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...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | eBook Book |
| Language: | English |
| Published: |
Providence, Rhode Island
American Mathematical Society
2021
|
| Edition: | 1 |
| Series: | Memoirs of the American Mathematical Society |
| Subjects: | |
| ISBN: | 9781470449360, 1470449366 |
| ISSN: | 0065-9266, 1947-6221 |
| Online Access: | Get full text |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| 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. |
|---|---|
| AbstractList | 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. |
| Author | Savov, Mladen Patie, Pierre |
| Author_xml | – sequence: 1 fullname: Patie, Pierre – sequence: 2 fullname: Savov, Mladen |
| BackLink | https://cir.nii.ac.jp/crid/1130008896098930960$$DView record in CiNii |
| BookMark | eNpFkUtLw0AUhUdtxbZ24T8IKLiKvfOeWWqpVSi4UNyGSXKnRJtMzbQg_noTIri5Z3E_zn2cKRk1oUFCrijcUbCwqLEOC8q5OiFzqw0VGoTSkolTMqFW6FQxRs_-e8JyBSMyAVAytUypMZkyYAzAAJXnZEqZltwyo_QFmcdY5SC0NlIoOyHr1z0Wh9btEvzeuyZWoYlJ8Em3Uxpx51NXfoSqOSRbbLDDqh8sk43bHrFtMYlYV9s2HPfxkoy920Wc_-mMvD-u3pZP6eZl_by836SOdVN5yq2VpSidVt1JkDPgXnlqkFJTAjdFYdBLLKXhUpTGF4o5rpFyr4UUeWH5jNwOxvs2fB0xHjLMQ_gssOmvyFYPS6WtAd2TNwPZVFVWVH2llEP3FmMVWGM5dNph1wPm6jh4xYxC1keR9VFkfRT8Fx-AcKg |
| CitedBy_id | crossref_primary_10_1214_22_AOP1577 crossref_primary_10_1090_tran_9519 crossref_primary_10_1007_s10959_022_01176_y crossref_primary_10_1090_proc_16644 crossref_primary_10_5802_aif_3600 |
| ContentType | eBook Book |
| Copyright | Copyright 2021 American Mathematical Society |
| Copyright_xml | – notice: Copyright 2021 American Mathematical Society |
| DBID | RYH |
| DEWEY | 512.27 |
| DOI | 10.1090/memo/1336 |
| DatabaseName | CiNii Complete |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISBN | 9781470467524 1470467526 |
| EISSN | 1947-6221 |
| Edition | 1 |
| ExternalDocumentID | EBC6798079 BC11379616 10_1090_memo_1336 |
| GroupedDBID | --Z -~X 123 4.4 85S ABPPZ ACNCT ACNUO AEGFZ AENEX ALMA_UNASSIGNED_HOLDINGS DU5 P2P RMA WH7 YNT YQT AABBV ABARN ABQKM ABQPQ ADVEM AERYV AFOJC AHWGJ AJFER BBABE CZZ GEOUK RYH |
| ID | FETCH-LOGICAL-a20473-3995d4da766750b203f6f18e118d038cc8ef5ed58354d8fc62a37e13f7454bc93 |
| ISBN | 9781470449360 1470449366 |
| ISICitedReferencesCount | 13 |
| ISICitedReferencesURI | http://www.webofscience.com/api/gateway?GWVersion=2&SrcApp=Summon&SrcAuth=ProQuest&DestLinkType=CitingArticles&DestApp=WOS_CPL&KeyUT=0000059202&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| ISSN | 0065-9266 |
| IngestDate | Wed Dec 10 12:30:49 EST 2025 Fri Jun 27 01:01:06 EDT 2025 Thu Aug 14 15:25:31 EDT 2025 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | intertwining functional equations Spectral theory Laguerre polynomials asymptotic analysis infinitely divisible distribution Markov semigroups convergence to equilibrium non-self-adjoint integro-differential operators Bernstein functions Hilbert sequences special functions |
| LCCN | 2022008015 |
| LCCallNum_Ident | QA320 .P38 2021 |
| Language | English |
| LinkModel | OpenURL |
