Some classes of operators on spaces of analytic functions

The aim of this project is to investigate the following classes of operators acting on spaces of analytic functions: - Carleson embeddings for vector-measures on vector-valued Bergman spaces; - Integration operators on Fock spaces with radial weights. The first direction aims to provide Carleson embedding theorems for operator-valued measures, thus answering a fundamental question concerning vector-valued Bergman spaces. This problem is related to the investigation of the Bergman projection for vector-valued functions and to the study of Hankel operators with operator-valued symbols. For these two problems recent papers co-authored by the principal investigator show that, in the setting of vector-valued Bergman spaces, the scalar results have natural generalizations even in the case when the "target- space" has infinite dimension. This is in contrast to the situation on vector-valued Hardy spaces, where the analogous results fail in the infinite-dimensional case. The gained expertise makes it realistic to expect that, while the classical Carleson embedding theorem for Hardy spaces does not extend to operator-measures, an appropriate analogue of this theorem can be obtained for vector-valued Bergman spaces. The second objective is the study of Volterra-type integration operators and of generalized Cesaro operators on Fock spaces with radial weights. We are concerned with boundedness, compactness and spectral properties, building on recent progress on these problems in somewhat simpler settings. The recent contributions of the principal investigator in this direction represent a good starting point for this part of the project. The methods of investigation consist of a combination of tools from complex analysis, functional analysis, operator theory and harmonic analysis.

Mathematics is an evolution from the human brain, most of which originally arose out of the physical world, and which subsequently began to ask internal questions that are not immediately connected to the external reality, but whose answers, together with the machinery that emerged in the process, often turn out to be useful for understanding the outside world. These internal questions are the object of abstract mathematics, a vast area the present project is a minuscule/microscopic bit of. The objects of our study are the so-called operators. 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. One of the main outcomes of this project is obtaining so-called Carleson embedding theorems for specific vector-valued spaces of analytic functions, which are useful tools in the investigation of these abstract structures, being also related to the concepts used in Pisier's achievement mentioned above. Another outcome is the study of different classes of operators acting on Fock-type spaces. Finally, a quite interesting, unexpected achievement of the project is the observation that a purely complex analytic approach can be used to tackle a problem arising in water waves, namely, the description of particle paths in fluid flows.This is yet another confirmation that, it often happens, that methods developed in abstract mathematics, subsequently turn out to be useful in applications to the real world