We note that the well-known result of von Neumann (Contrib Theory Games 2:5–12, 1953) is not valid for all doubly substochastic operators on discrete Lebesgue spaces \(\ell ^p(I)\), \(p\in [1,\infty )\). This fact lead us to distinguish two classes of these operators. Precisely, the class of increasable doubly substochastic operators on \(\ell ^p(I)\) is isolated with the property that an analogue of the Von Neumann result on operators in this class is true. The submajorization relation \(\prec _s\) on the positive cone \(\ell ^p(I)^+\), when \(p\in [1,\infty )\), is introduced by increasable substochastic operators and it is provided that submajorization may be considered as a partial order. Two different shapes of linear preservers of submajorization \(\prec _s\) on \(\ell ^1(I)^+\) and on \(\ell ^p(I)^+\), when I is an infinite set, are presented.

我们注意到 von Neumann (Contrib Theory Games 2:5–12, 1953) 的著名结果不适用于离散勒贝格空间\(\ell ^p(I)\) , \(p \in [1,\infty )\)。这一事实使我们将这些运算符分为两类。准确地说，\(\ell ^p(I)\) 上的可增双次随机算子类与这样的性质隔离，即此类中的算子上的冯诺依曼结果的类似物为真。正锥体上的次大化关系\(\prec _s\) \(\ell ^p(I)^+\)，当\(p\in [1,\infty )\), 是由可增加的亚随机算子引入的，并且提供了 submajorization 可以被认为是偏序。当I是无穷大时，在\(\ell ^1(I)^+\)和\(\ell ^p(I)^+\) 上的两个不同形状的次大化线性保持器\(\prec _s\)设置，呈现。