Doubly stochastic operators obtained from positive operators

Charles Ray Hobby and Ronald Pyke

Vol. 15 (1965), No. 1, 153–157

DOI: 10.2140/pjm.1965.15.153

Abstract

A recent result of Sinkhorn [3] states that for any square matrix $A$ of positive elements, there exist diagonal matrices $D_1$ and $D_2$ with positive diagonal elements for which $D_{1}A{D}_2$ is doubly stochastic. In the present paper, this result is generalized to a wide class of positive operators as follows.

Let $\left(\Omega ,\mathfrak{A},\lambda \right)$ be the product space of two probability measure spaces $\left({\Omega }_i,{\mathfrak{A}}_i,{\lambda }_i\right)$. Let $f$ denote a measurable function on $\left(\Omega ,\mathfrak{A}\right)$ for which there exist constants $c$, $C$ such that $0<c\leqq f\leqq C<\infty$. Let $K$ be any nonnegative, two-dimensional real valued continuous function defined on the open unit square, $\left(0,1\right)\times \left(0,1\right)$, for which the functions $K\left(u,\cdot \right)$ and $K\left(\cdot ,v\right)$ are strictly increasing functions with strict ranges $\left(0,\infty \right)$ for each $u$ or $v$ in (0,1). Then there exist functions $h:{\Omega }_1\to E_1$ and $g:{\Omega }_2\to E_1$ such that

$${\int \; <!--nolimits-->}_{{\Omega }_2}f\left(x,v\right)K\left(h\left(x\right),g\left(v\right)\right)d{\lambda }_2\left(v\right)=1={\int \; <!--nolimits-->}_{{\Omega }_1}f\left(u,y\right)K\left(h\left(u\right),g\left(y\right)\right)d{\lambda }_1\left(u\right),$$

almost everywhere $-\left(\lambda \right)$.

Mathematical Subject Classification

Primary: 60.60

Milestones

Received: 13 February 1964

Published: 1 March 1965