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Izv. RAN. Ser. Mat., 2009, Volume 73, Issue 2, Pages 3–68 (Mi izv2643)  

This article is cited in 48 scientific papers (total in 48 papers)

Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations

A. G. Baskakov

Voronezh State University

Abstract: We study linear differential operators with unbounded operator-valued coefficients acting on homogeneous spaces of functions on the semi-axis. We obtain necessary and sufficient conditions for such operators to be invertible or Fredholm. Essential use is made of the spectral theory of difference relations and semigroups of linear relations (multi-valued linear operators).

Keywords: linear differential operators, spectrum of an operator, homogeneous spaces of functions, Green function, Fredholm operator.

DOI: https://doi.org/10.4213/im2643

Full text: PDF file (1029 kB)
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English version:
Izvestiya: Mathematics, 2009, 73:2, 215–278

Bibliographic databases:

UDC: 517.937+517.983
MSC: 47A06, 47A53, 47D06, 34D09, 34G10, 35P05
Received: 28.03.2007
Revised: 21.11.2007

Citation: A. G. Baskakov, “Spectral analysis of differential operators with unbounded operator-valued coefficients, difference relations and semigroups of difference relations”, Izv. RAN. Ser. Mat., 73:2 (2009), 3–68; Izv. Math., 73:2 (2009), 215–278

