Journal of applied mathematics & informatics

한국전산응용수학회 (The Korean Society for Computational and Applied Mathematics)
권호 표지
DOI QR코드

DOI QR Code

AN IDENTITY ON STANDARD OPERATOR ALGEBRA

  • SHUJAT, FAIZA (Department of Mathematics, Faculty of Science, Taibah University)
  • 투고 : 2022.01.16
  • 심사 : 2022.05.29
  • 발행 : 2022.09.30
PDF 다운로드
⟨ 이전 논문 다음 논문 ⟩

초록

The purpose of this research is to find an extension of the renowned Chernoff theorem on standard operator algebra. Infact, we prove the following result: Let H be a real (or complex) Banach space and 𝓛(H) be the algebra of bounded linear operators on H. Let 𝓐(H) ⊂ 𝓛(H) be a standard operator algebra. Suppose that D : 𝓐(H) → 𝓛(H) is a linear mapping satisfying the relation D(AnBn) = D(An)Bn + AnD(Bn) for all A, B ∈ 𝓐(H). Then D is a linear derivation on 𝓐(H). In particular, D is continuous. We also present the limitations on such identity by an example.

키워드

참고문헌

  1. A.Z. Ansari, On identities with additive mappings in rings, Iranian Journal of Mathematical Sciences and informatics 15 (2020), 125-133. https://doi.org/10.29252/ijmsi.15.1.125
  2. A.Z. Ansari and F. Shujat, Jordan ∗-derivations on Standard Operator Algebras, Filomat (2020) Preprint.
  3. M. Bresar, Jordan derivations on semiprime rings, Proc. Amer. Math. Soc. 104 (1988), 1003-1006. https://doi.org/10.1090/S0002-9939-1988-0929422-1
  4. M. Bresar, Jordan mappings of semiprime rings, J. Algebra 127 (1989), 218-228. https://doi.org/10.1016/0021-8693(89)90285-8
  5. P.R. Chernoff, Representations, automorphisms and derivations of some Operator Algebras, Journal Funct. Anal. 2 (1973), 275-289. https://doi.org/10.1016/0022-1236(73)90080-3
  6. J. Cusack, Jordan derivations on rings, Proc. Amer. Math. Soc. 53 (1975), 321-324. https://doi.org/10.1090/S0002-9939-1975-0399182-5
  7. P. Semrl, Jordan ∗-derivations of standard operator algebras, Proc. Amer. Math. Soc. 120 (1994), 3347-3350.
  8. P. Semrl, Ring derivations on standard operator algebras, J. Funct. Anal. 112 (1993), 318-324. https://doi.org/10.1006/jfan.1993.1035
  9. J. Vukman, Some remarks on derivations in semiprime rings and standard operator algebras, Glasnik Mat. 46 (2011), 43-48. https://doi.org/10.3336/gm.46.1.07
  10. A.M. Sinclair, Automatic continuity of linear operators, London Math. Soc. Lecture Note 21, Cambridge University Press, Cambridge, London, New York and Melbourne, 1976.
  11. J. Vukman, Identities related to derivations and centralizers on standard operator algebras, Taiwanese J. Math. 11 (2007), 255-265. https://doi.org/10.11650/twjm/1500404650
  12. V. Runde, Range inclusion results for derivations on noncommutative Banach algebras, Studia Math. 105 (1993), 159-172. https://doi.org/10.4064/sm-105-2-159-172
  13. B. Zalar, On centralizers of semiprime rings, Comment. Math. Univ. Carol. 32 (1991), 609-614.