Relations between different types of spectra and spectral characterizations

Basit, Bolis; Günzler, Hans

doi:10.1007/s00233-007-9042-4

Cited by 5 publications

(11 citation statements)

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“…In [8] a concept reduced spectrum of not necessarily uniformly continuous functions on the whole line is also defined. Note that the uniform closedness condition (see [8, Definition 3.1 and Theorem 4.3]) seems to be too restrictive.…”

Section: Functions On the Half Linementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…1 After the first version [27] of this paper appeared we learned that in [8] related concepts are discussed. However, the related results stated in [8] seem to be different and incomplete. For example, in addition to the uniform closedness condition which seems to be very strict, Proposition 4.1 and Theorem 4.3 obviously assume that the reduced spectrum of the considered function is non-empty.…”

Section: Introductionmentioning

confidence: 99%

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A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations

Minh

2009

Journal of Differential Equations

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Section: Functions On the Half Linementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations

Minh

2009

Journal of Differential Equations

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“…In (3.4) we consider a more general spectrum sp A,V (F ), the reduced spectrum of F ∈ L 1 loc (J, X) relative to (A, V ), where V ⊂ L 1 (R), a spectrum closely related to the one defined in (3.4 * ) which was first studied in [10]. We are able to strengthen further the improvements made by Chill ([17, Lemma 1.16] and [18, Proposition 1.3, Theorem 1.5, Corollary 1.7]).…”

mentioning

confidence: 99%

“…In [10], this property was called C ub -invariance. In the proof of Theorem 2.2.4 in [6, p. 13], it is shown that if J = R + and A satisfies (3.1), then [3]).…”

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confidence: 99%

Comparison of spectra of absolutely regular distributions and applications

Basit¹,

Pryde²

2014

Rocky Mountain J. Math.

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We study the reduced Beurling spectra sp A,V (F ) of functions F ∈ L 1 loc (J, X) relative to certain function spaces A ⊂ L ∞ (J, X) and V ⊂ L 1 (R) and compare them with other spectra including the weak Laplace spectrum. Here J is R + or R and X is a Banach space. If F belongs to the space S ′ ar (J, X) of absolutely regular distributions and has uniformly continuous indefinite integral with 0 ∈ sp A,S(R) (F ) (for example if F is slowly oscillating and A is {0} or C 0 (J, X)),for all ψ ∈ S(R). We show that tauberian theorems for Laplace transforms follow from results about reduced spectra. Our results are more general than previous ones and we demonstrate this through examples 26 August 2011 §1. IntroductionThe goal of this paper 1 is to study the asymptotic behaviour of certain locally integrable functions F : J → X where J denotes R or R + and X is a complex Banach space. Such a study has a long history. It is motivated by Loomis's theorem (see [25] and [24, Theorem 4, p. 92, p. 97]) which gives spectral conditions under which a function F : R → C is almost periodic and by the tauberian theorem of Ingham (see [22] and [2, Theorem 4.9.5, p. 326]) which gives conditions under which lim t→∞ F (t) = 0. Many notions of the spectrum of a function have since been introduced in order to obtain (vector valued) extensions of these results and we will review and compare some of these in this paper. In particular we develop the reduced spectrum sp A (F ) of F relative to various closed subspaces A of BU C(J, X), a spectrum that was introduced before in this context (see [24, Chapter 6.4, p. 91], [6], [7], [2, p. 371], [8] and [17]). Typically the reduced spectrum 1 AMS subject classification 2010: Primary 47A10, 44A10 Secondary 47A35, 43A60.

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Existence of bounded uniformly continuous mild solutions onRof evolution equations and their asymptotic behaviour

Basit

Günzler

2013

Journal of Mathematical Analysis and Applications

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We prove that u ′ = Au + φ has on R a mild solution u φ ∈ BU C(R, X) (that is bounded and uniformly continuous), where A is the generator of a C 0semigroup on the Banach space X with resolvent satisfying ||R(it,As a consequence it is shown that if F is the space of almost periodic, almost automorphic, bounded Levitan almost periodic or certain classes of recurrent functions and φ as above is such that M h φ := (1/h) h 0 φ(·+s) ds ∈ F for each h > 0, then u φ ∈ F ∩BU C(R, X). These results seem new and strengthen several recent theorems.

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Relations between different types of spectra and spectral characterizations

Cited by 5 publications

References 29 publications

A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations

A spectral theory of continuous functions and the Loomis–Arendt–Batty–Vu theory on the asymptotic behavior of solutions of evolution equations

Comparison of spectra of absolutely regular distributions and applications

Existence of bounded uniformly continuous mild solutions onRof evolution equations and their asymptotic behaviour

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