Date: Tuesday, November 26, 2024

Time: 14.00-15.00 (Istanbul) = 12.00-13.00 (Ghent) = 16.00-17.00 (Almaty)

Zoom link: https://us02web.zoom.us/j/6678270445?pwd=SFNmQUIvT0tRaHlDaVYrN3l5bzJVQT09, 

Conference ID: 667 827 0445, Access code: 1

Speaker: 

Assoc. Prof. Dr. Maksim V. Kukushkin

National Research University Higher School of Economics (HSE), Moscow, Russia 

Institute of Applied Mathematics and Automation, Russian Academy of Sciences, Nalchik, Russia

Title: On the infinitesimalness of the summation order in the Abell-Lidskii sense for the trace class

Abstract: 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.

References:

[1]   Agranovich M.S. Elliptic operators on closed manifolds. Partial differential equa-tions. VI. Elliptic operators on closed manifolds. Encycl. Math. Sci.; translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya, 63, (1990), 5129.

[2]  Agranovich M.S. On series with respect to root vectors of operators associated with  forms having symmetric principal part. Functional Analysis and its applications, 28, (1994), 151- 167.

[3]  Agranovich M.S., Katsenelenbaum B.Z., Sivov A.N., Voitovich N.N. Generalized method of eigenoscillations in the diffraction theory. Zbl 0929.65097 Weinheim: Wiley-VCH., 1999.

[4]  Agranovich M.S., Markus A.S. On Spectral Properties of Elliptic Pseudo-Differential Operators Far from Self-Adjoint Ones. Z. Anal. Anwend., 8 (1989), No.3, 237260.

[5]   Gohberg I.C., Krein M.G. Introduction to the theory of linear non-selfadjoint opera-tors in a Hilbert space. Moscow: Nauka, Fizmatlit, 1965.

[6]  Katsnelson V.E. Conditions under which systems of eigenvectors of some classes of operators form a basis. Funct. Anal. Appl., 1, No.2 (1967), 122-132.

[7]   Krein M.G. Criteria for completeness of the system of root vectors of a dissipative operator. Amer. Math. Soc. Transl. Ser., Amer. Math. Soc., Providence, RI, 26, No.2 (1963), 221-229.

[8]  Krein M.G. On the theory of linear non-selfadjoint operators. Reports of the USSR Academy of Sciences, 130, No.2 (1960), 254-256.

[9]  Kukushkin M.V. Asymptotics of eigenvalues for differential operators of fractional order.Fract. Calc. Appl. Anal. 22, No. 3 (2019), 658–681, arXiv:1804.10840v2 [math.FA]; DOI:10.1515/fca-2019-0037.

[10]                     Kukushkin M.V. Cauchy Problem for an Abstract Evolution Equation of Fractional Order. Fractal Fract. (2023), 7, 111.

[11]                       Kukushkin M.V. Schatten Index of the Sectorial Operator via the Real Component of Its Inverse. Mathematics (2024), 12, 540.

[12]                      Kukushkin M.V. On the essential decreasing of the summation order in the Abel-Lidskii sense. arXiv:2306.13547v3 [math.FA].

[13]                      Lidskii V.B. Summability of series in terms of the principal vectors of nonselfadjoint operators. Tr. Mosk. Mat. Obs., 11, (1962), 3-35.

[14]                      Lidskii V.B. Conditions for completeness of a system of root subspaces for nonselfadjoint operators with discrete spectra. Amer. Math. Soc. Transl. Ser., Amer. Math. Soc., Providence, RI., 34 (1963), No. 2, 241-281.

[15]                      Lidskii V.B. On the Fourier series expansion on the major vectors of a nonselfadjoint elliptic operator. Tr. Mosk. Mat. Obs., 57(99), (1962), 137-150. 

[16]                      Markus A.S. On the basis of root vectors of a dissipative operator. Soviet Math. Dokl., 1 (1960), 599-602.

[17]                      Markus A.S. Expansion in root vectors of a slightly perturbed selfadjoint operator. Soviet Math. Dokl., 3 (1962), 104-108.

[18]                      Markus A.S., Matsaev V.I. Operators generated by sesquilinear forms and their spectral asymptotics. Linear operators and integral equations, Mat. Issled., Stiintsa, Kishinev, 61 (1981), 86-103.

[19]                      Shkalikov A.A. Perturbations of selfadjoint and normal operators with a discrete spectrum. Russian Mathematical Surveys, 71, Issue 5(431) (2016), 113-174.

Biography: 

Maksim V. Kukushkin has obtained a degree in Mathematics in 2002 and PhD degree in Mathematics in 2016, last of them from Ministry of Education and Science of Russian Federation (Russia). After completion of his studies, he had a position of Associate Professor in Saint-Petersburg State University of Aerospace Instrumentation, Emperor Alexander I Saint Petersburg State Transport University, Kabardino-Balkarian Scientific Center, RAS, Moscow State University of Civil Engineering. His topics of research deal mostly with the spectral theory of non-selfadjoint operators, the applications to the semigroup theory, as well as to evolution equations in the abstract Hilbert space.