Additive maps preserving the fixed points of Jordan products of operators

Document Type : Research Paper

Author

Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, P. O. Box 47416-1468, Babolsar, Iran.

10.22072/wala.2022.540575.1349

Abstract

Let $\mathcal{B(X)}$ be the algebra of all bounded linear operators on a complex Banach space $\mathcal{X}$. In this paper, we determine the form of a surjective additive map $\phi: \mathcal{B(X)} \rightarrow \mathcal{B(X)}$ preserving the fixed points of Jordan products of operators, i.e., $F(AoB) \subseteq F(\phi(A) o\phi(B))$, for every $A,B \in \mathcal{B(X)}$, where $AoB=AB+BA$, and $F(A)$ denotes the set of all fixed points of operator $A$.

Keywords


[1] M. Bre\v{s}ar and P. \v{S}emrl, On locally linearly dependent operators and derivations, Trans. Am. Math. Soc., 351 
     (1999), 1257-1275.
[2] G.M.A. Chebotar, W.-F. Ke, P.-H. Lee and N.-C. Wong, Mappings preserving zero products, Studia Math., 155 (2003), 
      77-94.
[3] M. Dobovi\v{s}ek, B. Kuzma, G. Le\v{s}njak, C.K. Li and T. Petek, Mappings that preserve pairs of operators with zero 
     triple Jordan Product, Linear Algebra Appl., 426 (2007), 255-279.
[4] G. Dolinar, S. Du, J. Hou and P. Legi\v{s}a, General preservers of invariant subspace lattices, Linear Algebra Appl., 429 
     (2008), 100-109.
[5] L. Fang, G. Ji and Y. Pang, Maps preserving the idempotency of products of operators, {\it Linear Algebra Appl.}, {\bf 426} (2007), 40-52.
[6] C.-K. Li, P. \v{S}emrl and N.-K. Tsing, Maps preserving the nilpotency of products of operators, Linear Algebra Appl., 
      424 (2007), 222-239.
[7] L. Moln\v{a}r, Non-linear Jordan triple automorphisms of sets of self-adjoint matrices and operators, Studia Math., 173
     (2006), 39-48.
[8] P. \v{S}emrl, Linear mappings that preserve operators annihilated by a poly-nomial, J. Oper. Theory, 36, (1996), 45-58.
[9] A. Taghavi and R. Hosseinzadeh, Maps preserving the dimension of fixed points of products of operators, Linear and 
     Multilinear Algebra, 62 (2013), 1285-1292.
[10] A. Taghavi, R. Hosseinzadeh and H. Rohi, Maps preserving the fixed points of sum of operators, Oper. Matrices, 9 
       (2015), 563-569.
[11] A. Taghavi and R. Hosseinzadeh, Maps preserving the fixed points of triple Jordan products of operators, Indag. 
       Math., 27 (2016), 850-854.
[12] M. Wang, L. Fang and G. Ji, Linear maps preserving idempotency of products or triple Jordan products of operators, 
       Linear Algebra Appl., 429 (2008), 181-189.