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
.