Operator algebras of multipliers on reproducing kernel Hilbert spaces

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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.