Joint Analysis / Probability Seminar
The aim of this talk is to provide a thorough spectral analysis of the class, denoted by K, of markovian self-similar Co-semigroup on L2. This class of operators, which are associated to non-self-adjoint and non-local generators, appears in various frameworks such as the study of growth-fragmentation equations, random planar maps and stable processes.
We resort to the concept of intertwining (a weak version of similarity) which turns out to be a natural and powerful technique to develop the spectral reduction of non-normal operators. We start by providing a (brief) historical account on the connection between spectral theory of linear operators and intertwining, including the notion of spectral operators introduced by Dunford. We proceed by showing that the intertwining orbit of each element in K is K itself, meaning that any two elements in K are intertwined with a closed linear (non-necessarily positive, neither bounded) operator. Relying on these commutation relationships, we characterize, for each semigroup in K, its spectrum which can be either point, continuous or residual. We end the talk by describing the associated (weak) eigenfunctions as (weak) Fourier kernels, the spectral reduction operator and its domain. (Joint work with M. Savov)