Existence and Uniqueness of Contractive Solutions of Some Riccati Equations

Adamjan, Vadim; Langer, H.; Tretter, Christiane

doi:10.1006/jfan.2000.3680

Cited by 41 publications

(61 citation statements)

References 7 publications

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“…The result is an immediate corollary to Theorems 4.1 and 4.9. We only notice that the role of the quantity δ in the bounds like (4.1) (see inequalities (4.18) The solution K satisfies the estimate (1) and K (2) are given by…”

Section: Theorem 49 Assume Hypothesis 47 Let the Sylvester Operatmentioning

confidence: 99%

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Albeverio

Мотовилов²,

Шкаликов

2009

Integr. equ. oper. theory

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ABSTRACT. Let A be a self-adjoint operator on a Hilbert space H. Assume that the spectrum of A consists of two disjoint components σ 0 and σ 1 . Let V be a bounded operator on H, off-diagonal and J-self-adjoint with respect to the orthogonal decomposition H = H 0 ⊕ H 1 where H 0 and H 1 are the spectral subspaces of A associated with the spectral sets σ 0 and σ 1 , respectively. We find (optimal) conditions on V guaranteeing that the perturbed operator L = A +V is similar to a selfadjoint operator. Moreover, we prove a number of (sharp) norm bounds on the variation of the spectral subspaces of A under the perturbation V . Some of the results obtained are reformulated in terms of the Krein space theory. As an example, the quantum harmonic oscillator under a PT -symmetric perturbation is discussed.

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Section: Theorem 49 Assume Hypothesis 47 Let the Sylvester Operatmentioning

confidence: 99%

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Albeverio

Мотовилов²,

Шкаликов

2009

Integr. equ. oper. theory

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“…Thus, the solution X(t) can be continued to [-a>,u>]. Obviously, this procedure can be continued to obtain a periodic solution of equation (1). This completes the proof.…”

Section: General Existence Criterionmentioning

confidence: 67%

“…which is analogous to the Cauchy formula for scalar analytic functions, defines an operator /(Mi) [1,8]. Thus we have …”

Section: Proof Let Ti Be a Cauchy Contour Around Cr(m\) Separating Amentioning

confidence: 85%

Periodic Solutions of the Riccati Equation in Banach Spaces

Baris¹,

Buraczewski²

2003

Demonstratio Mathematica

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Abstract. In this paper we study the problem of the existence and the construction of periodic solutions of the Riccati equation with continuous periodic coefficients defined on the real line with values in Banach space. IntroductionThe theory of invariant manifolds, especially the theory of center manifolds, yields an important contribution to the study of some systems of differential equations [2]- [11]. The main theorems of the invariant manifold theory have been already proved, generally speaking, for quasi-linear systems with a block-diagonal structure of their linear parts.For this reason an important problem arises how to construct a transformation, which transforms an arbitrary system of differential equations to such a form. However, this transformation significantly complicates the nonlinear part of the system. On the other hand, some results of the invariant manifold theory may be obtained for so-called systems of special form, which are differential systems with block-triangular structure of the linear part [4;5], [11]. In this connection a crucial question arises: how to construct a transformation, which transforms this system to a system of special form. To this end, we must construct a solution of the corresponding differential Riccati equation. We also note that systems of special form can be easily transformed to systems with block-diagonal structure of their linear parts. If the given and obtained systems are periodic, then the solution of the Riccati equation is also periodic. This motivates the study of periodic solutions of the Riccati equation.1991 Mathematics Subject Classification: 34A05, 34C25, 34G20.

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“…An analog of Theorem 3 for the case without gap, that is, for sup σ ≤ inf Σ or sup Σ ≤ inf σ is also known (see [3] and [15]). …”

Section: Theorem 3 Suppose That the Self-adjoint Bounded Perturbatiomentioning

confidence: 96%

Perturbation of spectra and spectral subspaces

Kostrykin

Makarov

Мотовилов

2006

Trans. Amer. Math. Soc.

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Abstract. We consider the problem of variation of spectral subspaces for linear self-adjoint operators with emphasis on the case of off-diagonal perturbations. We prove a number of new optimal results on the shift of the spectrum and obtain (sharp) estimates on the norm of the difference of two spectral projections associated with isolated parts of the spectrum of the perturbed and unperturbed operators, respectively.

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Existence and Uniqueness of Contractive Solutions of Some Riccati Equations

Abstract: In this paper we establish criteria for the existence and uniqueness of contractive solutions K of the Riccati equation KBK+KA&DK&C=0 under the assumption that the spectra of A and D are disjoint. Academic Press

Cited by 41 publications

References 7 publications

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Bounds on Variation of Spectral Subspaces under J-Self-adjoint Perturbations

Periodic Solutions of the Riccati Equation in Banach Spaces

Perturbation of spectra and spectral subspaces

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