Operator algebras of multipliers on reproducing kernel Hilbert spaces

我的研究计划的目的是对称为运算符的数学对象进行分类。算子是一种变换，它具有非常严格的性质，即线性。例如，绘制作用于笛卡尔平面上的线性变换的图形会得到三维空间中的另一个平面。算子的这个简单性质使它们易于用数学工具进行分析。另一方面，算子可以有效地用于描述自然科学和工程中发生的许多重要现象。事实上，它们是量子力学中出现的基本物体。因此，有必要为它们发展一个坚实的数学理论。从我的研究中得出的预期分类将为理解算子提供一种方便的方法，并且可以使其他领域的数学家和科学家受益。建立一般理论是一个复杂的问题，因为操作符表现出巨大的多样性。幸运的是，在应用程序中经常遇到的许多常见操作符显示了某种程度的额外结构，并且基于这种给定的额外结构进行分类变得更易于管理。实现它的想法是识别代表的某个具体模型子类。如果做得适当，一般算符的研究可以（在某种精确意义上）简化为对这些更具体的模型的研究，然后可以从中提取出广泛的结论。具体算符通常被选择作用于由函数组成的空间，这使它们对数学家来说非常熟悉。事实上，这个想法并不新鲜，多年来，在专注于单个运营商的情况下，已经取得了很大的效果。然而，为了描述两个自然系统之间的相互作用，我们需要捕捉一对算子的行为。关于这类配对的模型的最新知识仍然相当初级，改进它是我将追求的主要目标。***上述项目有两个主要特点，它们将与我的研究生共同承担。首先，需要选择足够丰富的具体代表集合，以便对一个重要和实用的一般算子族进行建模。通过将感兴趣的两个操作符组合在一起，将所有与它们自然相关的操作符组合到一个操作符代数中，并对结果对象执行分析，可以检测到这种丰富性。然后，第二步是非常详细地探索和检查已确定的具体代表。我们检查的精确性质将取决于所需的信息，但一个典型的例子可能涉及所谓的算子对频谱，它对应于可以在实验室中进行的可观测量的测量

The aim of my research program is to classify mathematical objects called operators. An operator is a transformation having a very rigid property known as linearity. Plotting the graph of a linear transformation acting on the Cartesian plane results in another plane in three dimensional space, for instance. This simple property of operators makes them amenable to analysis using mathematical tools. On the other hand, operators can be fruitfully used to describe many important phenomena occurring in natural science and engineering. Indeed, they are the basic objects appearing in quantum mechanics for example. Consequently, it is desirable to develop a solid mathematical theory for them. The anticipated classification resulting from my research would yield a convenient method for understanding operators, and could benefit both mathematicians and scientists in other fields.***Building a general theory is a complex problem since operators exhibit tremendous variety. Fortunately, many common operators that are frequently encountered in applications display some level of additional structure, and basing a classification on this given extra structure becomes a more manageable endeavour. The idea to achieve it is to identify a certain concrete model subclass of representatives. If done appropriately, the study of the general operators can be reduced (in some precise sense) to the study of these more concrete models and broad conclusions can then be extracted thereof. The concrete operators are usually chosen to act on spaces consisting of functions, making them very familiar to mathematicians. In fact, this idea is not new and has been exploited to great effect for many years now in the case where the focus is on a single operator. To describe interactions between two natural systems however, one needs to capture the behaviour of a pair of operators. The state of the art knowledge on models for such pairs is still rather rudimentary, and improving it is the principal objective I will pursue.***There are two main features in the program described above, and they will be undertaken jointly with my graduate students. First, the collection of concrete representatives needs to be chosen to be rich enough in order for it to model a significant and practical family of general operators. This richness will be detected by grouping the two operators of interest together will all the ones that are naturally related to them into an operator algebra, and performing an analysis of the resulting object. The second step is then to explore and examine in great detail the concrete representatives that have been identified. The precise nature of our examination will depend on the information that is required, but a typical example could involve the so-called spectrum of the pair of operators which corresponds to the measurements of an observable quantity that can be made in a laboratory.**