On operators preserving commutativity (Q1072093)

Let L(X) be the algebra of all bounded operators on a non-trivial complex Banach space X and \(F: L(X)\to L(X)\) a bijective linear operator such that F and \(F^{-1}\) both send commuting pairs of operators into commuting pairs. Then, either \(F(A)=\sigma UAU^{-1}+p(A)I\), or \(F(A)=\sigma UA'U^{-1}+p(A)I\), where p is a linear functional on L(X), U is a bounded linear bijective operator between the appropriate two spaces, \(\sigma\) is a complex constant, and A' is the adjoint of A. The form of an operator F for which F and \(F^{-1}\) both send projections of rank one into projections of rank one is also determined.