Open Access

Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation


Let $H$ be a finite dimensional Hilbert space and let $[H]$ be the set of all li ear operators in $H$. We consider first-order symmetric system $J y'-B(t)y=\Lambda(t) f(t)$ with $[H]$-valued coefficients defined on an interval $[a,b) $ with the regular endpoint $a$. It is assumed that the corresponding minimal relation $T_{\rm min}$ has maximally possible deficiency index $n_+(T_{\rm min})=\dim H$. The main result is a parametrization of all characteristic matrices and pseudospectral (spectral) functions of a given system by means of a Nevanlinna type boundary parameter $\tau$. Similar parametrization for regular systems has earlier been obtained by Langer and Textorius. We also show that the coefficients of the parametrization form the matrix $W(\lambda)$ with the properties similar to those of the resolvent matrix in the extension theory of symmetric operators.

Key words: First-order symmetric system, characteristic matrix, spectral function, pseudospectral function, Fourier transform.

Full Text

Article Information

TitleCharacteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation
SourceMethods Funct. Anal. Topology, Vol. 21 (2015), no. 1, 76-98
MathSciNet MR3407922
zbMATH 06533469
MilestonesReceived 08/08/2014; Revised 26/11/2014
CopyrightThe Author(s) 2015 (CC BY-SA)

Google Scholar Metrics

Citing articles in Google Scholar
Similar articles in Google Scholar

Export article

Save to Mendeley

Citation Example

Vadim Mogilevskii, Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation, Methods Funct. Anal. Topology 21 (2015), no. 1, 76-98.


@article {MFAT749,
    AUTHOR = {Mogilevskii, Vadim},
     TITLE = {Characteristic matrices and spectral functions of first order symmetric systems with maximal deficiency index of the minimal relation},
   JOURNAL = {Methods Funct. Anal. Topology},
  FJOURNAL = {Methods of Functional Analysis and Topology},
    VOLUME = {21},
      YEAR = {2015},
    NUMBER = {1},
     PAGES = {76-98},
      ISSN = {1029-3531},
  MRNUMBER = {MR3407922},
 ZBLNUMBER = {06533469},
       URL = {},


