Vol. 58, No. 1, 1975

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The range of a normal derivation

Barry E. Johnson and James Patrick Williams

Vol. 58 (1975), No. 1, 105–122

The inner derivation δA implemented by an element A of the algebra (t) of bounded linear operators on the separable Hilbert space is the map X AX XA(X ∈ℬ(X)). The main result of this paper is that when A is normal, range inclusion (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).

Mathematical Subject Classification 2000
Primary: 47B15
Received: 19 November 1973
Revised: 21 May 1974
Published: 1 May 1975
Barry E. Johnson
James Patrick Williams