Semigroups of linear operators

点击翻译按钮获取中文摘要

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.