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  In the recent century the problem of root vectors system completeness related to non-selfadjoint operators is undergone a serious attention by such mathematicians as Markus A.S. [16], [17], Lidskii V.B. [14], Krein M.G. [7], Katsnelson V.E. [6], Matsaev V.I.[18], Agranovich M.S. [2] and others. In consequence, there appeared a fundamental concept in the framework of abstract spectral theory including propositions on summation of spectral decompositions (series on root vectors) in a generalized sense such as Abel-Lidskii, Riesz, Bari, senses [2],[5]. The problem of decreasing the summation order in the Abell-Lidskii sense was formulated by Lidskii V.B. 1962 [15] for a case corresponding to the self adjoint elliptic operator perturbed by a non-selfadjoint operator. More generally, the problem was considered by Katsnelson V.E. 1967 [3] for perturbations of a positive self adjoint operator under the strong subordination condition [19]. In 1994, Agaranovich M.S. proved that the summation order can be decreased to some positive number in the case corresponding to an operator with the numerical range of values contained in the domain of the parabolic type [2] (what is an essential restriction in comparison with the sectorial condition). However, a problem on the lower bound of the summation order has still not been solved.