#### 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
##### 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. Let $K$ be any nonnegative, two-dimensional real valued continuous function defined on the open unit square, $\left(0,1\right)×\left(0,1\right)$, 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

${\int }_{{\Omega }_{2}}f\left(x,v\right)K\left(h\left(x\right),g\left(v\right)\right)\phantom{\rule{0.3em}{0ex}}d{\lambda }_{2}\left(v\right)=1={\int }_{{\Omega }_{1}}f\left(u,y\right)K\left(h\left(u\right),g\left(y\right)\right)\phantom{\rule{0.3em}{0ex}}d{\lambda }_{1}\left(u\right),$

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

Primary: 60.60