On maps preserving $C$-symmetric triple Jordan product of pairs of operators

Abstract

A bounded linear operator $T$, which operates on a complex separable Hilbert space $H$, is referred to as $C$-symmetric if $T=CT^*C$, where $C$ represents a conjugate-linear isometric involution on $H$. In this paper, we thoroughly investigate mappings $\Phi:\lh\to\lh$, where $\lh$ denotes the algebra of all bounded linear operators on $H$, that fulfill the following condition:
$$
ABA \mbox{ is } C\mbox{-symmetric}\;\Longleftrightarrow\; \Phi(A)\Phi(B)\Phi(A) \mbox{ is } C\mbox{-symmetric}
$$
for all $A,B\in\lh$ and conjugations $C$ on $H$.