On the Infinitesimalness of the Summation Order in the Abell-Lidskii Sense for the Trace Class

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 of the summation order in the Abell-Lidskii sense was formulated by Lidskii V.B. 1962 [15] for a case corresponding to the selfadjoint 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 selfadjoint 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 containing 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 not been still solved.
In this report we will show that the summation order in the Abell-Lidskii sense can be decreased to an arbitrary small positive value in the case corresponding to the sectorial operator belonging to the trace class. In addition, we construct a qualitative theory of summation in the Abell-Lidkii sense and produce relevant applications in the theory of pseudo-differential operators.
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