대한수학회지 (Journal of the Korean Mathematical Society)

  • 제59권4호
  • /
  • Pages.789-804
  • /
  • 2022
  • /
  • 0304-9914(pISSN)
  • /
  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
권호 표지
DOI QR코드

DOI QR Code

C*-ALGEBRAIC SCHUR PRODUCT THEOREM, PÓLYA-SZEGŐ-RUDIN QUESTION AND NOVAK'S CONJECTURE

  • 투고 : 2021.10.10
  • 심사 : 2022.03.22
  • 발행 : 2022.07.01
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

Striking result of Vybíral [51] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vybíral used this result to prove the Novak's conjecture. In this paper, we define Schur product of matrices over arbitrary C*-algebras and derive the results of Schur and Vybíral. As an application, we state C*-algebraic version of Novak's conjecture and solve it for commutative unital C*-algebras. We formulate Pólya-Szegő-Rudin question for the C*-algebraic Schur product of positive matrices.

키워드

과제정보

Remark 4.4 is due to the reviewer and I thank the reviewer for this remark. I'm grateful to Prof. Adam H. Fuller, Department of Mathematics, Ohio University, USA for going through the arXiv version of this paper and pointing out that Theorem 2.2 is proved for certain C*-subalgebras of the concrete C*-algebra B(𝓗) (the space of all bounded linear operators on Hilbert space 𝓗) in his paper [10] (in a different way), which I was not aware (see Theorem 2.3 in [10]). In an email communication he showed me further that his arguments in can be invoked for arbitrary commutative C*-algebras. However, the definition of Schur product in this paper and in [10] differs for arbitrary C*-algebras. I thank Dr. P. Sam Johnson, Department of Mathematical and Computational Sciences, National Institute of Technology Karnataka (NITK), Surathkal for his help and some discussions. The author is supported by Indian Statistical Institute, Bangalore, through the J. C. Bose Fellowship of Prof. B. V. Rajarama Bhat. He thanks Prof. B. V. Rajarama Bhat for the Post Doc position.

