Additive mappings decreasing rank one (Q1601628)

Let \(X\) be a Banach space on a real or complex field \(\mathbb{F}\). By \({\mathcal B}(X)\) and \({\mathcal F}(X)\) denote respectively the algebra of all bounded linear operators on \(X\) and the subalgebra of all finite rank operators in \({\mathcal B}(X)\). The author characterizes the structure of additive maps \(\Phi\) from \({\mathcal F}(X)\) into itself which map rank-1 operators into operators of rank at most one. He proves that such \(\Phi\) is either of the form \(\Phi (x\otimes f)=Ax\otimes Cf\) or \(\Phi (x\otimes f)=Cx\otimes Af\), where \(A:X\rightarrow X\) (\(X^\prime \)) and \(C:X^{\prime }\rightarrow X^{\prime }\) (\(X\)) are \(h\)-quasilinear and \(h\) is a ring homomorphism of \(\mathbb{F}\).