Positive commutators [formula omitted

Abstract

We consider commutators of the form R ƒ, g = i[ƒ(P), g(Q)] , where P, Q are an irreducible canonical pair of selfadjoint operators in Hilbert space ([ P, Q] = − i), and where ƒ, g are real-valued functions with ƒ′, g′ ϵ L 1( R) . A necessary and sufficient condition for R ƒ, g to be of rank one [zero] is given. Some necessary and some sufficient conditions are given for R ƒ, g ⩾ 0 . For example, it is necessary that both ƒ, g are strictly monotone (unless R ƒ, g = 0 ). If K a is the set of bounded functions ƒ such that ƒ has an analytic continuation to a strip ¦ Im z¦ < a with ( Im z)( Im ƒ(z)) ⩾ 0 , then it is sufficient that ƒ ϵ K a and g ϵ K b with ab ⩾ π 2 .