Romashchenko G. On similarity of Volterra operators

In the dissertation, sufficient conditions for the similarity of the Volterra operators in the Lebesgue space and Sobolev space of vector-functions are studied. In scalar Sobolev space, necessary and sufficient conditions for the Volterra operator to be similar to the integration operator or to the operator of fractional integration are obtained. For a pair of convolution Volterra operators a criterion for the simultaneously similarity is given. In the Sobolev spaces with real positive index the lattices of invariant and hyperinvariant subspaces of the operator of fractional integration are described. The operator algebras, commutant, and double commutant are investigated too. Conditions for a system of eigenvectors and generalized eigenvectors of perturbed Volterra operator to be complete and minimal in the Sobolev spaces are obtained.