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Spectral expansions of non-self-adjoint generalized Laguerre semigroups
About this Title
Pierre Patie and Mladen Savov
Publication: Memoirs of the American Mathematical Society
Publication Year:
2021; Volume 272, Number 1336
ISBNs: 978-1-4704-4936-0 (print); 978-1-4704-6752-4 (online)
DOI: https://doi.org/10.1090/memo/1336
Published electronically: September 24, 2021
Keywords: Spectral theory,
non-self-adjoint integro-differential operators,
Markov semigroups,
intertwining,
convergence to equilibrium,
asymptotic analysis,
infinitely divisible distribution,
Hilbert sequences,
Laguerre polynomials,
Bernstein functions,
functional equations,
special functions
Table of Contents
Chapters
- Acknowledgments
- 1. Introduction and main results
- 2. Strategy of proofs and auxiliary results
- 3. Examples
- 4. New developments in the theory of Bernstein functions
- 5. Fine properties of the density of the invariant measure
- 6. Bernstein-Weierstrass products and Mellin transforms
- 7. Intertwining relations and a set of eigenfunctions
- 8. Co-eigenfunctions: existence and characterization
- 9. Uniform and norms estimates of the co-eigenfunctions
- 10. The concept of reference semigroups: ${{\mathrm {L}}^{2}(\nu )}$-norm estimates and completeness of the set of co-eigenfunctions
- 11. Hilbert sequences, intertwining and spectrum
- 12. Proof of Theorems , and
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.- N.I. Akhiezer, The classical moment problem, Oliver and Boyd, 1965.
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