On the problem of the similarity for one class of not self-conjugate operators

The criteria of the similarity of an operator to a self-conjugate one has been formulated. The not self-conjugate operator A of the Friedrichs model in the space L₂(R,M) has been considered. The condition of the similarity of the weakly perturbed multiplication operator A to the self-conjugate in the terms of the functions determining the perturbation having the rank equal to 1 has been described. It has been established that the operator A has been similar to the self-conjugate one if the singular integral operator T has been bounded. The following formula has been correct A=L-[T,L], where L is the multiplication operator on a independent variable in the space L₂(R,M).