborate discussion on Redheffer products we refer to [ [12]. The co-isometric case is obtained by applying the statement for the isometric case to M * 1 and M * 2 , and finally the unitary case follows from the result when M 1 and M 2 are both isometric and co-isometric.…”

Section: Then {X B C D} and { X B C D} Are Unitarily Equivalenmentioning

confidence: 99%

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Redheffer Representations and Relaxed Commutant Lifting

Horst

2010

Complex Anal. Oper. Theory

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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 D 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.

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Section: Then {X B C D} and { X B C D} Are Unitarily Equivalenmentioning

confidence: 99%

“…and later appeared in the encompassing commutant lifting theory of Sz.-Nagy-Foiaş [26] and D. Sarason [30]; cf., [12,13]. They also play an important role in linear system theory [33].…”

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confidence: 99%

Redheffer Representations and Relaxed Commutant Lifting

Horst

2010

Complex Anal. Oper. Theory

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“…It has a growing impact on different branches of mathematics and theoretical physics as invariant subspace theory [27], interpolation theory [3] or prediction theory for stationary stochastic processes [9]. Among different proofs for the above mentioned theorem there is one strongly connected to our approach.…”

Section: Introductionmentioning

confidence: 99%

Positive Definite Functions and Sebestyén's Operator Moment Problem

Sebestyén

Popovici

2005

Glasgow Math. J.

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Abstract. Given a sequence {A n } n∈‫ޚ‬ + of bounded linear operators between complex Hilbert spaces H and K we characterize the existence of a contraction (resp. isometry, unitary operator, shift) T on K such thatSuch moment problems are motivated by their connection with the dilatability of positive operator measures having applications in the theory of stochastic processes.The solutions, based on the fact that a certain operator function attached to T is positive definite on ‫,ޚ‬ extend the ones given by Sebestyén in [18], [19] or, recently, by Jabłoński and Stochel in [8]. Some applications, containing new characterizations for isometric, unitary operators, orthogonal projections or commuting pairs having regular dilation, conclude the paper.2000 Mathematics Subject Classification. Primary 47A57, 43A35; Secondary 47A20, 60G10, 47B99.

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“…. , n, with F (λ) ≤ 1 for λ ∈ D. An elegant answer to this problem was given by G. Pick (for the case N = 1; the extension to N > 1 was noted later -we refer to [4] for an account of classical interpolation theory from a modern viewpoint). Pick's condition is simply that the block matrix [(I − W * i W j )/(1 − λ i λ j )] n i,j=1 be nonnegative semidefinite:…”

Section: Introductionmentioning

confidence: 99%