n-strongly quasi-invariant measure on double coset spaces

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., $f(aob) subseteq f(phi(a) ophi(b))$, for every $a,b in mathcal{b(x)}$, where $aob=ab+ba$, and $f(a)$ denotes the set of all fixed points of operator $a$.