Semigroups of linear operators

在数学和精确科学中，我们经常要处理对象的变换。 如果我们根据一个规则转换一个对象，然后根据另一个规则转换结果，最终的结果通常取决于转换应用于对象的顺序。换句话说，如果我们用符号A和B表示两个变换，那么组合变换AB和BA通常是完全不同的。如果它们相同，即，如果AB = BA，我们说这两个变换可交换。不难接受，当人们处理和计算交换变换时，生活要容易得多。因此，在我们的研究中有一个有趣的问题是：什么条件保证一个给定的转换集合的成员相互交换？由于这种理想的状态很少发生，我们必须满足于“次佳之事”。“这可以用各种方式来解释，所有这些都是我们感兴趣的。当处理某些非交换变换的集合时，一个这样的折衷是所谓的“同时三角化”，它保留了交换性的许多优良性质，并产生关于变换结构的信息。

In mathematics, and in exact sciences, we often have to deal with transformations of objects. If we transform an object according to one rule and then transform the result according to another rule, the final outcome usually depends on the order in which the transformations are applied to the object. In other words, if we use the symbols A and B for the two transformations, then the combined transformations AB and BA are usually quite different. If they are the same, i.e., if AB = BA, we say the two transformations commute. It is not hard to accept that life is much easier when one deals, and computes, with commuting transformations. Thus one question of interest in our studies is: what conditions guarantee that members of a given collection of transformations mutually commute? Since this desired state of affairs seldom occurs, we have to be satisfied with "the next best thing." This can be interpreted in various ways, all of which are of interest to us. One such compromise, when dealing with certain collections of non-commuting transformations, is what is called "simultaneous triangularizability," which retains many of the nice properties of commutativity and yields information about the structure of transformations.