Pairs of continuous linear bijective maps preserving fixed products of operators

Abstract

Let X X be a complex Banach space, and denote by B ( X ) \mathcal {B}(X) the algebra of all bounded linear operators on X X . Let C , D ∈ B ( X ) C,D\in \mathcal {B} \left ( X\right ) be fixed operators. In this paper, we characterize linear, continuous and bijective maps φ \varphi and ψ \psi on B ( X ) \mathcal {B}\left ( X\right ) for which there exist invertible operators T 0 , W 0 ∈ B ( X ) T_0, W_0 \in \mathcal { B}(X) such that φ ( T 0 ) , ψ ( W 0 ) ∈ B ( X ) \varphi (T_0), \psi (W_0) \in \mathcal {B}(X) are both invertible, having the property that φ ( A ) ψ ( B ) = D \varphi \left ( A\right ) \psi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C in B ( X ) \mathcal {B}(X) . As a corollary, we deduce the form of linear, bijective and continuous maps φ \varphi on B ( X ) \mathcal {B}(X) having the property that φ ( A ) φ ( B ) = D \varphi \left ( A\right ) \varphi \left ( B\right ) =D in B ( X ) \mathcal {B}(X) whenever A B = C AB=C .