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Mat. Zametki, 1998, Volume 63, Issue 5, Pages 709–716 (Mi mz1337)  

This article is cited in 22 scientific papers (total in 23 papers)

Three-term recurrence relations with matrix coefficients. The completely indefinite case

A. G. Kostyuchenkoa, K. A. Mirzoevb

a M. V. Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Moscow State Aviation Technological University

Abstract: In the space $\ell_p^2$ of vector sequences, we consider the symmetric operator $L$ generated by the expression $(lu)_j:=B_ju_{j+1}+A_ju_j+B_{j-1}^*u_{j-1}$, where $u_{-1}=0$, $u_0,u_1,\ldots\in\mathbb C^p$, $A_j$ and $B_j$ are $p\times p$ matrices with entries from $\mathbb C$, $A_j^*=A_j$, and the inverses $B_j^{-1}$ ($j=0,1,…$) exist. We state a necessary and sufficient condition for the deficiency numbers of the operator $L$ to be maximal; this corresponds to the completely indefinite case for the expression $l$. Tests for incomplete indefiniteness and complete indefiniteness for $l$ in terms of the coefficients $A_j$ and $B_j$ are derived.

DOI: https://doi.org/10.4213/mzm1337

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English version:
Mathematical Notes, 1998, 63:5, 624–630

Bibliographic databases:

UDC: 517.984+517.929
Received: 15.11.1996
Revised: 29.12.1997

Citation: A. G. Kostyuchenko, K. A. Mirzoev, “Three-term recurrence relations with matrix coefficients. The completely indefinite case”, Mat. Zametki, 63:5 (1998), 709–716; Math. Notes, 63:5 (1998), 624–630

Citation in format AMSBIB
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\paper Three-term recurrence relations with matrix coefficients. The completely indefinite case
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\yr 1998
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\pages 709--716
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\transl
\jour Math. Notes
\yr 1998
\vol 63
\issue 5
\pages 624--630
\crossref{https://doi.org/10.1007/BF02312843}
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    This publication is cited in the following articles:
    1. A. G. Kostyuchenko, K. A. Mirzoev, “Generalized Jacobi Matrices and Deficiency Numbers of Ordinary Differential Operators with Polynomial Coefficients”, Funct. Anal. Appl., 33:1 (1999), 25–37  mathnet  crossref  crossref  mathscinet  zmath  isi
    2. A. G. Kostyuchenko, K. A. Mirzoev, “Complete Indefiniteness Tests for Jacobi Matrices with Matrix Entries”, Funct. Anal. Appl., 35:4 (2001), 265–269  mathnet  crossref  crossref  mathscinet  zmath  isi
    3. Allahverdiev, BP, “Dilation and functional model of dissipative operator generated by an infinite Jacobi matrix”, Mathematical and Computer Modelling, 38:10 (2003), 989  crossref  mathscinet  zmath  isi  scopus  scopus
    4. Osipov, AS, “On the Hellinger theorem and l(p)-properties of solutions of difference equations”, Journal of Difference Equations and Applications, 9:9 (2003), 841  crossref  mathscinet  zmath  isi  scopus  scopus
    5. Allahverdiev, BP, “Extensions of the symmetric operator generated by an infinite Jacobi matrix”, Mathematical and Computer Modelling, 37:9–10 (2003), 1093  crossref  mathscinet  zmath  isi  scopus  scopus
    6. Clark, S, “On Weyl-Titchmarsh theory for singular finite difference Hamiltonian systems”, Journal of Computational and Applied Mathematics, 171:1–2 (2004), 151  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    7. Clark, S, “Trace formulas and Borg-type theorems for matrix-valued Jacobi and Dirac finite difference operators”, Journal of Differential Equations, 219:1 (2005), 144  crossref  mathscinet  zmath  adsnasa  isi  scopus
    8. M. S. Agranovich, V. M. Buchstaber, R. S. Ismagilov, B. S. Kashin, B. S. Mityagin, S. P. Novikov, V. A. Sadovnichii, A. G. Sergeev, Ya. G. Sinai, A. A. Shkalikov, “Anatolii Gordeevich Kostyuchenko (obituary)”, Russian Math. Surveys, 65:4 (2010), 767–780  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    9. Clark S., Zemanek P., “On a Weyl-Titchmarsh Theory for Discrete Symplectic Systems on a Half Line”, Appl. Math. Comput., 217:7 (2010), 2952–2976  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. K. A. Mirzoev, T. A. Safonova, “Singulyarnye operatory Shturma–Liuvillya s negladkimi potentsialami v prostranstve vektor-funktsii”, Ufimsk. matem. zhurn., 3:3 (2011), 105–119  mathnet  zmath
    11. Nagy B., “Multiplicities, Generalized Jacobi Matrices, and Symmetric Operators”, J. Operat. Theor., 65:1 (2011), 211–232  mathscinet  zmath  isi  elib
    12. Mirzoev K.A., Safonova T.A., “Singulyarnye operatory shturma–liuvillya s potentsialom-raspredeleniem v prostranstve vektor-funktsii”, Doklady Akademii nauk, 441:2 (2011), 165–168  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    13. Allahverdiev B.P., “Extensions of Symmetric Second-Order Difference Operators with Matrix Coefficients”, J. Differ. Equ. Appl., 19:5 (2013), 839–849  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    14. K. A. Mirzoev, T. A. Safonova, “On the Deficiency Index of the Vector-Valued Sturm–Liouville Operator”, Math. Notes, 99:2 (2016), 290–303  mathnet  crossref  crossref  mathscinet  isi  elib
    15. A. S. Kostenko, M. M. Malamud, D. D. Natyagajlo, “Matrix Schrödinger Operator with $\delta$-Interactions”, Math. Notes, 100:1 (2016), 49–65  mathnet  crossref  crossref  mathscinet  isi  elib
    16. Braeutigam I.N., Mirzoev K.A., “Deficiency numbers of operators generated by infinite Jacobi matrices”, Dokl. Math., 93:2 (2016), 170–174  crossref  mathscinet  zmath  isi  elib  scopus
    17. Osipov A., “On the Completely Indeterminate Case For Block Jacobi Matrices”, Concr. Operators, 4:1 (2017), 48–57  crossref  mathscinet  zmath  isi
    18. Swiderski G., “Spectral Properties of Block Jacobi Matrices”, Constr. Approx., 48:2 (2018), 301–335  crossref  mathscinet  isi  scopus
    19. I. N. Braeutigam, K. A. Mirzoev, “On the defect numbers of operators generated by Jacobian matrices with operator entries”, St. Petersburg Math. J., 30:4 (2019), 621–638  mathnet  crossref  isi  elib
    20. Intissar A., Intissar J.-K., “A Complete Spectral Analysis of Generalized Gribov-Intissar'S Operator in Bargmann Space”, Complex Anal. Oper. Theory, 13:3 (2019), 1481–1510  crossref  isi
    21. V. S. Budyka, M. M. Malamud, “On the Deficiency Indices of Block Jacobi Matrices Related to Dirac Operators with Point Interactions”, Math. Notes, 106:6 (2019), 1009–1014  mathnet  crossref  crossref  isi  elib
    22. I. N. Braeutigam, D. M. Polyakov, “Asymptotics of eigenvalues of infinite block matrices”, Ufa Math. J., 11:3 (2019), 11–28  mathnet  crossref  isi
    23. V. S. Budyka, M. M. Malamud, “Self-Adjointness and Discreteness of the Spectrum of Block Jacobi Matrices”, Math. Notes, 108:3 (2020), 445–450  mathnet  crossref  crossref
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