참고문헌

  1. T. Ando, Concavity of certain maps on positive definite matrices and applications to Hadamard products, Linear Algebra Appl. 26 (1979), 203-241. https://doi.org/10.1016/0024-3795(79)90179-4
  2. R. B. Bapat and M. K. Kwong, A generalization of A ◦ A-1 ≥ I, Linear Algebra Appl. 93 (1987), 107-112. https://doi.org/10.1016/S0024-3795(87)90315-6
  3. A. Belton, D. Guillot, A. Khare, and M. Putinar, Matrix positivity preservers in fixed dimension. I, Adv. Math. 298 (2016), 325-368. https://doi.org/10.1016/j.aim.2016.04.016
  4. A. Belton, D. Guillot, A. Khare, and M. Putinar, A panorama of positivity. I: Dimension free, in Analysis of operators on function spaces, 117-164, Trends Math, Birkhauser/Springer, Cham, 2019. https://doi.org/10.1007/978-3-030-14640-5_5
  5. A. Belton, D. Guillot, A. Khare, and M. Putinar, A panorama of positivity. II: Fixed dimension, in Complex analysis and spectral theory, 109-150, Contemp. Math., 743, Centre Rech. Math. Proc, Amer. Math. Soc., RI, 2020. https://doi.org/10.1090/conm/743/14958
  6. B. Blackadar, K-theory for operator algebras, Mathematical Sciences Research Institute Publications, 5, Springer-Verlag, New York, 1986. https://doi.org/10.1007/978-1-4613-9572-0
  7. J. P. R. Christensen and P. Ressel, Functions operating on positive definite matrices and a theorem of Schoenberg, Trans. Amer. Math. Soc. 243 (1978), 89-95. https://doi.org/10.2307/1997755
  8. M. Fiedler, Uber eine Ungleichung fur positiv definite Matrizen, Math. Nachr. 23 (1961), 197-199. https://doi.org/10.1002/mana.1961.3210230307
  9. M. Fiedler and T. L. Markham, An observation on the Hadamard product of Hermitian matrices, Linear Algebra Appl. 215 (1995), 179-182. https://doi.org/10.1016/0024-3795(93)00087-G
  10. A. H. Fuller, Nonself-adjoint semicrossed products by abelian semigroups, Canad. J. Math. 65 (2013), no. 4, 768-782. https://doi.org/10.4153/CJM-2012-051-8
  11. K. Grove and G. K. Pedersen, Diagonalizing matrices over C(X), J. Funct. Anal. 59 (1984), no. 1, 65-89. https://doi.org/10.1016/0022-1236(84)90053-3
  12. D. Guillot, A. Khare, and B. Rajaratnam, Preserving positivity for rank-constrained matrices, Trans. Amer. Math. Soc. 369 (2017), no. 9, 6105-6145. https://doi.org/10.1090/tran/6826
  13. C. S. Herz, Fonctions operant sur les fonctions definies-positives, Ann. Inst. Fourier (Grenoble) 13 (1963), 161-180. https://doi.org/10.5802/aif.137
  14. C. Heunen and M. L. Reyes, Diagonalizing matrices over AW*-algebras, J. Funct. Anal. 264 (2013), no. 8, 1873-1898. https://doi.org/10.1016/j.jfa.2013.01.022
  15. A. Hinrichs, D. Krieg, E. Novak, and J. Vybiral, Lower bounds for the error of quadrature formulas for Hilbert spaces, J. Complexity 65 (2021), Paper No. 101544, 20 pp. https://doi.org/10.1016/j.jco.2020.101544
  16. A. Hinrichs and J. Vybiral, On positive positive-definite functions and Bochner's Theorem, J. Complexity 27 (2011), no. 3-4, 264-272. https://doi.org/10.1016/j.jco.2011.01.002
  17. R. A. Horn, The theory of infinitely divisible matrices and kernels, Trans. Amer. Math. Soc. 136 (1969), 269-286. https://doi.org/10.2307/1994714
  18. R. A. Horn, The Hadamard product, in Matrix theory and applications (Phoenix, AZ, 1989), 87-169, Proc. Sympos. Appl. Math., 40, AMS Short Course Lecture Notes, Amer. Math. Soc., Providence, RI, 1990. https://doi.org/10.1090/psapm/040/1059485
  19. R. A. Horn and C. R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, 1994.
  20. R. A. Horn and C. R. Johnson, Matrix Analysis, second edition, Cambridge University Press, Cambridge, 2013.
  21. K. K. Jensen and K. Thomsen, Elements of KK-theory, Mathematics: Theory & Applications, Birkhauser Boston, Inc., Boston, MA, 1991. https://doi.org/10.1007/978-1-4612-0449-7
  22. C. R. Johnson, Partitioned and Hadamard product matrix inequalities, J. Res. Nat. Bur. Standards 83 (1978), no. 6, 585-591. https://doi.org/10.6028/jres.083.039
  23. R. V. Kadison, Diagonalizing matrices over operator algebras, Bull. Amer. Math. Soc. (N.S.) 8 (1983), no. 1, 84-86. https://doi.org/10.1090/S0273-0979-1983-15091-7
  24. R. V. Kadison, Diagonalizing matrices, Amer. J. Math. 106 (1984), no. 6, 1451-1468. https://doi.org/10.2307/2374400
  25. I. Kaplansky, Projections in Banach algebras, Ann. of Math. (2) 53 (1951), 235-249. https://doi.org/10.2307/1969540
  26. I. Kaplansky, Modules over operator algebras, Amer. J. Math. 75 (1953), 839-858. https://doi.org/10.2307/2372552
  27. G. G. Kasparov, Hilbert C*-modules: theorems of Stinespring and Voiculescu, J. Operator Theory 4 (1980), no. 1, 133-150.
  28. A. Khare, Sharp nonzero lower bounds for the Schur product theorem, Proc. Amer. Math. Soc. 149 (2021), no. 12, 5049-5063. https://doi.org/10.1090/proc/15555
