Kyungpook Mathematical Journal

  • 제64권1호
  • /
  • Pages.15-30
  • /
  • 2024
  • /
  • 1225-6951(pISSN)
  • /
  • 0454-8124(eISSN)
경북대학교 자연과학대학 수학과 (Department of Mathematics, Kyungpook National University)
권호 표지
DOI QR코드

DOI QR Code

Generalized Inverses and Solutions to Equations in Rings with Involution

  • Yue Sui (Department of Mathematics, Yangzhou University) ;
  • Junchao Wei (Department of Mathematics, Yangzhou University)
  • 투고 : 2021.05.04
  • 심사 : 2022.07.25
  • 발행 : 2024.03.31
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

In this paper, we focus on partial isometry elements and strongly EP elements on a ring. We construct characterizing equations such that an element which is both group invertible and MP-invertible, is a partial isometry element, or is strongly EP, exactly when these equations have a solution in a given set. In particular, an element a ∈ R# ∩ R† is a partial isometry element if and only if the equation x = x(a†)*a† has at least one solution in {a, a#, a†, a*, (a#)*, (a†)*}. An element a ∈ R#∩R† is a strongly EP element if and only if the equation (a†)*xa† = xa†a has at least one solution in {a, a#, a†, a*, (a#)*, (a†)*}. These characterizations extend many well-known results.

키워드

과제정보

This work was supported by Jiangsu Students' Innovation and Entrepreneurship Training Program [202011117097Y] and the National Nature Science Foundation of China [11471282].

참고문헌

  1. A.Ben-Israel and T.N.E. Greville, Generalized Inverses: Theory and Applications, 2nd. ed., Springer, New York, 2003.
  2. A. Dajic, Common solutions of linear equations in a ring, with applications, Eletron. Linear Algebra, 30(2015), 66-79.
  3. R. E. Hartwig, Block generalized inverses, Arch. Rational Mech. Anal., 61(1976), 197-251.
  4. S. Karanasios, EP elements in rings and semigroup with involution and C*-algebras, C*-algebras, Serdica Math. J., 41(2015), 83-116.
  5. J. J. Koliha and P. Patr'icio, Elements of rings with equal spectral idempotents, J. Aust. Math. Soc. 72(1)(2002), 137-152.
  6. D. Mosic and D. S. Djordjevic, Moore-Penrose-invertible normal and Hermitian elements in rings, Linear Algebra Appl., 431(2009), 732-745.
  7. D. Mosic and D. S. Djordjevic, Partial isometries and EP elements in rings with involution, Electron. J. Linear Algebra, 18(2009), 761-772.
  8. D. Mosic and D. S. Djordjevic, Further results on partial isometries and EP elements in rings with involution, Math. Comput. Modelling, 54(1-2)(2011), 460-465.
  9. D. Mosic, D. S. Djordjevic, J. J. Koliha, EP elements in rings, Linear Algebra Appl., 431(2009), 527-535.
  10. R. Penrose, A generalized inverse for matrices, Proc. Cambridge Philos. Soc., 51(1955), 406-413.
  11. Y. C. Qu, J. C. Wei and H. Yao, Characterizations of normal elements in ring with involution, Acta. Math. Hungar., 156(2)(2018), 459-464.
  12. S. Z. Xu, J. L. Chen and J. Benitez, EP elements in rings with involution, Bull. Malay. Math. Soc. Ser., 42(2019), 3409-3426.
  13. Z. C. Xu, R. J. Tang and J. C. Wei, Strongly EP elements in a ring with involution, Filomat, 34(6)(2020), 2101-2107.
  14. R. J. Zhao, H. Yao and J. C. Wei, Characterizations of partial isometries and two special kinds of EP elements, Czechoslovak Math. J., 70(2020), 539-551.
  15. D. D. Zhao and J. C. Wei, Strongly EP elements in rings with involution, J. Algebra Appl., 21(5)(2022), 2250088.