Let
L(X) be the algebra of all bounded linear operators on a complex Banach space
X. We describe surjective linear maps
ϕ on
L(X) that satisfy
for every
x∈X and
c490618cfd493b4bd335f84309015" title="Click to view the MathML source">T∈L(X). We also describe surjective linear maps
ϕ on
L(X) that satisfy
for every
x∈X and
c490618cfd493b4bd335f84309015" title="Click to view the MathML source">T∈L(X). Furthermore, we characterize maps
ϕ (not necessarily linear nor surjective) on
L(X) which satisfy
for every
x∈X and
T,S∈L(X).