| MergedId | FETCHMERGED-LOGICAL-a20473-3995d4da766750b203f6f18e118d038cc8ef5ed58354d8fc62a37e13f7454bc93 |
| Notes | Includes bibliographical references (p. 177-182) July 2021, volume 272, number 1336 (sixth of 7 numbers) |
| OCLC | 1275392867 |
| PQID | EBC6798079 |
| PageCount | 196 |
| ParticipantIDs | proquest_ebookcentral_EBC6798079 nii_cinii_1130008896098930960 ams_ebooks_10_1090_memo_1336 |
| PublicationCentury | 2000 |
| PublicationDate | [2021] |
| PublicationDateYYYYMMDD | 2021-01-01 |
| PublicationDate_xml | – year: 2021 text: [2021] |
| PublicationDecade | 2020 |
| PublicationPlace | Providence, Rhode Island |
| PublicationPlace_xml | – name: Providence, Rhode Island – name: Providence, R.I – name: Providence |
| PublicationSeriesTitle | Memoirs of the American Mathematical Society |
| PublicationYear | 2021 |
| Publisher | American Mathematical Society |
| Publisher_xml | – name: American Mathematical Society |
| SSID | ssib047785469 ssj0002594674 ssj0008047 |
| Score | 2.725023 |
| Snippet | 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... |
| SourceID | proquest nii ams |
| SourceType | Publisher |
| SubjectTerms | Laguerre polynomials Nonselfadjoint operators Spectral theory (Mathematics) |
| TableOfContents | 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) |
| Title | Spectral expansions of non-self-adjoint generalized Laguerre semigroups |
| URI | https://www.ams.org/memo/1336/ https://cir.nii.ac.jp/crid/1130008896098930960 https://ebookcentral.proquest.com/lib/[SITE_ID]/detail.action?docID=6798079 |
| Volume | 272 |
| WOSCitedRecordID | wos0000059202&url=https%3A%2F%2Fcvtisr.summon.serialssolutions.com%2F%23%21%2Fsearch%3Fho%3Df%26include.ft.matches%3Dt%26l%3Dnull%26q%3D |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://cvtisr.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwtV1Nb9MwGLZY4QAnPkVhQzlwQ9Ec2_XHdVUZEtPoYUy7RY7toKA1mZquqvj1vG_qJKg7IA5crDZK7eR92ub9eh4T8lEXQRclxbZyY1OhZlmqXelSpkwhmWPWd9yq6wt1ealvbswyCtm03XYCqq71bmfu_ivUcAzARursP8A9TAoH4DWADiPADuOBRzy8jaQOJE0i3z7s4Cfe9h1uEN-nbbgtU-t_NlW9wU2TMRFV_QJf88L-uA_rdfjUhlXVMTwGH3tpY-1iWeEZQyrGbpttl0a9tT7yyGLWgGUHWYOxHDSIw6L4SDOojwwhZiYUFcLwver_gz9carBDcRVWDaYAOD-Qte4elGfzLOPKyEwekSMlIUR-fL749v3rkAuDEAw3Pel4d3E12ctx9av3qlCGnuJqp7hWp47bwlhX1YNnaecgXD0nEySNvCCPQv2SPBtvt31FzntckhGXpCmTQ1ySP3BJelySEZfX5Prz4mr-JY3bV6SWUaF4iqxhL7xVEqIyWjDKS1lmOkBM5ynXzulQzoKfYe7N69JJZrkKGS-VmInCGf6GTOBawluSKKuNl1ZIUcCMjGkn4XNK-MAdk0pOyTFYIu8K7G2-byygORoqR0NNyQmYKHcVjhlWMLG7TVID3irGsFOS9MbbzxE7g_PF2RzLdFSZd3-Z4j15On7Pjslks74PJ-SJ226qdv0hIv4bEmw9UA |
| linkProvider | ProQuest Ebooks |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.title=Spectral+expansions+of+non-self-adjoint+generalized+Laguerre+semigroups&rft.au=Patie%2C+Pierre&rft.au=Savov%2C+Mladen&rft.date=2021-01-01&rft.pub=American+Mathematical+Society&rft.isbn=9781470449360&rft_id=info:doi/10.1090%2Fmemo%2F1336&rft.externalDocID=BC11379616 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=0065-9266&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=0065-9266&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=0065-9266&client=summon |