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Funktsional. Anal. i Prilozhen., 2004, Volume 38, Issue 4, Pages 82–86 (Mi faa129)  

Brief communications

Regular Mittag-Leffler Kernels and Volterra Operators

G. M. Gubreev

South Ukrainian State K. D. Ushynsky Pedagogical University

Abstract: We give the definition of an abstract Mittag-Leffler kernel $\mathcal{E}_\rho$ ranging in a separable Hilbert space $\mathfrak{H}$. In the simplest case, $\mathcal{E}_\rho(z)$ can be expressed via the Mittag-Leffler function $E_\rho(z,\mu)$. The kernel $\mathcal{E}_\rho$ is said to be $c$-regular if it generates an integral transform of Fourier–Dzhrbashyan type and $d$-regular if its range contains an unconditional basis of $\mathfrak{H}$. We give a complete description of $d$- and $c$-regular kernels, which permits us to answer a question posed by M. Krein. An application to the problem on the similarity of a rank one perturbation of a fractional power of a Volterra operator to a normal operator is considered.

Keywords: Mittag-Leffler kernel, Mittag-Leffler function, Fourier–Dzhrbashyan transform, rank one perturbation, Volterra operator, fractional power

DOI: https://doi.org/10.4213/faa129

Full text: PDF file (188 kB)
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English version:
Functional Analysis and Its Applications, 2004, 38:4, 305–308

Bibliographic databases:

UDC: 517.43+513.88
Received: 20.02.2003

Citation: G. M. Gubreev, “Regular Mittag-Leffler Kernels and Volterra Operators”, Funktsional. Anal. i Prilozhen., 38:4 (2004), 82–86; Funct. Anal. Appl., 38:4 (2004), 305–308

Citation in format AMSBIB
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\paper Regular Mittag-Leffler Kernels and Volterra Operators
\jour Funktsional. Anal. i Prilozhen.
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\pages 82--86
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\jour Funct. Anal. Appl.
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