PAIRS OF CONTINUOUS LINEAR BIJECTIVE MAPS PRESERVING FIXED PRODUCTS OF OPERATORS

Let X be a complex Banach space, and denote by 13 ( X ) the algebra of all bounded linear operators on X . Let C, D E 13 ( X ) be fixed operators. In this paper, we characterize linear, continuous and bijective maps phi and psi on 13 ( X ) for which there exist invertible operators T 0 , W 0 E 13 ( X ) such that phi ( T 0 ) , psi ( W 0 ) E 13 ( X ) are both invertible, having the property that phi ( A ) psi ( B ) = D in 13 ( X ) whenever AB = C in 13 ( X ). As a corollary, we deduce the form of linear, bijective and continuous maps phi on 13 ( X ) having the property that phi ( A ) phi ( B ) = D in 13 ( X ) whenever AB = C .