Classes of Linear Operators Vol. I

Gohberg, Israel; Goldberg, Seymour; Kaashoek, M. A.

doi:10.1007/978-3-0348-7509-7

Cited by 532 publications

(299 citation statements)

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“…In a similar manner, if k is supported in R-, using (2.30), (3.6), Proposition 2.3, and Corollary 3.2, we obtain that R(A) extends to a function that is continuous on C +, is matrices for all A E R (see e.g. [23]). We will use W~: to denote the subalgebra of those functions Z(A) for which z(c 0 has support in R i and )4) q %o to denote the subalgebra of those functions Z(A) for which Zoo : 0 and z(a) has support in R +.…”

Section: Similarly Ilk(x) Is Supported In the Left Half Line R- Thementioning

confidence: 69%

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

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A direct and inverse scattering theory on the full line is developed for a class of firstorder selfadjoint 2n • 2n systems of differential equations with integrable potential matrices. Various properties of the corresponding scattering matrices including unitarity and canonical Wiener-Hopf factorization are established. The Marchenko integral equations are derived and their unique solvability is proved. The unique recovery of the potential from the solutions of the Marchenko equations is shown. In the case of rational scattering matrices, state space methods are employed to construct the scattering matrix from a reflection coefficient and to recover the potential explicitly.

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Section: Similarly Ilk(x) Is Supported In the Left Half Line R- Thementioning

confidence: 69%

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Aktosun

Klaus

Mee

2000

Integr equ oper theory

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“…Hence, T X is isometrically isomorphic to the Laurent operator defined by the matrix (function) X, and thus it follows from Corollary XXIII.2.5 of Gohberg et al (1990) that the spectrum of T X coincides with its essential spectrum, which consists of only nonnegative numbers, as mentioned above. Hence, T X is nonnegative definite.…”

Section: Theorem 6 Suppose That the Assumptions In Theorem 4 Are Satimentioning

confidence: 75%

Separator-type robust stability theorem of sampled-data systems allowing noncausal LPTV scaling

Hagiwara

2009

Automatica

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This paper derives a necessary and sufficient condition for robust stability of sampleddata systems, which is stated by using the notion of separators that are dealt with in an operator-theoretic framework. Such operator-theoretic treatment of separators provides a new perspective, which we call noncausal linear periodically time varying scaling and leads to reducing conservativeness in robust stability analysis. A numerical example is given to demonstrate the results.

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“…Note that (i) and (iii) directly follow from theorem 4.1 in section I.4 of [21]. It is easy to show that the adjoint of any solution to (6.3) or (6.4) is also a solution to the same equation, and hence the unique solutions Q(0; 0) and N(0) must be self-adjoint.…”

Section: Q(x; T)a + a † Q(x; T)mentioning

confidence: 82%

Marchenko Equations and Norming Constants of the Matrix Zakharov-Shabat System

Demontis¹,

Mee²

2008

Operators and Matrices

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A method is given to construct globally analytic (in space and time) exact solutions to the focusing cubic nonlinear Schrödinger equation on the line. An explicit formula and its equivalents are presented to express such exact solutions in a compact form in terms of matrix exponentials. Such exact solutions can alternatively be written explicitly as algebraic combinations of exponential, trigonometric and polynomial functions of the spatial and temporal coordinates.

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Classes of Linear Operators Vol. I

Cited by 532 publications

References 0 publications

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Direct and inverse scattering for selfadjoint Hamiltonian systems on the line

Separator-type robust stability theorem of sampled-data systems allowing noncausal LPTV scaling

Marchenko Equations and Norming Constants of the Matrix Zakharov-Shabat System

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