Unitary Representations and Complex Analysis
Unitary Representations and Complex Analysis
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酉表示和复分析
DOI：
10.1007/978-3-540-76892-0_5
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
D. Vogan
中科院分区：
文献类型：
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作者：
D. Vogan
我要说的很大程度上取决于表示论和线性代数之间的类比，所以让我从回顾线性代数的一些想法开始。线性代数的一个目标是抽象地理解向量空间V的所有可能的线性变换。线性变换最简单的例子是一维空间上的标量乘法。光谱理论试图从这个例子中建立更一般的变换。在无限维向量空间的情况下，在V上引入一个拓扑，并要求它是连续的，这是有用和有趣的。V有许多可能的选择，并且为特定问题选择正确的V可能是微妙而重要的(例如，当T是作用于函数空间的微分运算符时)。
Much of what I will say depends on analogies between representation theory and linear algebra, so let me begin by recalling some ideas from linear algebra. One goal of linear algebra is to understand abstractly all possible linear transformationsTof a vector spaceV. The simplest example of a linear transformation is multiplication by a scalar on a one-dimensional space. Spectral theory seeks to build more general transformations from this example. In the case of infinite-dimensional vector spaces, it is useful and interesting to introduce a topology onV, and to require thatTbe continuous. It often happens (as in the case whenTis a differential operator acting on a space of functions) that there are many possible choices ofV, and that choosing the right one for a particular problem can be subtle and important.