Geodesic
Regular Mittag-Leffler kernels and spectral decomposition of a~class of non-selfadjoint operators
Izvestiya. Mathematics , Tome 69 (2005) no. 1, pp. 15-57

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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. 
@article{IM2_2005_69_1_a1,
     author = {G. M. Gubreev},
     title = {Regular {Mittag-Leffler} kernels and spectral decomposition of a~class of non-selfadjoint operators},
     journal = {Izvestiya. Mathematics },
     pages = {15--57},
     publisher = {mathdoc},
     volume = {69},
     number = {1},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2005_69_1_a1/}
}
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G. M. Gubreev. Regular Mittag-Leffler kernels and spectral decomposition of a~class of non-selfadjoint operators. Izvestiya. Mathematics , Tome 69 (2005) no. 1, pp. 15-57. http://geodesic.mathdoc.fr/item/IM2_2005_69_1_a1/