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. S. Bichegkuev, “O spektre raznostnykh i differentsialnykh operatorov v vesovykh prostranstvakh”, Funkts. analiz i ego pril., 44:1 (2010), 80–83  mathnet  crossref  mathscinet  zmath
    2. Baskakov A.G., Sintyaev Yu.N., “Finite-difference operators in the study of differential operators: solution estimates”, Differ. Equ., 46:2 (2010), 214–223  crossref  mathscinet  zmath  isi  elib  elib
    3. V. B. Didenko, “On the Spectral Properties of Differential Operators with Unbounded Operator Coefficients Determined by a Linear Relation”, Math. Notes, 89:2 (2011), 224–237  mathnet  crossref  crossref  mathscinet  isi
    4. M. S. Bichegkuev, S. V. Besaeva, “Spectral properties of difference and differential operators in weighted spaces”, Russian Math. (Iz. VUZ), 55:2 (2011), 13–17  mathnet  crossref  mathscinet  elib
    5. M. S. Bichegkuev, “On conditions for invertibility of difference and differential operators in weight spaces”, Izv. Math., 75:4 (2011), 665–680  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. V. M. Bruk, “On linear relations generated by a differential expression and by a Nevanlinna operator function”, Zhurn. matem. fiz., anal., geom., 7:2 (2011), 115–140  mathnet  mathscinet  zmath  elib
    7. Baskakov A.G., Kobychev K.S., “Estimates for the embedding operator of a sobolev space of periodic functions and for the solutions of differential equations with periodic coefficients”, Differ. Equ., 47:5 (2011), 609–619  crossref  mathscinet  zmath  isi  elib  elib
    8. Nazaruk E.M., Romanovskaya A.M., “Ob eksponentsialnoi dikhotomii reshenii zadachi koshi dlya sistemy differentsialno-raznostnykh uravnenii s postoyannymi koeffitsientami”, Vestnik omskogo universiteta, 2011, no. 4, 45–49  elib
    9. S. V. Maryushenkov, “Estimation of the norm of an operator inverse to a differential one”, Russian Math. (Iz. VUZ), 56:1 (2012), 43–47  mathnet  crossref  mathscinet
    10. A. V. Pechkurov, “Bisectorial operator pencils and the problem of bounded solutions”, Russian Math. (Iz. VUZ), 56:3 (2012), 26–35  mathnet  crossref  mathscinet
    11. V. M. Bruk, “Linear relations generated by an integral equation with Nevanlinna operator measure”, Russian Math. (Iz. VUZ), 56:10 (2012), 1–14  mathnet  crossref  mathscinet
    12. A. G. Baskakov, N. S. Kaluzhina, “Beurlings theorem for functions with essential spectrum from homogeneous spaces and stabilization of solutions of parabolic equations”, Math. Notes, 92:5 (2012), 587–605  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    13. Bichegkuev M.S., “On the exponential dichotomy and spectral properties of difference operators related to the howland semigroup”, Differ. Equ., 48:6 (2012), 769–778  crossref  mathscinet  zmath  zmath  isi  elib  elib
    14. Pokutnyi A.A., “Bounded solutions of linear and weakly nonlinear differential equations in a Banach space with an unbounded operator in the linear part”, Differ. Equ., 48:6 (2012), 809–819  crossref  mathscinet  zmath  zmath  isi  elib
    15. V. B. Didenko, “On the continuous invertibility and the Fredholm property of differential operators with multi-valued impulse effects”, Izv. Math., 77:1 (2013), 3–19  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    16. V. M. Bruk, “Invertible linear relations generated by an integral equation with a Nevanlinna measure”, Russian Math. (Iz. VUZ), 57:2 (2013), 13–24  mathnet  crossref
    17. A. G. Baskakov, “Analysis of linear differential equations by methods of the spectral theory of difference operators and linear relations”, Russian Math. Surveys, 68:1 (2013), 69–116  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    18. Dmitry Todorov, “Generalizations of analogs of theorems of Maizel and Pliss and their application in shadowing theory”, DCDS-A, 33:9 (2013), 4187  crossref  mathscinet  zmath  isi
    19. S. V. Maryushenkov, “Estimate for the norm of the inverse of a differential operator”, Diff. Eq., 49:3 (2013), 392  crossref  mathscinet  zmath  isi  elib  elib
    20. M. S. Bichegkuev, “Spectral analysis of difference and differential operators in weighted spaces”, Sb. Math., 204:11 (2013), 1549–1564  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    21. A. G. Chshiev, “The Gearhart–Prüss Theorem for a Class of Degenerate Semigroups of Operators”, Math. Notes, 94:3 (2013), 400–413  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    22. Baskakov A.G. Krishtal I.A., “On Completeness of Spectral Subspaces of Linear Relations and Ordered Pairs of Linear Operators”, J. Math. Anal. Appl., 407:1 (2013), 157–178  crossref  mathscinet  zmath  isi  elib
    23. Teresa Álvarez, Aymen Ammar, Aref Jeribi, “On the essential spectra of some matrix of linear relations”, Math. Meth. Appl. Sci., 37:5 (2014), 620–644  crossref  mathscinet  zmath
    24. R. K. Romanovskii, E. M. Nazaruk, “On the Dichotomy of Linear Autonomous Systems of Functional-Differential Equations”, Math. Notes, 95:1 (2014), 116–121  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    25. M. S. Bichegkuev, “Spectral Analysis of Differential Operators with Unbounded Operator Coefficients in Weighted Spaces of Functions”, Math. Notes, 95:1 (2014), 15–21  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    26. V. M. Bruk, “On the Characteristic Operator of an Integral Equation with a Nevanlinna Measure in the Infinite-Dimensional Case”, Zhurn. matem. fiz., anal., geom., 10:2 (2014), 163–188  mathnet  crossref  mathscinet
