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