Semigroups of linear operators

在数学和精确科学中，我们经常不得不处理对象的变换。如果我们根据一条规则转换对象，然后根据另一条规则转换结果，最终结果通常取决于将转换应用于对象的顺序。换句话说，如果我们将符号A和B用于两个变换，那么组合变换AB和BA通常是非常不同的。如果它们是相同的，即如果AB=BA，我们说这两个变换是可交换的。不难接受的是，当人们处理和计算通勤转换时，生活会容易得多。因此，我们研究中感兴趣的一个问题是：什么条件保证给定变换集合的成员相互通勤？由于这种情况很少发生，我们只能满足于“次好的事情”。这可以用各种方式来解释，所有这些都是我们感兴趣的。在处理某些非交换变换集合时，一种这样的折衷是所谓的“同时三角化”，它保留了许多良好的交换性属性，并产生了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.