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

Summary: "This survey is concerned with the spectral theory of Volterra operators A_n=bigoplus^n_jb_jJ^{alpha_j}, alpha_j>0, which are direct sums of multiples of fractional order Riemann-Liouville operators J^{alpha_j}. We discuss the lattices of invariant and hyperinvariant subspaces of operators A_n, as well as their commutants, double commutants, and other operator algebras related to A_n. We describe the sets of extended eigenvalues and the corresponding eigenvectors of the operators J^alpha. 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 J^alpha is discussed too."