  1. N. I. Akhiezer, The Classical Moment Problem, Oliver and Boyd, Edinburgh-London, 1965.
  2. Sergio Albeverio, Mark Malamud, Vadim Mogilevskii, On Titchmarsh-Weyl functions and eigenfunction expansions of first-order symmetric systems, Integral Equations Operator Theory 77 (2013), no. 3, 303-354.  MathSciNet CrossRef
  3. Damir Z. Arov, Harry Dym, Bitangential direct and inverse problems for systems of integral and differential equations, Cambridge University Press, Cambridge, 2012.  MathSciNet CrossRef
  4. F. V. Atkinson, Discrete and continuous boundary problems, Academic Press, New York-London, 1964.  MathSciNet
  5. V. M. Bruk, Linear relations in a space of vector-valued functions, Mat. Zametki 24 (1978), no. 4, 499-511, 590.  MathSciNet
  6. V. A. Derkach, S. Hassi, M. M. Malamud, H. S. V. Snoo de, Generalized resolvents of symmetric operators and admissibility, Methods Funct. Anal. Topology 6 (2000), no. 3, 24-55.  MathSciNet
  7. V. A. Derkach, M. M. Malamud, Generalized resolvents and the boundary value problems for Hermitian operators with gaps, J. Funct. Anal. 95 (1991), no. 1, 1-95.  MathSciNet CrossRef
  8. Aad Dijksma, Heinz Langer, Henk Snoo de, Eigenvalues and pole functions of Hamiltonian systems with eigenvalue depending boundary conditions, Math. Nachr. 161 (1993), 107-154.  MathSciNet CrossRef
  9. Nelson Dunford, Jacob T. Schwartz, Linear operators. Part II, John Wiley & Sons, Inc., New York, 1988.  MathSciNet
  10. Charles T. Fulton, Parametrizations of Titchmarshs $m(\lambda )$-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51-63.  MathSciNet
  11. I. C. Gohberg, M. G. Krein, Theory and applications of Volterra operators in Hilbert space, American Mathematical Society, Providence, R.I., 1970.  MathSciNet
  12. M. L. Gorbacuk, On spectral functions of a second order differential equation with operator coefficients, Ukrain. Mat. \v Z. 18 (1966), no. 2, 3-21.  MathSciNet
  13. Granichnye zadachi dlya differentsialno-operatornykh uravnenii, Akad. Nauk Ukrainy, Inst. Mat., Kiev, 1991.  MathSciNet
  14. D. B. Hinton, J. K. Shaw, Parameterization of the $M(\lambda )$ function for a Hamiltonian system of limit circle type, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 3-4, 349-360.  MathSciNet CrossRef
  15. I. S. Kats, Linear relations generated by a canonical differential equation of dimension 2, and eigenfunction expansions, Algebra i Analiz 14 (2002), no. 3, 86-120.  MathSciNet
  16. F. Atkinson, Diskretnye i nepreryvnye granichnye zadachi, Izdat. ``Mir'', Moscow, 1968.  MathSciNet
  17. A. M. Khol′kin, Description of selfadjoint extensions of differential operators of arbitrary order on an infinite interval in the absolutely indeterminate case, Teor. Funktsi\u\i\ Funktsional. Anal. i Prilozhen. (1985), no. 44, 112-122.  MathSciNet CrossRef
  18. V. I. Kogan, F. S. Rofe-Beketov, On square-integrable solutions of symmetric systems of differential equations of arbitrary order, Proc. Roy. Soc. Edinburgh Sect. A 74 (1974/75), 5-40 (1976).  MathSciNet
  19. M. G. Krein, Sh. N. Saakyan, Some new results in the theory of resolvents of Hermitian operators, Soviet Math. Dokl. 7 (1966), 1086-1089.
  20. H. Langer, B. Textorius, $L$-resolvent matrices of symmetric linear relations with equal defect numbers;\ applications to canonical differential relations, Integral Equations Operator Theory 5 (1982), no. 2, 208-243.  MathSciNet CrossRef
  21. H. Langer, B. Textorius, Spectral functions of a symmetric linear relation with a directing mapping. I, Proc. Roy. Soc. Edinburgh Sect. A 97 (1984), 165-176.  MathSciNet CrossRef
  22. H. Langer, B. Textorius, Spectral functions of a symmetric linear relation with a directing mapping. II, Proc. Roy. Soc. Edinburgh Sect. A 101 (1985), no. 1-2, 111-124.  MathSciNet CrossRef
  23. Matthias Lesch, Mark Malamud, On the deficiency indices and self-adjointness of symmetric Hamiltonian systems, J. Differential Equations 189 (2003), no. 2, 556-615.  MathSciNet CrossRef
  24. M. M. Malamud, On a formula for the generalized resolvents of a non-densely defined Hermitian operator, Ukrain. Mat. Zh. 44 (1992), no. 12, 1658-1688.  MathSciNet CrossRef
  25. Vadim Mogilevskii, Boundary triplets and Krein type resolvent formula for symmetric operators with unequal defect numbers, Methods Funct. Anal. Topology 12 (2006), no. 3, 258-280.  MathSciNet
  26. V. I. Mogilevskii, On exit space extensions of symmetric operators with applications to first order symmetric systems, Methods Funct. Anal. Topology 19 (2013), no. 3, 268-292.  MathSciNet
  27. Tim Mogilevskii, On generalized resolvents and characteristic matrices of first-order symmetric systems, Methods Funct. Anal. Topology 20 (2014), no. 4, 328-348.  MathSciNet
  28. V. I. Mogilevskii, On spectral and pseudospectral functions of first-order symmetric systems; arXiv:1407.5398v1 [math.FA] 21 Jul 2014.
  29. M. A. Naimark, Lineinye differentsialnye operatory, Izdat. ``Nauka'', Moscow, 1969.  MathSciNet
  30. Bruce Call Orcutt, CANONICAL DIFFERENTIAL EQUATIONS, ProQuest LLC, Ann Arbor, MI, 1969.  MathSciNet
  31. A. L. Sakhnovich, Spectral functions of a second-order canonical system, Mat. Sb. 181 (1990), no. 11, 1510-1524.  MathSciNet
  32. A. V. Straus, On generalized resolvents and spectral functions of differential operators of even order, Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957), 785-808.  MathSciNet

All Issues