    27. A. G. Baskakov, N. S. Kaluzhina, D. M. Polyakov, “Slowly varying on infinity semigroups of operators”, Russian Math. (Iz. VUZ), 58:7 (2014), 1–10  mathnet  crossref
    28. V. M. Bruk, “On Linear Relations Generated by an Integro-Differential Equation with Nevanlinna Measure in the Infinite-Dimensional Case”, Math. Notes, 96:1 (2014), 10–25  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    29. S. V. Besaeva, “On the spectra of difference relations and differential operators on weighted spaces of sequences and functions”, St. Petersburg Math. J., 26:4 (2015), 499–513  mathnet  crossref  mathscinet  isi  elib  elib
    30. V. M. Bruk, “Invertibility of linear relations generated by integral equation with operator measures”, Ufa Math. J., 6:4 (2014), 48–59  mathnet  crossref
    31. M. S. Bichegkuev, “Ob obratimosti raznostnykh otnoshenii i differentsialnykh operatorov”, Vestnik Voronezhskogo gosudarstvennogo universiteta. Ser. Fizika. Matematika, 2014, no. 4, 109–115  zmath  elib
    32. A. Yu. Duplishcheva, “About Conditions of Invertibility of Difference Operators of the Second Order”, J. Math. Sci., 213:6 (2016), 832–837  mathnet  crossref
    33. A. G. Chshiev, “Ob obraschenii polugruppy operatorov”, Vladikavk. matem. zhurn., 16:2 (2014), 79–89  mathnet
    34. A. V. Pechkurov, “An Example in the Theory of Bisectorial Operators”, Math. Notes, 97:2 (2015), 243–248  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    35. A. Yu. Duplischeva, “O differentsialnykh operatorakh i matritsakh vtorogo poryadka”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 15:1 (2015), 31–37  mathnet  crossref  elib
    36. A. G. Baskakov, V. B. Didenko, “Spectral analysis of differential operators with unbounded periodic coefficients”, Diff Equat, 51:3 (2015), 325  crossref  mathscinet  zmath
    37. A. G. Baskakov, A. Yu. Duplishcheva, “Difference operators and operator-valued matrices of the second order”, Izv. Math., 79:2 (2015), 217–232  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    38. Aymen Ammar, Toka Diagana, Aref Jeribi, “Perturbations of Fredholm linear relations in Banach spaces with application to <mml:math altimg="si1.gif" display="inline" overflow="scroll" xmlns:xocs="http://www.elsevier.com/xml/xocs/dtd" xmlns:xs="http://www.w3.org/2001/XMLSchema" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns="http://www.elsevier.com/xml/ja/dtd" xmlns:ja="http://www.elsevier.com/xml/ja/dtd" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:tb="http://www.elsevier.com/xml/common/table/dtd" xmlns:sb="http://www.elsevier.com/xml/common/struct-bib/dtd" xmlns:ce="http://www.elsevier.com/xml/common/dtd" xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:cals="http://www.elsevier.com/xml/common/cals/dtd" xmlns:sa="http://www.elsevier.com/xml/common/struct-aff/dtd"><mml:mn>3</mml:mn><mml:mo>×</mml:mo><mml:mn>3</mml:mn></mml:math>-block matrices of linear relations”, Arab Journal of Mathematical Sciences, 2015  crossref
    39. A. G. Baskakov, “Estimates for the Green's function and parameters of exponential dichotomy of a hyperbolic operator semigroup and linear relations”, Sb. Math., 206:8 (2015), 1049–1086  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    40. M. S. Bichegkuev, “Lyapunov Transformation of Differential Operators with Unbounded Operator Coefficients”, Math. Notes, 99:1 (2016), 24–36  mathnet  crossref  crossref  mathscinet  isi  elib
    41. Baskakov A.G. Kabantsova L.Yu. Kostrub I.D. Smagina T.I., “Linear differential operators and operator matrices of the second order”, Differ. Equ., 53:1 (2017), 8–17  crossref  mathscinet  zmath  isi  scopus
    42. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v prostranstvakh Stepanova”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 17:2 (2017), 172–182  mathnet  crossref  elib
    43. V. M. Bruk, “Boundary value problems for integral equations with operator measures”, Probl. anal. Issues Anal., 6(24):1 (2017), 19–40  mathnet  crossref  elib
    44. I. I. Strukova, “Garmonicheskii analiz periodicheskikh na beskonechnosti funktsii v odnorodnykh prostranstvakh”, Vestn. Volgogr. gos. un-ta. Ser. 1, Mat. Fiz., 2017, no. 2(39), 29–38  mathnet  crossref
    45. Baskakov A.G., Krishtal I.A., “Spectral Analysis of Abstract Parabolic Operators in Homogeneous Function Spaces, II”, Mediterr. J. Math., 14:4 (2017), UNSP 181  crossref  mathscinet  isi
    46. A. G. Baskakov, V. B. Didenko, “On invertibility states of differential and difference operators”, Izv. Math., 82:1 (2018), 1–13  mathnet  crossref  crossref  adsnasa  isi  elib
    47. D. B. Didenko, “Spectral Properties of the Operators $AB$ and $BA$”, Math. Notes, 103:2 (2018), 196–208  mathnet  crossref  crossref  isi  elib
    48. Ammar A., “Some Results on Semi-Fredholm Perturbations of Multivalued Linear Operators”, Linear Multilinear Algebra, 66:7 (2018), 1311–1332  crossref  mathscinet  zmath  isi
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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