Spectral expansions of non-self-adjoint generalized Laguerre semigroups
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...
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| Hauptverfasser: | , |
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| Format: | E-Book Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Providence, Rhode Island
American Mathematical Society
2021
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| Ausgabe: | 1 |
| Schriftenreihe: | Memoirs of the American Mathematical Society |
| Schlagworte: | |
| ISBN: | 9781470449360, 1470449366 |
| ISSN: | 0065-9266, 1947-6221 |
| Online-Zugang: | Volltext |
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Inhaltsangabe:
- Acknowledgments -- Introduction and main results -- Strategy of proofs and auxiliary results -- Examples -- New developments in the theory of Bernstein functions -- Fine properties of the density of the invariant measure -- Bernstein-Weierstrass products and Mellin transforms -- Intertwining relations and a set of eigenfunctions -- Co-eigenfunctions: existence and characterization -- Uniform and norms estimates of the co-eigenfunctions -- The concept of reference semigroups: <inline-formula content-type="math/mathml"> L 2 ( ν<!-- ν --> ) {{\mathrm {L}}^{2}(\nu )} </inline-formula>-norm estimates and completeness of the set of co-eigenfunctions -- Hilbert sequences, intertwining and spectrum -- Proof of Theorems , and
- 7.3. Proofs of Theorem ??? (???) and (???) -- 7.4. Proof of the uniqueness of the invariant measure -- 7.5. Proof of Theorem ??? -- Chapter 8. Co-eigenfunctions: existence and characterization -- 8.1. Mellin convolution equations: distributional and classical solutions -- 8.2. Existence of co-eigenfunctions: Proof of Theorem ??? -- 8.3. The case ∈\Ne_{∞,∞}. -- 8.4. The case ∈\Ne_{∞}∖\Nii -- 8.5. The case ∈\Ne^{ }_{∞}. -- Chapter 9. Uniform and norms estimates of the co-eigenfunctions -- 9.1. Proof of Theorem 2.1.5 (1) via a classical saddle point method -- 9.2. Proof of Theorem 2.1.5 (2) via the asymptotic behaviour of zeros of the derivatives of -- 9.3. Proof of Theorem ??? (???) through Phragmén-Lindelöf principle -- Chapter 10. The concept of reference semigroups: \Lnu-norm estimates and completeness of the set of co-eigenfunctions -- 10.1. Estimates for the \lnu norm of \nun -- 10.2. Completeness of (\nun)_{ ≥0} in \lnu -- Chapter 11. Hilbert sequences, intertwining and spectrum -- 11.1. Proof of Theorem ??? -- Chapter 12. Proof of Theorems ???, ??? and ??? -- 12.1. Proof of Theorem 1.3.1 (2) -- 12.2. Proof of Theorem ??? (???) -- 12.3. Heat kernel expansion -- 12.4. Expansion of the adjoint semigroup: Proof of Theorem ??? -- 12.5. Proof of of Theorem ???: Rate of convergence to equilibrium -- Bibliography -- Back Cover
- Cover -- Title page -- Acknowledgments -- Chapter 1. Introduction and main results -- 1.1. Characterization and properties of gL semigroups -- 1.2. Definition and properties of subsets of \Ne -- 1.3. Eigenvalue expansion and regularity of the gL semigroups -- 1.4. Convergence to equilibrium -- 1.5. Hilbert sequences and spectrum -- 1.6. Plan of the paper -- 1.7. Notation, conventions and general facts -- Chapter 2. Strategy of proofs and auxiliary results -- 2.1. Outline of our methodology -- 2.2. Proof of Theorem ??? (???) -- 2.3. Additional basic facts on gL semigroups -- Chapter 3. Examples -- Chapter 4. New developments in the theory of Bernstein functions -- 4.1. Review and basic properties of Bernstein functions -- 4.2. Products of Bernstein functions: new examples -- 4.3. Useful estimates of Bernstein functions on \C₊ -- Chapter 5. Fine properties of the density of the invariant measure -- 5.1. A connection with remarkable self-decomposable variables -- 5.2. Fine distributional properties of _{ } -- 5.3. Proof of Theorem ??? (???) -- 5.4. Small asymptotic behaviour of \nuh and of its successive derivatives -- 5.5. Proof of Theorem ??? -- 5.6. Proof of Theorem ??? -- 5.7. End of proof of Theorem ??? -- Chapter 6. Bernstein-Weierstrass products and Mellin transforms -- 6.1. Exponential functional of subordinators -- 6.2. The functional equations (???) and (???) on \R₊ -- 6.3. Proof of Theorem ??? -- 6.4. Proof of Proposition 6.1.2 -- 6.5. Proof of Theorem ??? (???): Bounds for ᵩ -- 6.6. Large asymptotic behaviours of ᵩ along imaginary lines -- 6.7. Proof of Theorem ??? (???) -- 6.8. Proof of Theorem 6.0.2 (2b): Examples of large asymptotic estimates of | ᵩ| -- Chapter 7. Intertwining relations and a set of eigenfunctions -- 7.1. Proof of Theorem ??? -- 7.2. End of the proof of the intertwining relation (7.3)