Hostname: page-component-cb9f654ff-pvkqz Total loading time: 0 Render date: 2025-08-24T05:45:49.238Z Has data issue: false hasContentIssue false
  • English
  • Français

CHARACTERIZATIONS OF JORDAN DERIVATIONS ON STRONGLY DOUBLE TRIANGLE SUBSPACE LATTICE ALGEBRAS

Part of: Special classes of linear operators

Published online by Cambridge University Press:  21 July 2011

YUN-HE CHEN  and
JIAN-KUI LI
YUN-HE CHEN
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China (email: heyunchen@hotmail.com)
JIAN-KUI LI*
Affiliation:
Department of Mathematics, East China University of Science and Technology, Shanghai 200237, PR China (email: jiankuili@yahoo.com)
*
✉For correspondence; e-mail: jiankuili@yahoo.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let 𝒟 be a strongly double triangle subspace lattice on a nonzero complex reflexive Banach space X and let δ:Alg 𝒟→Alg 𝒟 be a linear mapping. We show that δ is Jordan derivable at zero, that is, δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A) whenever AB+BA=0 if and only if δ has the form δ(A)=τ(A)+λA for some derivation τ and some scalar λ. We also show that if the dimension of X is greater than 2, then δ satisfies δ(AB+BA)=δ(A)B+Aδ(B)+δ(B)A+Bδ(A)whenever AB=0if and only if δ is a derivation.

Keywords

derivationJordan derivationsubspace lattice

MSC classification

Secondary: 47B47: Commutators, derivations, elementary operators, etc.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 84 , Issue 2 , October 2011 , pp. 300 - 309
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

Footnotes

This work is supported by NSF of China.

References

[1]Chebotar, M., Ke, W. and Lee, P., ‘Maps characterized by action on zero products’, Pacific J. Math. 216 (2004), 217–228.Google Scholar
[2]Halmos, P., ‘Reflexive lattices of subspaces’, J. Lond. Math. Soc. 4(2) (1971), 257–263.CrossRefGoogle Scholar
[3]Jiao, M. and Hou, J., ‘Additive maps derivable or Jordan derivable at zero point on nest algebras’, Linear Algebra Appl. 432 (2010), 2984–2994.Google Scholar
[4]Jing, W., Lu, S. and Li, P., ‘Characterisations of derivations on some operator algebras’, Bull. Aust. Math. Soc. 66 (2002), 227–232.CrossRefGoogle Scholar
[5]Kadison, R. and Ringrose, J., Fundamentals of the Theory of Operator Algebras, Vols. I and II (Academic Press, London, 1983, 1986).Google Scholar
[6]Lambrou, M. and Longstaff, W. E., ‘Finite rank operators leaving double triangles invariant’, J. Lond. Math. Soc. 45 (1992), 153–168.CrossRefGoogle Scholar
[7]Longstaff, W. E., ‘Strongly reflexive lattices’, J. Lond. Math. Soc. 11 (1975), 491–498.CrossRefGoogle Scholar
[8]Longstaff, W. E., ‘Non-reflexive double triangles’, J. Aust. Math. Soc. 35 (1983), 349–356.CrossRefGoogle Scholar
[9]Pang, Y. and Yang, W., ‘Derivations and local derivations on strongly double triangle subspace lattice algebras’, Linear Multilinear Algebra 58 (2010), 855–862.Google Scholar
[10]Zhao, S. and Zhu, J., ‘Jordan all-derivable points in the algebra of all upper triangular matrices’, Linear Algebra Appl. 433 (2010), 1922–1938.Google Scholar