SPECTRAL PROPERTIES OF THE OPERATOR EQUATION AX + XB = Y

Arendt, Wolfgang; Räbiger, Frank; Sourour, A. R.

doi:10.1093/qmath/45.2.133

Cited by 50 publications

(46 citation statements)

References 5 publications

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“…This result is due to G. Lumer and M. Rosenblum [34]. Equality (3.5) also holds if only one of the entries A 0 and A 1 is a bounded operator [8]. In this case (3.5) implies that if the spectra A 0 and A 1 are disjoint then 0 ∈ spec(S) and hence the operator S is boundedly invertible.…”

Section: Operator Sylvester Equationmentioning

confidence: 88%

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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: Operator Sylvester Equationmentioning

confidence: 88%

“…It is known that in general the spectrum of S is larger than the (numerical) difference between the spectra of A 0 and A 1 . More precisely, provided that spec(A 0 ) = C or spec(A 1 ) = C always the following inclusion holds [8]:…”

Section: Operator Sylvester Equationmentioning

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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“…Some useful results for the solvability of the algebraic Sylvester equation (57) with a bounded operator F ∈ H based on spectral properties of the operators A and E in various special cases can be found in the work of Arendt et al (1994).…”

Section: Has a Unique Solution Z ∈ H If And Only If F ∈ H −1 Moreovmentioning

confidence: 99%

Infinite-dimensional Sylvester equations: Basic theory and application to observer design

Emirsajlow

2012

International Journal of Applied Mathematics and Computer Science

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This paper develops a mathematical framework for the infinite-dimensional Sylvester equation both in the differential and the algebraic form. It uses the implemented semigroup concept as the main mathematical tool. This concept may be found in the literature on evolution equations occurring in mathematics and physics and is rather unknown in systems and control theories. But it is just systems and control theory where Sylvester equations widely appear, and for this reason we intend to give a mathematically rigorous introduction to the subject which is tailored to researchers and postgraduate students working on systems and control. This goal motivates the assumptions under which the results are developed. As an important example of applications we study the problem of designing an asymptotic state observer for a linear infinitedimensional control system with a bounded input operator and an unbounded output operator.

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“…(i) 1 ∈ ρ(T (1)); (ii) 2πki ∈ ρ(A) for every k ∈ Z and sup k∈Z (2πki − A) −1 < ∞; (iii) for every 1-periodic continuous function f : R → H, there exists a unique 1-periodic mild solution of the equation (1) u (t) = Au(t) + f (t).…”

Section: Theorem 1 the Following Are Equivalentmentioning

confidence: 99%

A new proof and generalizations of Gearhart’s theorem

Phóng

2007

Proc. Amer. Math. Soc.

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Abstract. Let H be a Hilbert space, let AP (R, H) be the space of almost periodic functions from R to H, and let A be a closed densely defined linear operator on H. For a closed subset Λ ⊂ R, let M (Λ) be the subspace of AP (R, H) consisting of functions with spectrum contained in Λ. We prove that the following properties are equivalent: (i) for every function f ∈ M (Λ) there exists a unique mild solution..} yields a new proof of the well-known Gearhart's spectral mapping theorem.

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SPECTRAL PROPERTIES OF THE OPERATOR EQUATION AX + XB = Y

Cited by 50 publications

References 5 publications

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

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

Infinite-dimensional Sylvester equations: Basic theory and application to observer design

A new proof and generalizations of Gearhart’s theorem

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