Chasing traces of analyticity with applications to operator theory

My proposed investigation falls under the umbrella of research related to the uncertainty principle in Fourier analysis. The aim is to study the spectra, i.e.

Fourier transforms, of functions which satisfy a one-sided decay condition, and certain multi-dimensional generalizations. An ultimate form of one-sided decay of a function is its complete vanishing on a half-axis.

In this case, the fundamental Jensen´s inequality of complex analysis states that the corresponding Fourier transform will satisfy certain global logarithmic integrability properties which are typical of analytic functions.

My initial investigations of the subject have revealed some fascinating phenomena: certain one-sided decay conditions on a function ensure that the spectrum of such a function "clumps up", in the sense that the Fourier transform lives on a bunch of intervals, and on these intervals local versions of logarithmic integrability properties are satisfied.

My aim is to develop further the spectral theory of one-sided decaying functions, and investigate also possible multi-dimensional version of such a result, for instance by investigeting spectra of functions defined in the plane and decaying in an angle.

This research is motivated by its multiple applications to modern operator theory: theories of subnormal operators and de Branges-Rovnyak spaces.

In these applications, functions with one-sided decay appear, and revealing their spectral structure is the key to a deeper understanding.