Let
de7e0dc6f9c83" title="Click to view the MathML source">L(X)den">de"> be the algebra of all boun
ded linear operators on a complex Banach space
X. We
describe surjective linear maps
ϕ on
de7e0dc6f9c83" title="Click to view the MathML source">L(X)den">de"> that satisfy
for every
de80a5bf6f8136a953e91f47a14f0" title="Click to view the MathML source">x∈Xden">de"> and
T∈L(X)den">de">. We also
describe surjective linear maps
ϕ on
de7e0dc6f9c83" title="Click to view the MathML source">L(X)den">de"> that satisfy
for every
de80a5bf6f8136a953e91f47a14f0" title="Click to view the MathML source">x∈Xden">de"> and
T∈L(X)den">de">. Furthermore, we characterize maps
ϕ (not necessarily linear nor surjective) on
de7e0dc6f9c83" title="Click to view the MathML source">L(X)den">de"> which satisfy
for every
de80a5bf6f8136a953e91f47a14f0" title="Click to view the MathML source">x∈Xden">de"> and
de0891bade88db17bf407ac2ea" title="Click to view the MathML source">T,S∈L(X)den">de">.