Metric Constrained Interpolation, Commutant Lifting and Systems

Foiaş, Ciprian; Frazho, A. E.; Gohberg, Israel; Kaashoek, M. A.

doi:10.1007/978-3-0348-8791-5

Cited by 126 publications

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“…The additional assumption Φ 11 (0)F (0) < 1 is necessary and sufficient for the inverse to be well defined for each λ ∈ D, and it is well known that R Φ [F ] ∈ S(U, Y) for any F ∈ S(E, E ′ ) which satisfies this constraint. Redheffer maps of this type play an important role in the theory of metric constrained interpolation and system and control theory, cf., [1,7,8] and the references given there. Typically in applications Φ 11 (0) < 1, or even Φ 11 (0) = 0, such that R Φ is defined on the whole of S(E, E ′ ).…”

Section: Theorem 02 Let F G ∈ S(u Y) Then T F ≺ T G If and Only mentioning

confidence: 99%

A pre-order and an equivalence relation on Schur class functions and their invariance under linear fractional transformations

Horst¹

2014

J. Operator Theory

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Abstract. Motivated by work of Yu.L. Shmul'yan a pre-order and an equivalence relation on the set of operator-valued Schur class functions are introduced and the behavior of Redheffer linear fractional transformations (LFTs) with respect to these relations is studied. In particular, it is shown that Redheffer LFTs preserve the equivalence relation, but not necessarily the pre-order. The latter does occur under some additional assumptions on the coefficients in the Redheffer LFT. IntroductionIn a 1980 paper [17] Yu.L. Shmul'yan introduced a pre-order relation on the set of Hilbert space contractions, and showed that linear fractional maps of Redheffer type, as initially studied by R.M. Redheffer in [13,14], preserve this pre-order.To be more precise, let H 1 and H 2 be Hilbert spaces. With L 1 (H 1 , H 2 ) we denote the set of contractions from H 1 to H 2 , and, given C ∈ L 1 (H 1 , H 2 ), we write D C for the defect operator D C = (I − C * C) (H 1 , H 2 ). (Actually,in [17] this is denoted by B ≺ A, however, more in line with other pre-orders on L 1 (H 1 , H 2 ), the order is reversed in [11], and we will adopt the notation of [11] here.) Section 1 contains a detailed discussion of the pre-order ≺ and the corresponding equivalence relation ∼. To give an idea, the set of strict contractions form an equivalence class, and strict contractions dominate all other contractions, and if A ∈ L 1 (H 1 , H 2 ) is an isometry or co-isometry, then A forms an equivalence class by itself and A is dominated by no other contraction than itself, see Lemma 1.7 below. Now let K 1 and K 2 be two additional Hilbert spaces and let Shmul'yan proved the following result.

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Section: Theorem 02 Let F G ∈ S(u Y) Then T F ≺ T G If and Only mentioning

confidence: 99%

A pre-order and an equivalence relation on Schur class functions and their invariance under linear fractional transformations

Horst¹

2014

J. Operator Theory

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“…On the other hand, given a closed subspace M of K such that decomposition (8) holds, then M is the range of an orthogonal projection in K. In this case, (M, ⟨⋅, ⋅⟩) is again a Krein space. Unlike in the Hilbert space case, the decomposition (8) need not hold for a given closed subspace.…”

Section: Preliminariesmentioning

confidence: 99%

“…Extensions of this theorem to an indefinite setting are given in [1][2][3][4][5]. In [6] (see also [7,8]), a time-variant version of the commutant lifting theorem is developed. This time-variant version is called the three-chain completion theorem and is used to solve a number of nonstationary norm constrained interpolation problems on Hilbert spaces.…”

Section: Introductionmentioning

confidence: 99%

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An Extension Theorem for a Sequence of Krein Space Contractions

Wanjala

2018

International Journal of Mathematics and Mathematical Sciences

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“…By the solvability condition of the standard NP interpolation problem [12], the constrained interpolation problem has a solution if and only if there exists T ∈ T (q, p) such that…”

Section: Theorem 1 There Exists a Solution To Problem 1 If And Only Ifmentioning

confidence: 99%

Multirate Periodic Systems and Constrained Analytic Function Interpolation Problems

Chai¹,

Qiu

2005

SIAM J. Control Optim.

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Multirate periodic systems and some related constrained analytic function interpolation problems are studied in this paper. After showing how to convert a general multirate periodic system to an equivalent LTI system with a structural constraint, we formulate some analytic function interpolation problems with such constraint which can find various applications in the study of multirate and periodic systems. Both the solvability conditions and characterization of all solutions are presented to these constrained interpolation problems.

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Metric Constrained Interpolation, Commutant Lifting and Systems

Cited by 126 publications

References 0 publications

A pre-order and an equivalence relation on Schur class functions and their invariance under linear fractional transformations

A pre-order and an equivalence relation on Schur class functions and their invariance under linear fractional transformations

An Extension Theorem for a Sequence of Krein Space Contractions

Multirate Periodic Systems and Constrained Analytic Function Interpolation Problems

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