Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients

Баскаков, А. Г.; Strukova, I. I.; Trishina, I. A.

doi:10.1134/s0037446618020052

Cited by 10 publications

(5 citation statements)

References 25 publications

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“…The space of F -almost-periodic vectors from X will be denoted by AP F X and the space of Fperiodic vectors of period ω -by P F ω (X ). We note that periodic and almost periodic at infinity functions that were considered, e. g., in [35,36,38] are examples of X c -(almost-)periodic functions in the module (X , T ), where X = L ∞ (R, X ) and T = M as in Example 2.4.…”

Section: Definition 26mentioning

confidence: 99%

“…It is not hard to see that if F is a closed submodule of M, then the spaces AP F M and P F ω (M), ω > 0, are also A-harmonious spaces and Banach algebras (see Definition 2.20). An operator V ∈ M = M 0 = End L 2 from Example 3.3 is M c -almost-periodic if the function v is almost periodic at infinity [35,36,38].…”

Section: Periodic and Almost Periodic Operatorsmentioning

confidence: 99%

“…The list above represents only a small portion of useful homogeneous spaces. Other examples can be found in [23], [26], [36], etc.…”

Section: Stepanov Spacesmentioning

confidence: 99%

“…It is clear from Definition 6.1 that every homogeneous Banach space X is a non-degenerate Banach module (X , S) and, hence, a harmonious space according to Definition 2.6. We also note that the terminology of homogeneous spaces is relatively new and the corresponding definitions in [20], [23], and [36] are slightly different from Definition 6.1. For example, Definition 6.1 is more general than [20, Defintion 2.1], and [36, Definition 2.1] requires injectivity of the embedding X ֒→ S 1 instead of the non-degeneracy Property 5.…”

Section: Stepanov Spacesmentioning

confidence: 99%

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Method of similar operators in the problem of bi-invariant subspaces

Баскаков

Garkavenko

Krishtal

et al. 2022

Итоги Науки И Техники Серия «Современная Математика И Ее Приложения Тематические Обзоры»

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We consider similarity transformations of a perturbed linear operator A − B in a complex Banach space X , where the unperturbed operator A is a generator of a Banach L1(R)-module and the perturbation operator B is a bounded linear operator. The result of the transformation is a simpler operator A − B0. For example, if A is a differentiation operator and B is an operator of multiplication by an operator-valued function, then B0 is an operator of multiplication by a function that is a restriction of an entire function of exponential type and could be 0 in some cases. As a consequence, we derive the spectral invariance of the operator A − B in a large class of spaces. The study is based on a widely applicable modification of the method of similar operators that is also presented in the paper. This non-traditional modification is rooted in the spectral theory of Banach L1(R)-modules.

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Section: Definition 26mentioning

confidence: 99%

Section: Periodic and Almost Periodic Operatorsmentioning

confidence: 99%

“…The list above represents only a small portion of useful homogeneous spaces. Other examples can be found in [23], [26], [36], etc.…”

Section: Stepanov Spacesmentioning

confidence: 99%

Section: Stepanov Spacesmentioning

confidence: 99%

See 2 more Smart Citations

Method of similar operators in the problem of bi-invariant subspaces

Баскаков

Garkavenko

Krishtal

et al. 2022

Итоги Науки И Техники Серия «Современная Математика И Ее Приложения Тематические Обзоры»

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“…The set of almost periodic at infinity functions from C b,u (J,X) is denoted by AP ∞ (J, X) and studied in [3,4,11,13]. Definitions 10 and 12 imply that every function x ∈ C b, u (R, X) almost periodic in the Bohr sense (x ∈ AP (R, X), see [5,15]) is almost periodic at infinity.…”

Section: Definition 12mentioning

confidence: 99%

On the almost periodic at infinity functions from homogeneous spaces

Баскаков¹,

Струков²,

Strukova³

2018

Issues of Analysis

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We consider homogeneous spaces of functions defined on the real axis (or semi-axis) with values in a complex Banach space. We study the new class of almost periodic at infinity functions from homogeneous spaces. The main results of the article are connected to harmonic analysis of those functions. We give four definitions of an almost periodic at infinity function from a homogeneous space and prove them to be equivalent. We also introduce the concept of a Fourier series with slowly varying at infinity coefficients (neither necessarily constant nor necessarily having a limit at infinity). It is proved that the Fourier coefficients of almost periodic at infinity function from a homogeneous space (not necessarily continuous) can be chosen continuous. Moreover, they can be extended on C to bounded entire functions of exponential type. Besides, we prove the summability of Fourier series by the method of Bochner-Fejer. The results were received with essential use of isometric representations and Banach modules theory.

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On Distributions That Are Almost Periodic at Infinity

Strukov

2022

J Math Sci

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Solutions Almost Periodic at Infinity to Differential Equations With Unbounded Operator Coefficients

Cited by 10 publications

References 25 publications

Method of similar operators in the problem of bi-invariant subspaces

Method of similar operators in the problem of bi-invariant subspaces

On the almost periodic at infinity functions from homogeneous spaces

On Distributions That Are Almost Periodic at Infinity

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