Surjective maps preserving the reduced minimum modulus of products

Abstract

Suppose $\mathfrak{B}(H)$ is the Banach algebra of all bounded linear operators on a Hilbert space $H$ with $\dim(H)\geq 3$. Let $\gamma(.)$ denote the reduced minimum modulus of an operator. We charaterize surjective maps $\varphi$ on $\mathfrak{B}(H)$ satisfying $$\gamma(\varphi(T)\varphi(S))=\gamma(T S)\;\;\;(T, S\in \mathfrak{B}(H)).$$ Also, we give the general form of surjective maps on $\mathfrak B(H)$ preserving the reduced minimum modulus of Jordan triple products of operators.