On properties of the second type matrix ball $B_{m,n}^{(2)}$ from space ${\mathbb C}^{n}[m\times m]$

Uktam S. Rakhmonov ; Jonibek Sh. Abdullayev

Žurnal Sibirskogo federalʹnogo universiteta. Matematika i fizika, Tome 15 (2022) no. 3, pp. 329-342

Résumé

The automorphisms of the matrix ball associated with the classical domains of the second type are described in this paper. The properties of the second type matrix ball $B_{m,n}^{(2)}$ are studied.

Détail

Keywords: classical domain, automorphism of the matrix ball, Shilov's boundary.
Mots-clés : matrix ball, volume

Bibliographie

[1] V.S. Vladimirov, Methods of functions of several complex variables, Nauka, M., 1964 | MR

[2] B.V. Shabat, Introduction to Complex Analysis, v. II, Functions of Several Variables, Nauka. Physical and mathematical literature, M., 1985 | MR

[3] W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^{n}$, Springer–Verlag, New York–Berlin–Heidelberg, 1980 | MR

[4] S.G. Krantz, Harmonic and complex analysis in several variables, Springer Monographs in Mathematics, Cham, Switzerland, 2017 | DOI | MR | Zbl

[5] E. Cartan, “Sur les domaines bornes homogenes de l'espace de $n$ variables complexes”, Abh. Math. Sem. Univ. Hamburg, 11 (1935), 116–162 | DOI | MR

[6] C.L. Siegel, Automorphic functions of several complex variables, Inostrannaya Literatura, M., 1954 | MR

[7] I.I. Pjateckii–Sapiro, Geometry of classical domains and the theory of automorphic functions, Fizmatgiz, M., 1961 | MR

[8] Hua Luogeng, Harmonic analysis of functions of several complex variables, in classical domains, IL, M., 1963 | MR

[9] A.G. Sergeev, On matrix and Reinhardt domains, Inst. Mittag–Leffler, Stockholm, 1988

[10] G. Khudayberganov, Spectral decomposition for functions from several matrices, TashSU, Tashkent, 1993 (in Russian)

[11] G. Khudayberganov, B.B. Khidirov, U.S. Rakhmonov, “Automorphisms of matrix balls”, Acta NUUz, 2010, no. 3, 205–210 (in Russian)

[12] G. Khudayberganov, A.M. Kytmanov, B.A. Shaimkulov, Complex analysis in matrix domains, Monograph., Siberian Federal University, Krasnoyarsk, 2011 (in Russian)

[13] G. Khudayberganov, A.M. Khalknazarov, J. Sh.Abdullayev, “Laplace and Hua Luogeng operators”, Russian Mathematics (Izvestiya VUZ. Matematika), 64:3 (2020), 66–71 | DOI | MR | Zbl

[14] S. Kosbergenov, “Holomorphic automorphisms and the Bergman integral for a matrix ball”, Dokl. AN RUz, 1998, no. 1, 7–10 (In Russian) | MR | Zbl

[15] S. Kosbergenov, “On a multidimensional boundary Morera theorem for the matrix ball”, Izv. Vyssh. Uchebn. Zaved. Mat., 2001, no. 4, 28–32 | MR | Zbl

[16] A. Khalknazarov, “The volume of the matrix ball in the space of matrices”, Uzbek Mathematical Journal, 2012, no. 3, 135–139 (in Russian) | MR

[17] S. Kosbergenov, “On the Carleman formula for a matrix ball”, Russian Math. (Iz. VUZ), 43:1 (1999), 72–75 | DOI | MR | Zbl

[18] G. Khudayberganov, U.S. Rakhmonov, Z.Q. Matyakubov, “Integral formulas for some matrix domains”, Contemporary Mathematic, 662, AMS, 2016, 89–95 | DOI | MR | Zbl

[19] G. Khudayberganov, U.S. Rakhmonov, “The Bergman and Cauchy-Szegő kernels for matrix ball of the second type”, Journal of Siberian Federal University. Mathematics $\$ Physics, 7:3 (2014), 305–310 | MR | Zbl

[20] G. Khudayberganov, B.P. Otemuratov, U.S. Rakhmonov, “Boundary Morera theorem for the matrix ball of the third type”, Journal of Siberian Federal University. Mathematics Physics, 11:1 (2018), 40–45 | DOI | MR | Zbl

[21] G. Khudayberganov, U.S. Rakhmonov, “Carleman Formula for Matrix Ball of the Third Type, Algebra”, Complex Analysis, and Pluripotential Theory, USUZCAMP 2017, Springer Proceedings in Mathematics Statistics, 264, Springer, Cham, 2018, 101–108 | DOI | MR | Zbl

[22] U.S. Rakhmonov, J. Sh.Abdullayev, “On volumes of matrix ball of third type and generalized Lie balls”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 29:4 (2019), 548–557 | DOI | MR | Zbl

[23] G. Khudayberganov, J. Sh.Abdullayev, “Relationship between the Kernels Bergman and Cauchy-Szegő in the domains $\tau ^{+} \left(n-1\right)$ and $\Re _{IV}^{n} $”, Journal of Siberian Federal University. Mathematics $\$ Physics, 13:5 (2020), 559–567 | DOI | MR | Zbl

[24] Th.Th. Voronov, “On volumes of classical supermanifolds”, Sbornik: Mathematics, 207:11 (2016), 1512–1536 | DOI | MR | Zbl

[25] Hua Luogeng, “On the theory of automorphic functions of a matrix variable I-geometrical basis”, American Journal of Mathematics, 66:3 (1944), 470–488 | DOI | MR | Zbl

[26] G. Khudayberganov, J. Sh.Abdullayev, “Holomorphic continuation into a matrix ball of functions defined on a piece of its skeleton”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 31:2 (2021), 296–310 | DOI | MR | Zbl

[27] B.T. Kurbanov, “Morera's theorem for some unbounded domains”, Proceedings of the international conference “Mathematical models and methods of their research”, v. 2, Krasnoyarsk, 2001, 49–51

[28] G. Khudayberganov, Z.K. Matyakubov, “Integral formulas in Siegel domains”, Uzbek mathematical journal, 2013, no. 3, 116–128 (in Russian) | MR

[29] G. Khudayberganov, Z.K. Matyakubov, “Bergman's formula for an unbounded matrix domain”, Vestnik NUUz, 2013, no. 4.1, 36–39 (in Russian)

[30] J. Sh.Abdullayev, “An analogue of Bremermann's theorem on finding the Bergman kernel for the Cartesian product of the classical domains ${{\Re }_{I}}\left( m,k \right)$ and ${{\Re }_{II}}\left( n \right)$”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2020, no. 3, 88–96 | Zbl

[31] G. Khudayberganov, J. Sh.Abdullayev, “The boundary Morera theorem for domain ${{\tau }^{+}}\left( n-1 \right)$”, Ufimsk. Mat. Zh., 13:3 (2021), 196–210 | MR | Zbl

[32] Dzh.K. Tishabaev, “Invariant metrics and indicatrices of bounded domains in ${{\mathbb{C}}^{n}}$”, Siberian Math. J., 30:1 (1989), 166–168 | DOI | MR | Zbl