Vol. 15, No. 1, 1965

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Doubly stochastic operators obtained from positive operators

Charles Ray Hobby and Ronald Pyke

Vol. 15 (1965), No. 1, 153–157

A recent result of Sinkhorn [3] states that for any square matrix A of positive elements, there exist diagonal matrices D1 and D2 with positive diagonal elements for which D1AD2 is doubly stochastic. In the present paper, this result is generalized to a wide class of positive operators as follows.

Let (Ω,A,λ) be the product space of two probability measure spaces (Ωi,Ai,λi). Let f denote a measurable function on (Ω,A) for which there exist constants c, C such that 0 < c f C < . Let K be any nonnegative, two-dimensional real valued continuous function defined on the open unit square, (0,1) × (0,1), for which the functions K(u,) and K(,v) are strictly increasing functions with strict ranges (0,) for each u or v in (0,1). Then there exist functions h : Ω1 E1 and g : Ω2 E1 such that

Ω2f(x,v)K(h(x),g(v))dλ2(v) = 1 =Ω1f(u,y)K(h(u),g(y))dλ1(u),

almost everywhere (λ).

Mathematical Subject Classification
Primary: 60.60
Received: 13 February 1964
Published: 1 March 1965
Charles Ray Hobby
Ronald Pyke