A derivation on an algebra 𝒜 is
a linear transformation δ on 𝒜 with the property δ(XY ) = Xδ(Y ) + δ(X)Y
for all X,Y ∈𝒜. If 𝒜 = ℬ(ℋ) is the Banach algebra of all bounded linear
operators on a complex separable infinitedimensional Hilbert space ℋ then it is
known that every derivation δ on 𝒜 is inner, that is, there is a bounded
operator A on ℋ such that δ(X) = AX − XA = δA(X) for all X ∈ℬ(ℋ).
(See [8].) In the present note simple necessary and sufficient conditions are
obtained that (i) the range ℛ(δA) be dense in the weak and ultraweak operator
topologies; (ii) the norm closure of the range contain the ideal 𝒦 of compact
operators on ℋ, (iii) the set of commutalors BX − XB where B belongs to the
C∗-algebra generated by A and X is arbitrary be weakly or ultraweakly dense in
ℬ(ℋ). The commutant of the range of a derivation is also computed and
it is shown that the ranges of any two nonzero derivations have nonzero
intersection.
