The inner derivation δA
implemented by an element A of the algebra ℬ(ℋt) of bounded linear operators on
the separable Hilbert space ℋ is the map X → AX − XA(X ∈ℬ(X)). The main
result of this paper is that when A is normal, range inclusion ℛ(δB) ⊂ℛ(δA) is
equivalent to the condition that B = f(A) where Λ(z,w) = (f(z) − f(w))(z − w)−1
(taken as 0 when z = w) has the property that Λ(z,w)t(z,w) is a trace class kernel
on L2(μ) whenever t(z,w) is such a kernel. Here μ is the dominating scalar valued
spectral measure of A constructed in multiplicity theory. In order that a Borel
function f satisfy this condition it is necessary that f be equal almost everywhere to
a Lipschitz function with derivative in σ(A) at each limit point of σ(A) and it is
sufficient (for A self-adjoint) that f ∈ C(3)(R).
