Estimates for the Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations

Баскаков, А. Г.

doi:10.1070/sm2015v206n08abeh004489

Cited by 17 publications

(12 citation statements)

References 40 publications

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“…We observe that the method of similar operator is usually employed in the spectral analysis of various classes of differential (see [37] - [40]) and difference (see [41], [42]) operators. While presenting the method, we shall base on works [21], [22].…”

Section: Methods Of Similar Operatorsmentioning

confidence: 99%

Fourier method for first order differential equations with involution and groups of operators

Баскаков¹,

Ускова²

2018

Ufimskii Matematicheskii Zhurnal

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In the paper we study a mixed problem for a first-order differential equation with an involution. It is written by means of a differential operator with an involution acting in the space functions square integrable on a finite interval. We construct a similarity transform of this operator in an operator being an orthogonal direct sum of an operator of finite rank and operators of rank 1. The method of our study is the method of similar operators. Theorem on similarity serves as the basis for constructing groups of operators, whose generator is the original operator. We write out asymptotic formulae for groups of operators. The constructed group allows us to introduce the notion of a mild solution, and also to describe the mild solutions to the considered problem. This serves to justify the Fourier method. Almost periodicity of bounded mild solutions is established. The proof of almost periodicity is based on the asymptotic representation of the spectrum of a differential operator with an involution.

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Section: Methods Of Similar Operatorsmentioning

confidence: 99%

Fourier method for first order differential equations with involution and groups of operators

Баскаков¹,

Ускова²

2018

Ufimskii Matematicheskii Zhurnal

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“…Note that, in the recent paper [18], the method of similar operators was also used to estimate the Green function of hyperbolic semigroups of operators.…”

Section: Corollary 1 Suppose That θ ∈ {0 1} and The Following Condimentioning

confidence: 99%

Spectral properties of the Hill operator

Баскаков

Polyakov

2016

Math Notes

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Let L 2 [0, ω] be the Hilbert space of complex measurable (classes of) functions whose modulus is square-summable with inner product of the formWe consider the Hill differential operatorIt is assumed that the potential v belongs to the Hilbert space L 2 [0, ω] and its Fourier series expansionis used everywhere below. The operator L θ , θ ∈ [0, 1], can be expressed asThe operator V is the operator of multiplication by the potential v with domainThe operator L 0 θ is a self-adjoint nonnegative operator with compact resolvent.The spectrum σ(L 0 θ ) of the operator L 0 θ is described as follows:(a) for the case θ ∈ {0, 1}, the eigenvalues are of the form λ n = λ n,θ = π(2n + θ) ω 2 ,

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“…The function G is called [7] Green's function of the bounded solutions problem for equation (2). The main property of Green's function is described in the following theorem.…”

Section: Green's Functionmentioning

confidence: 99%

The Gelfand–Shilov Type Estimate for Green’s Function of the Bounded Solutions Problem

Kurbatov

Kurbatova

2017

Qual. Theory Dyn. Syst.

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Estimates for the Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations

Cited by 17 publications

References 40 publications

Fourier method for first order differential equations with involution and groups of operators

Fourier method for first order differential equations with involution and groups of operators

Spectral properties of the Hill operator

The Gelfand–Shilov Type Estimate for Green’s Function of the Bounded Solutions Problem

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