Expected scientific results of the project are described in detail in paragraph 9. In general, they are fundamental. In general, the project is aimed at developing mathematical methods of the qualitative theory of non-commutative analysis. Its implementation leads to the comprehensive development of the classical theory, and also determines new areas of research.

During the implementation of the project, the scientific qualifications of the research group in the field of non-commutative analysis that meet international requirements will increase.

The project will increase the interest of domestic scientists-mathematicians in the field of higher education and science in research in this area, the competitiveness of teams in these areas, and therefore, will positively affect the further development of mathematical research and higher education in the country. At the same time, the expected social and economic effect of the project will be observed in the future when studying the theoretical problems of non-commutative analysis and the possible practical problems that lead to them, using the results, methods and ideas of the project. We are confident that the Project will open up new opportunities to strengthen scientific cooperation at the international level. As follows from the review of preliminary works, research in the direction of the Project is intensively carried out in USA, French, Japan, China, etc. We have experience in scientific cooperation with scientists from the universities of Franche-Comté (France), the Wuhan Institute of Physics and Mathematics, the Chinese Academy of Sciences, Xinjian University (China), University of New South Wales (Australia) and others. Thus, the realization of the Project is fully consistent with the objectives of the State Program for the Development of Education and Science of the Republic of Kazakhstan for 2020-2025: “Improving the global competitiveness of Kazakhstani education and science … The increase in the contribution of science to the socio-economic development of the country”.

The first direction of the project is aimed to study matrix of measurable operators and Young type inequalities of the generalized singular number of measurable operators associated with a semi-finite von Neumann algebra.

Task 1. We study the relationship between the generalized singular numbers of matrix of measurable operators and the generalized singular numbers of its elements and its applications.

Task 2. We study Young type inequalities of the generalized singular numbers of measurable operators associated with a semi-finite von Neumann algebra.

The second direction of the project is aimed to study commutative and noncommutative weak Orlicz spaces and noncommutative versions of weak Orlicz space associated with states and weights.

Task 3. We study commutative and noncommutative weak Orlicz spaces.

Task 4.We study noncommutative versions of weak Orlicz space associated with states and weights.

The third direction of the project is aimed to study atomic decomposition of noncommuta- tive martingale symmetric spaces and asymmetric Doob inequalities of noncommutative martin- gales.

Task 5. We study atomic description for the noncommutative martingale symmetric spaces.

Task 6. We study asymmetric forms of Doob maximal inequality of noncommutative martingales for symmetric spaces.

As a result of research on the project, will be received the relationship between the generalized singular numbers of matrix of measurable operators and the generalized singular numbers of its elements and its applications, we prove Young type inequalities for the generalized singular number of measurable operators associated with a semi-finite von Neumann algebra. Will be obtain properties of commutative and non-commutative weak Orlicz spaces, noncommutative versions of weak Orlicz space associated with states and weights. Will be obtained atomic decomposition for the noncommutative martingale symmetric spaces and its applications. Expect to receive asymmetric Doob inequalities for noncommutative martingales for symmetric spaces. All of the listed expected Project results are new. The analysis shows that the expected results of the Project will significantly advance in the qualitative theory of noncommutative Orlicz space and martingales theory.

The research project is planned for 2021-2023. According to the approved calendar plan of the project, currently two scientific articles have been published in scientific journals, which are included in the Web of Science database in 2 (second) quartiles, and also participated in four international scientific conferences and presented reports.

1. Raxida Ahat, Madi Raikhan, Submajorization inequalities for matrices of τ-measurable operators // Linear and multilinear algebra. (соавторы:)– 2020. Online First published

2. T. N. Bekjan and M. N. Ospanov, On Young-type inequalities of measurable operators // Linear and Multilinear Algebra, -2021. DOI: 10.1080/03081087.2021.192087

3. Madi Raikhan, Azhar Uatayeva. Submajorisation inequalities for matrices of measurable operators // «Академик Қалменов Тынысбек Шәріпұлының 75 жылдығына арналған Қазақстан ғылым қызметкерлері күніне арналған дәстүрлі халықаралық сәуір ғылыми конференциясы» халықаралық конференциясының материалдар жинағына арналған тезис, 5-8 сәуір 2021 ж., (Алматы, Қазақстан).

4. M. Raikhan, Zh. Uspanova, On submajorization inequalities for measurable operators // International Conference «Problems of modern mathematics and its applications».–Bishkek, Kyrgyzstan. – 2021.

5. Turdebek N. Bekjan, Young type inequalities of measurable operators // International Conference «Уфимская осенняя математическая школа-2021».–Уфа, Башкортстан. – 2021.

