Analysis of operators on spaces of holomorphic functions

An operator is a mapping of one set into another, each of which has a certain structure. Operators play a central role in several branches of physics and engineering, and, in particular, modern operator theory has initially developed as the natural language of quantum mechanics. Many operators from quantum mechanics have useful realizations on spaces of analytic functions. One of the most significant examples in this sense is the Fock/Segal- Bargmann space, which finds its origins in the analysis of the harmonic oscillator through its decomposition into the Fock boson creation-annihilation operators. Subsequently, more elaborated structures were introduced, namely vector-valued spaces of analytic functions, which involve functions with values in an infinite dimensional space. Apart from being motivated by applications to engineering, these structures brought along new mathematical insights. For instance, they played a crucial role in the solution of a famous long-standing open problem in operator theory, the so-called Halmos problem, which was achieved by the French mathematician Gilles Pisier using a special class of operators, called Hankel operators. Hankel operators have numerous applications to various areas, including Control Theory, Prediction Theory, Approximation Theory, Stationary Processes. The aim of this project is to study several classes of operators (including Hankel and Toeplitz operators) acting on vector- valued Fock spaces, a natural topic which has been less pursued so far, which would complement recent studies and would lead to a better understanding of the vector-valued setting. A stark contrast between the behaviour of these structures in the finite-dimensional, respectively infinite-dimensional frameworks, has been revealed in the recent literature, a contrast which, for instance, strongly resonates in the solution of the celebrated Halmos problem. More precisely, the relationship between many basic concepts becomes much more intricate and it is, so far, less understood in infinite dimension. During the 24 months of the project a postdoctoral researcher under the PI`s supervision will endeavour to delve deeper into some of these problems and open questions.

An operator is a mapping of one set into another, each of which has a certain structure. Operators play a central role in several branches of physics and engineering, and, in particular, modern operator theory has initially developed as the natural language of quantum mechanics. Many operators from quantum mechanics have useful realizations on spaces of analytic functions. One of the most significant examples in this sense is the Fock/Segal-Bargmann space, which finds its origins in the analysis of the harmonic oscillator through its decomposition into the Fock boson creation-annihilation operators. Subsequently, more elaborated structures were introduced, namely vector-valued spaces of analytic functions, which involve functions with values in an infinite dimensional space. Apart from being motivated by applications to engineering, these structures brought along new mathematical insights. For instance, they played a crucial role in the solution of a famous long-standing open problem in operator theory, the so-called Halmos problem, which was achieved by the French mathematician Gilles Pisier using a special class of operators, called Hankel operators. Hankel operators have numerous applications to various areas, including Control Theory, Prediction Theory, Approximation Theory, Stationary Processes. The aim of this project was to study several classes of operators (including Hankel and Toeplitz operators) acting on vector-valued Fock spaces, a natural topic which has been less pursued so far, thus complementing recent studies and leading to a better understanding of the vector-valued setting. More precisely, the relationship between many basic concepts becomes much more intricate in infinite dimension. During the 24 months of the project, a postdoctoral researcher under the PI's supervision delved deeper into some of these problems and open questions.