Regular Mittag-Leffler kernels and spectral decomposition of a class of non-selfadjoint operators

We define abstract Mittag-Leffler kernels with values in a separable Hilbert space. A Mittag-Leffler kernel is said to be $c$-regular (resp. $d$-regular) if it generates an integral transform of Fourier–Dzhrbashyan type (resp. if the space has an unconditional basis consisting of values of the kernel). We give a complete description of $d$-regular and $c$-regular kernels, which enables us to answer a question of M. G. Krein. We apply the notion of a regular Mittag-Leffler kernel to construct the spectral decomposition for one-dimensional perturbations of fractional powers of dissipative Volterra operators.