A square matrix is said to be doubly stochastic if its elements are non-negative and all row sums and column sums are equal one. An important tool in the study of majorization for infinite dimensional spaces ${\ell }^p(I)$ are doubly stochastic operators. These operators are a generalization of doubly stochastic matrices. In this paper, we compare some properties of doubly stochastic operators in finite and infinite dimensions. We will see that if $D:{\ell }^p(I)\to {\ell }^p(I)$ is a doubly stochastic operator then $\Vert D\Vert \le 1$. Moreover, the existence of such an operator with $\Vert D\Vert <1$ is equivalent to $1<p<\infty$ and I is an infinite set. We discuss some other properties of doubly stochastic operators such as compactness and closedness and also, provide relevant applications of these operators in the existence of solutions for some infinite linear equations and functional equations.

如果方阵的元素为非负数且所有行和列总和均等于1，则称其为双随机的。研究无限维空间化的重要工具${\ell }^p（\mathrm{一世}）$是双重随机操作者。这些算子是双随机矩阵的推广。在本文中，我们在有限和无限维中比较了双随机算子的一些性质。我们将看到，如果$d：{\ell }^p（\mathrm{一世}）\to {\ell }^p（\mathrm{一世}）$ 是一个双重随机运算符 $\Vert d\Vert \le \mathrm{1个}$。而且，这样的运算符的存在$\Vert d\Vert <\mathrm{1个}$ 相当于 $\mathrm{1个}<p<\infty$而我是一个无限集。我们讨论了双随机算子的其他一些性质，例如紧致性和封闭性，并在存在一些无限线性方程和泛函方程的解的存在下，提供了这些算子的相关应用。