Spectral expansions of non-self-adjoint generalized Laguerre semigroups

TL;DR

This paper develops a spectral expansion for a class of non-self-adjoint Markov operators linked to generalized Laguerre semigroups, revealing decay behaviors and convergence rates in weighted Hilbert spaces.

Contribution

It introduces a novel spectral analysis approach for non-self-adjoint operators via intertwining with self-adjoint semigroups, extending beyond perturbation theory.

Findings

Spectral expansion in weighted Hilbert space for generalized Laguerre semigroups.

Identification of decay rates and hypocoercivity phenomena.

Smoothness properties of solutions and heat kernel derived.

Abstract

We provide the spectral expansion in a weighted Hilbert space of a substantial class of invariant non-self-adjoint and non-local Markov operators which appear in limit theorems for positive-valued Markov processes. We show that this class is in bijection with a subset of negative definite functions and we name it the class of generalized Laguerre semigroups. Our approach, which goes beyond the framework of perturbation theory, is based on an in-depth and original analysis of an intertwining relation that we establish between this class and a self-adjoint Markov semigroup, whose spectral expansion is expressed in terms of the classical Laguerre polynomials. As a by-product, we derive smoothness properties for the solution to the associated Cauchy problem as well as for the heat kernel. Our methodology also reveals a variety of possible decays, including the hypocoercivity type phenomena, for the speed of convergence to equilibrium for this class and enables us to provide an interpretation of these in terms of the rate of growth of the weighted Hilbert space norms of the spectral projections. Depending on the analytic properties of the aforementioned negative definite functions, we are led to implement several strategies, which require new developments in a variety of contexts, to derive precise upper bounds for these norms.