Regular Mittag-Leffler Kernels and Volterra Operators

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.