Characterization of compact linear integral operators in the space of functions of bounded variation. (English) Zbl 1482.47088
Summary: 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.
MSC:
| 47G10 | Integral operators |
| 47B07 | Linear operators defined by compactness properties |
| 26A45 | Functions of bounded variation, generalizations |
Keywords:
Abel operator; bounded variation; compact linear operator; convolution kernel; integral operator; space of functions of bounded variation; Volterra operator; weakly singular kernelReferences:
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