  29. A. Khare and T. Tao, On the sign patterns of entrywise positivity preservers in fixed dimension, Amer. J. Math. 143 (2021), no. 6, 1863-1929. https://doi.org/10.1353/ajm.2021.0049
  30. E. C. Lance, Hilbert C*-modules, London Mathematical Society Lecture Note Series, 210, Cambridge University Press, Cambridge, 1995. https://doi.org/10.1017/CBO9780511526206
  31. S. Liu, Inequalities involving Hadamard products of positive semidefinite matrices, J. Math. Anal. Appl. 243 (2000), no. 2, 458-463. https://doi.org/10.1006/jmaa.1999.6670
  32. S. Liu and G. Trenkler, Hadamard, Khatri-Rao, Kronecker and other matrix products, Int. J. Inf. Syst. Sci. 4 (2008), no. 1, 160-177.
  33. V. M. Manuilov, Diagonalizing operators in Hilbert modules over C*-algebras, J. Math. Sci. (New York) 98 (2000), no. 2, 202-244. https://doi.org/10.1007/BF02355448
  34. V. M. Manuilov and E. V. Troitsky, Hilbert C*-modules, translated from the 2001 Russian original by the authors, Translations of Mathematical Monographs, 226, American Mathematical Society, Providence, RI, 2005. https://doi.org/10.1090/mmono/226
  35. G. J. Murphy, C*-Algebras and Operator Theory, Academic Press, Inc., Boston, MA, 1990.
  36. E. Novak, Intractability results for positive quadrature formulas and extremal problems for trigonometric polynomials, J. Complexity 15 (1999), no. 3, 299-316. https://doi.org/10.1006/jcom.1999.0507
  37. E. Novak and H. Wozniakowski, Tractability of multivariate problems. Vol. 1, EMS Tracts in Mathematics, 6, European Mathematical Society (EMS), Zurich, 2008. https://doi.org/10.4171/026
  38. A. Oppenheim, Inequalities connected with definite hermitian forms, J. London Math. Soc. 5 (1930), no. 2, 114-119. https://doi.org/10.1112/jlms/s1-5.2.114
  39. W. L. Paschke, Inner product modules over B*-algebras, Trans. Amer. Math. Soc. 182 (1973), 443-468. https://doi.org/10.2307/1996542
  40. G. Polya and G. Szego, Aufgaben und Lehrsatze aus der Analysis. Band II, vierte Auflage, Heidelberger Taschenbucher, Band 74, Springer-Verlag, Berlin, 1971.
  41. M. Rid and B. Saimon, Methods of modern mathematical physics. 2: Harmonic analysis. Selfadjointness, translated from the English by A. K. Pogrebkov and V. N. Susko, Izdat. "Mir", Moscow, 1978.
  42. M. A. Rieffel, Induced representations of C*-algebras, Advances in Math. 13 (1974), 176-257. https://doi.org/10.1016/0001-8708(74)90068-1
  43. W. Rudin, Positive definite sequences and absolutely monotonic functions, Duke Math. J. 26 (1959), 617-622. http://projecteuclid.org/euclid.dmj/1077468771 https://doi.org/10.1215/S0012-7094-59-02659-6
  44. W. Rudin, Principles of Mathematical Analysis, third edition, International Series in Pure and Applied Mathematics, McGraw-Hill Book Co., New York, 1976.
  45. I. J. Schoenberg, Positive definite functions on spheres, Duke Math. J. 9 (1942), 96-108. http://projecteuclid.org/euclid.dmj/1077493072 https://doi.org/10.1215/S0012-7094-42-00908-6
  46. J. Schur, Bemerkungen zur Theorie der beschrankten Bilinearformen mit unendlich vielen Veranderlichen, J. Reine Angew. Math. 140 (1911), 1-28. https://doi.org/10.1515/crll.1911.140.1
  47. J. Stewart, Positive definite functions and generalizations, an historical survey, Rocky Mountain J. Math. 6 (1976), no. 3, 409-434. https://doi.org/10.1216/RMJ-1976-6-3-409
  48. G. P. H. Styan, Hadamard products and multivariate statistical analysis, Linear Algebra Appl. 6 (1973), 217-240. https://doi.org/10.1016/0024-3795(73)90023-2
  49. H. Vasudeva, Positive definite matrices and absolutely monotonic functions, Indian J. Pure Appl. Math. 10 (1979), no. 7, 854-858.
  50. G. Visick, A quantitative version of the observation that the Hadamard product is a principal submatrix of the Kronecker product, Linear Algebra Appl. 304 (2000), no. 1-3, 45-68. https://doi.org/10.1016/S0024-3795(99)00187-1
  51. J. Vybiral, A variant of Schur's product theorem and its applications, Adv. Math. 368 (2020), 107140. https://doi.org/10.1016/j.aim.2020.107140
  52. B.-Y. Wang and F. Zhang, Schur complements and matrix inequalities of Hadamard products, Linear and Multilinear Algebra 43 (1997), no. 1-3, 315-326. https://doi.org/10.1080/03081089708818531
  53. N. E. Wegge-Olsen, K-Theory and C*-Algebras, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1993.
  54. F. Zhang, Schur complements and matrix inequalities in the Lowner ordering, Linear Algebra Appl. 321 (2000), no. 1-3, 399-410. https://doi.org/10.1016/S0024-3795(00)00032-X
  55. F. Zhang, Matrix Theory, second edition, Universitext, Springer, New York, 2011. https://doi.org/10.1007/978-1-4614-1099-7