6. Turdebek N. Bekjan, Madi. Raikhan, Dual space of noncommutative weak Orlicz-Hardy space // International Conference «Уфимская осенняя математическая школа-2021».–Уфа, Башкортстан. – 2021.

**Raikhan Madi**, Scientific supervisor of the project, Chief Researcher, higher education (specialty “Mathematics”), candidate of physical and mathematical sciences, h- index – 1 (Scopus,

(https://www.scopus.com/authid/detail.uri?authorId=55945027400).

Scientific interests: operator theory, operator algebra, functional analysis. He has published 20 scientific papers, of which more than 5 works in the direction of the Project. He was the scientific supervisor of 1 grant financing projects for 2015-2019. Key publications related to the direction of the Project:

1. Raxida Ahat, Madi Raikhan, Submajorization inequalities for matrices of τ-measurable operators // Linear and multilinear algebra. (соавторы:)– 2020. Online First published

https://doi.org/10.1080/03081087.2020.1828248

2. Turdebek N. Bekjan, Raikhan Madi, A Beurling-Blecher-Labuschagne type theorem for Haagerup noncommutative Lp spaces, Banach Journal of Mathematical Analysis, Vol.15(2) (2021), Online First published,

https://doi.org/10.1007/s43037-021-00121-1

3. Bekjan Turdebek N., Raikhan Madi. Interpolation of Haagerup noncommutative Hardy spaces // Banach J. Math. Anal., advance publication. – 2019. – Vol.13(4). – P. 798-814.

(ISSN: 1735-8787, Web of Sciences, IF=0,927, Q2 (2018), https://doi.org/10.1215/17358787-2018-0026

4. K.S. Tulenov and M. Raikhan, Outer operators for the noncommutative symmetric Hardy spaces with finite subdiagonal algebras, Acta Mathematica Scientia 37(3)(2017), 799-805.

(ISSN: 0252-9602, Web of Sciences, IF=0,992, Q2 (2018), https://doi.org/10.1016/S0252-9602(17)30038-3

5. S. Junis and M. Raikhan, A-invariant subspaces of noncommutative Orlicz spaces, J. Xinjiang Univ. Nat. Sci.33(3)(2016), 301-306.

6. Turdebek N.Bekjan and Madi Raikhan, An Hadamard-type inequality, Linear Algebra and Applications 443 (2014), 228-234.

(ISSN: 0024-3795, Web of Sciences, IF=0,997, Q2 (2018), https://doi.org/10.1016/j.laa.2013.11.019

7. B.N. Biyarov and M. Raikhan, Nonselfadjoint correct restrictions and extensions with the real spectrum, AIP Conference Proceedings 1611, 138 (2014). https://doi.org/10.1063/1.4893818

**Bekjan Turdebek**, chief researcher, higher education (specialty “Mathematics”), PhD, Professor of Xinjiang University of China (Visiting professor of L.N. Gumilyov Eurasian National University). h- index – 9 (Scopus, https://www.scopus.com/authid/detail.uri?authorId=6506605539). the total number of citations is 219.

Scientific interests: functional analysis, noncommutative symmetric space theory, noncommutative theory of Hardy space, noncommutative theory of martingales.

He has published more than 40 articles on the scientific direction of the Project in rating mathematical journals.

Key publications related to the direction of the Project:

1. T. N. Bekjan, Duality for symmetric Hardy spaces of noncommutative martingales, Math. Z., 289 (2018), 787-802.

2. T. N. Bekjan and B. K. Sageman, A property of conditional expectation, Positivity,

22(5)(2018), 1359-1369.

3. T. N. Bekjan, Interpolation of noncommutative symmetric martingale spaces, J. Operator Theory 77 (2017), 245-259.

4. T. N. Bekjan, Szego type factorization of Haagerup noncommutative Hardy spaces, Acta Mathematica Scientia37(5)(2017), 1221-1229.

5. T. N. Bekjan, Z.Chen andA.Osekowski, Noncommutative maximal inequalities with associated with convex functions, Trans. Amer. Math. Soc. 369(1) (2017), 409–427.

6. T. N. Bekjan, Interpolation of noncommutative symmetric martingale spaces, J. Operator Theory 77 (2017), 245-259.

7. T. N. Bekjan, A submajorization of Carlen and Lieb convexity, Linear Algebra and pplications494 (2016), 23-31.

8. T. N. Bekjan, Noncommutative Hardy space associated with semi-finite subdiagonal algebras, J. Math. Anal. Appl. 429(2015), 1347-1369.

