Maps preserving operator pairs whose products are projections (Q5962287)

Let \(H\) be a complex Hilbert space with \(\dim H \geq 2\) and let \(B(H)\) be the algebra of all bounded linear operators on \(H\). The authors show that, if \(\phi: B(H)\to B(H)\) is a surjective map preserving operator pairs whose products are nonzero projections in both directions (i.e., \(\phi (A) \phi (B)\) is a nonzero projection if and only if \(AB\) is), then there exists a unitary or an anti-unitary operator \(U\) on \(H\) and a constant \(\lambda\) satisfying \(\lambda^2 =1\) such that \(\phi (T)= \lambda U^* T U\) for all \(T \in B(H)\). In [Acta Math.\ Sin., Chin.\ Ser.\ 53, No.\,2, 315--322 (2010; Zbl 1206.47033)], the authors considered linear maps satisfying the same property.NEWLINENEWLINENow let \(A,B \in B(H)\). Recall that the triple Jordan product of \(A\) and \(B\) is \(ABA\). Analogously, the authors show that, if \(\phi: B(H)\to B(H)\) is a surjective map preserving operator pairs whose triple Jordan products are nonzero projections in both directions, then there exists a unitary or an anti-unitary operator \(U\) on \(H\) and a constant \(\lambda\) satisfying \(\lambda^3 =1\) such that either \(\phi (T)= \lambda U^* T U\) for all \(T \in B(H)\) or \(\phi (T)= \lambda U^* T^* U\) for all \(T \in B(H)\).