On multipliers between bounded variation spaces

Chaparro, Héctor Camilo

doi:10.1215/20088752-2017-0047

Cited by 4 publications

(3 citation statements)

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“…Remark 17 Before we proceed to the proof of Theorem 16 let us notice that, since supp (g) is a closed subset of [0, 1], there are in fact only two possibilities in (b): either dim Ker M g = 0 , or dim Ker M g = +∞ , depending on whether supp (g) = [0, 1] or not. 16 We begin with the proof of (a). First, we show that dim Ker M g ≥ #Z g .…”

Section: Theorem 16mentioning

confidence: 99%

Multiplication operators in BV spaces

Reinwand

Kasprzak

2022

Annali di Matematica

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The aim of this paper is to provide necessary and sufficient conditions on the generator of a multiplication operator acting in the spaces of functions of bounded Young and Riesz variation so that it is, among other things, invertible, continuous, finite rank, compact, Fredholm or has closed range. Furthermore, we characterize various spectra of such operators and give some estimates on their measure of non-compactness.

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Section: Theorem 16mentioning

confidence: 99%

Multiplication operators in BV spaces

Reinwand

Kasprzak

2022

Annali di Matematica

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“…On the one hand, the paper is a continuation of the article [10], where necessary and sufficient conditions for continuity of linear integral operators in BV [a, b] were studied. On the other hand, it can be also regarded as a part of a larger whole, as various topological and algebraic properties of linear and nonlinear operators on BV [a, b] have been studied extensively recently (acting conditions as well as continuity of nonlinear superposition operators-also known as Nemytskii operators-were studied in, respectively, [6,32] and [22,29], while properties of multiplication operators and sets of multipliers were studied in [4,5,7,13]). The works of Maćkowiak and Gulgowski concerning continuity of Nemytskii operators in BV [a, b] (see [22,29]) are especially interesting here, since, together with the results from [10] and the compactness results from this paper, they enable more flexibility of assumptions while, for example, looking for solutions of integral equations in the space of functions of bounded variation (balancing the linear and nonlinear parts of the nonlinear operators associated with such problems is crucial here).…”

Section: Introductionmentioning

confidence: 99%

Characterization of compact linear integral operators in the space of functions of bounded variation

Kasprzak

2021

Ann. Fenn. Math.

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Although various operators in the space of functions of bounded variation have been studied by quite a few authors, no simple necessary and sufficient conditions guaranteeing compactness of linear integral operators acting in such spaces have been known. The aim of the paper is to fully characterize the class of kernels which generate compact linear integral operators in the BV-space. Using this characterization we show that certain weakly singular and convolution operators (such as the Abel and Volterra operators), when considered as transformations of BV [a, b], are compact. We also provide a detailed comparison of those new necessary and sufficient conditions with various other conditions connected with compactness of linear (integral) operators in the space of functions of bounded variation which already exist in the literature.

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“…[12]). There exist several papers devoted to the study of the multiplication operator, on L p spaces [13,18], on Lorentz spaces [2], on Orlicz-Lorentz spaces [6], on Weak L p spaces [9], on Cesàro spaces [15], on variable L p spaces [7], on Köthe sequence spaces [17], on Lorentz sequence spaces [8] and on bounded variation spaces [3,10]. For some of the history of the multiplication operator and open problems, see [16].…”

Section: Introductionmentioning

confidence: 99%