Summary Half-infinite lowertriangular matrixs. A classjof lcwvertriangular perturbations of a diagonal matrix in Hilbert space of sequences is considered. Some conditions are obtained in terms of perturbation matrix elements. They are sufficient for the operator associated with a perturbated matrix to be, similar to a selfadjoint operator. Thus the problem of whether the eigenvectors form the Riesz basis or not is resolved.