Spectral theory of fractional order integration operators, their direct sums, and similarity problem to these operators of their weak perturbations

This survey is concerned with the spectral theory of Volterra operators An = ⊕nj bjJαj, αj > 0, which are direct sums of multiples of fractional order Riemann- Liouville operators Jαj. We discuss the lattices of invariant and hyperinvariant subspaces of operators An, as well as their commutants, double commutants, and other operator algebras related to An. We describe the sets of extended eigenvalues and the corresponding eigenvectors of the operators Jα. The Gohberg-Krein conjecture on equivalence of unicellularity and cyclicity properties of a dissipative Volterra operator is also discussed. The problem of the similarity of the Volterra integral operators to the operators Jα is discussed too. © 2019 Walter de Gruyter GmbH, Berlin/Boston.