9. T.N. Bekjan, Noncommutative symmetric Hardy spaces, Inter.Equat. Oper.Th.81(2015), 191-212.

10. A. Adurexit and T. N. Bekjan, Noncommutative Orlicz-Hardy spaces, Acta Math. Sinica34B(2014), 1584-1592.

11. A. Adurexit and T. N. Bekjan, Noncommutative Orlicz-Hardy spaces associated with growth functions, J. Math. Anal. Appl420(2014) 824-834.

12. T. N. Bekjan and Z. Chen, Interpolation and-moment inequalities of noncommutative martingales, Probab. Theory Relat. Fields 152 (2012), 179-206.

13. T. N. Bekjan, Z. Chen, P. Liu, Y. Jiao, Noncommutative weak Orlicz spaces and martingale inequalities, Studia Math.204(3) (2011), 195-212.

14. T. N. Bekjan, Z. Chen, M. Perrin, and Z. Yin, Atomic decomposition and interpolation for Hardy spaces of noncommutative martingales, J. Funct. Anal.258 (2010), 2483-2505.

15. T.N. Bekjan, -inequalities of noncommutative martingales, Rocky Mountain J. Math.36 (2) (2006), 401-412.

**Kordan N. Ospanov**, chief researcher, higher education (specialty “Mathematics”), doctor of physical and mathematical sciences, professor of L.N.Gumilyov Eurasian National University.

Research interests: theory of partial differential equation, operator theory, functional analysis. h- index – 6 (Scopus, https://www.scopus.com/authid/detail.uri?authorId=8858354700), the total number of citations is 95.

The list of main publications related to the direction of the Project:

1. T. N. Bekjan and K. N. Ospanov, Complex interpolation of noncommutative Hardy spaces associated semifinite von Neumann algebra // Acta Math. Sci. 40B (1) (2020), 245-260.

2. T. N. Bekjan, K.N. Ospanov, On Outer Elements of Noncommutative Orlicz–Hardy Spaces // Russian Mathematics, 2016, Vol. 60, No. 12, 15–20.

3. T. N. Bekjan, K.N. Ospanov, Factorization properties of subdiagonal algebras // Functional Analysis and Its applications. 2016, Vol. 50, Issue 2, 146-149.

**Myrzagali N. Ospanov**, leading researcher, higher education (specialty “Mathematics”), candidate of physical and mathematical sciences, associated professor of L.N. Gumilyov Eurasian National University.

Research interests: theory of differential equation, theory of noncomutative symmetric spaces, operator theory. h- index – 1 (Scopus, https://www.scopus.com/authid/detail.uri?authorId=56367930000 ).

1. The list of main publications related to the direction of the Project (in peer-reviewed foreign scientific journals indexed on the Web of Science or Scopus database):

2. T. N. Bekjan and M. N. Ospanov, On products of noncommutative symmetric quasi Banach spaces and applications // Positivity, https://doi.org/10.1007/s11117-020-00753-x

3. T. N. Bekjan and M. N. Ospanov, Holder-type inequalities of measurable operators // Positivity 21(2017), 245-259.

4. M.N. Ospanov, Coercive estimates for solutions of one singular equation with the third-order partial derivative. AIP Conf. Proc. 1611.37 (2014).

5. M.N. Ospanov, The maximal regularity of the hyperbolic system. Advancements in Mathematical Sciences. AIP Conf. Proc. 1676, 020037-1-020037-5. doi:10.1063/1.4930463.

**Uataeva (Oshanova) Azhar**, Junior Researcher, higher education (specialty “Mathematics”), PhD student of the L.N. Gumilyov Eurasian National University.

Research interests: Non-commutative theory of spaces.

Key publications related to the direction of the Project:

1. S. Junis and A. Oshanova, On submajorization inequalities for matrices of measurable operators, DOI: 10.1007/s43036-020-00101-6, to appear Advances in Operator Theory. 2. T.N. Bekjan and A. Oshanova, Semifinite tracial subalgebras, Annals of Functional аnalysis, 8(4), (2017) 473-478

http://doi.org/10.1215/20088752-2017-0011

**Uspanova Zhulduzay Kenzhebekovna**, Junior Researcher, higher education, Master student of the L.N. Gumilyov Eurasian National University, (specialty “Mathematical and computer modelling”).

Research interests: Non-commutative theory of Spaces.

1. M. Raikhan, Zh. Uspanova, On submajorization inequalities for measurable operators // International Conference «Problems of modern mathematics and its applications».–Bishkek, Kyrgyzstan. – 2021.