Jordan isomorphisms and maps preserving spectra of certain operator products

Tom 184 / 2008

Jinchuan Hou, Chi-Kwong Li, Ngai-Ching Wong Studia Mathematica 184 (2008), 31-47 MSC: Primary 47B49, 47A12. DOI: 10.4064/sm184-1-2


Let $\mathcal{A}_1, \mathcal{A}_2$ be (not necessarily unital or closed) standard operator algebras on locally convex spaces $X_1, X_2$, respectively. For $k \ge 2$, consider different products $T_1* \cdots *T_k$ on elements in ${\cal A}_i$, which covers the usual product $T_1* \cdots *T_k = T_1\cdots T_k$ and the Jordan triple product $T_1*T_2 = T_2T_1T_2$. Let ${\mit\Phi} :\mathcal{A}_1\rightarrow\mathcal{A}_2$ be a (not necessarily linear) map satisfying $\sigma({\mit\Phi}(A_1)*\cdots *{\mit\Phi}(A_k)) =\sigma(A_1*\cdots *A_k)$ whenever any one of $A_i$'s has rank at most one. It is shown that if the range of ${\mit\Phi}$ contains all rank one and rank two operators then ${\mit\Phi}$ must be a Jordan isomorphism multiplied by a root of unity. Similar results for self-adjoint operators acting on Hilbert spaces are obtained.


  • Jinchuan HouDepartment of Mathematics
    Taiyuan University of Technology
    Taiyuan 030024, P.R. of China
  • Chi-Kwong LiDepartment of Mathematics
    The College of William & Mary
    Williamsburg, VA 13185, U.S.A.
  • Ngai-Ching WongDepartment of Applied Mathematics
    National Sun Yat-sen University
    and National Center for Theoretical Sciences
    Kaohsiung 80424, Taiwan
    Department of Mathematics
    The Chinese University of Hong Kong
    Hong Kong

Przeszukaj wydawnictwa IMPAN

Zbyt krótkie zapytanie. Wpisz co najmniej 4 znaki.

Przepisz kod z obrazka

Odśwież obrazek

Odśwież obrazek