Complex Analysis and Operator Theory

, Volume 12, Issue 4, pp 1027–1055 | Cite as

A New Method for Dissipative Dynamic Operator with Transmission Conditions



In this paper, we investigate the spectral properties of a boundary value transmission problem generated by a dynamic equation on the union of two time scales. For such an analysis we assign a suitable dynamic operator which is in limit-circle case at infinity. We also show that this operator is a simple maximal dissipative operator. Constructing the inverse operator we obtain some information about the spectrum of the dissipative operator. Moreover, using the Cayley transform of the dissipative operator we pass to the contractive operator which is of the class \(C_{0}.\) With the aid of the minimal function we obtain more information on the dissipative operator. Finally, we investigate other properties of the contraction such that multiplicity of the contraction, unitary colligation with basic operator and CMV matrix representation associated with the contraction.


Time scale Dissipative operator Cayley transform Completely non-unitary contraction Unitary colligation Characteristic function CMV matrix 

Mathematics Subject Classification

Primary 34B20 Secondary 34N05 47B44 


  1. 1.
    Gohberg, I.C., Krein, M.G.: Introduction to the theory of linear non-self-adjoint operators. In: Translations of Mathematical Monographs, vol. 18. (trans: Feinstein, A.) American Mathematical Society, Providence (1969)Google Scholar
  2. 2.
    Nagy, B., Foiaş, C.: Harmonic Analysis of Operators on Hilbert Space. Academia Kioda, Budapest (1970)MATHGoogle Scholar
  3. 3.
    Sz. Nagy, B., Foiaş, C., Bercovici, H., Kérchy, L.: Harmonic Analysis of Operators on Hilbert Space. Revised and Enlarged Edition. Springer, New York (2010)CrossRefMATHGoogle Scholar
  4. 4.
    Pavlov, B.S.: Dilation theory and spectral analysis of nonselfadjoint differential operators. In: Math. Programming and Related Questions (Proc. Seventh Winter School, Drogobych, 1974): Theory of Operators in Linear Spaces, Tsentral. Ekonom.-Mat. Inst. Akad. Nauk SSSR, Moscow, 1976, pp. 3–69; English transl in Amer. Math. Soc. Transl. (2) 115 (1980)Google Scholar
  5. 5.
    Lax, P.D., Phillips, R.S.: Scattering Theory. Academic press, New York (1967)MATHGoogle Scholar
  6. 6.
    Pavlov, B.S.: Spectral analysis of a dissipative singular Schr ödinger operator in terms of a functional model. Itogi Nauki Tekh. Ser. Sovrem. Probl. Math. Fundam Napravleniya 65 (1991), 95–163, English transl. in partial differential equations, 8 Encyc. Math. Sci. 65, 87–163 (1996)Google Scholar
  7. 7.
    Allahverdiev, B.P.: On dilation theory and spectral analysis of dissipative Schrödinger operators in Weyl’s limit-circle case. Math. USSR Izvestiya 36, 247–262 (1991)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Allahverdiev, B.P., Canoglu, A.: Spectral analysis of dissipative Schrodinger operators. Proc. R. Soc. Edinb. 127A, 1113–1121 (1997)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Allahverdiev, B.P.: A nonselfadjoint singular Sturm-Liouville problem with a spectral parameter in the boundary condition. Math. Nach. 278(7–8), 743–755 (2005)CrossRefMATHGoogle Scholar
  10. 10.
    Allahverdiev, B.P.: A dissipative singular Sturm–Liouville problem with a spectral parameter in the boundary condition. J. Math. Anal. Appl. 316, 510–524 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Allahverdiev, B.P., Bairamov, E., Ugurlu, E.: Eigenparameter dependent Sturm–Liouville problems in boundary conditions with transmission conditions. J. Math. Anal. Appl. 401, 388–396 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Uğurlu, E., Bairamov, E.: Spectral analysis of eigenparameter dependent boundary value transmission problems. J. Math. Anal. Appl. 413, 482–494 (2014)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Brodskiĭ, M.S.: Unitary operator colligations and their characteristic functions. Russ. Math. Surv. 33, 159–191 (1978)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Arlinskiĭ, Y., Golinskiĭ, L., Tsekanovskiĭ, E.: Contractions with rank one defect operators and truncated CMV matrices. J. Funct. Anal. 254, 154–195 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Akhiezer, N.I., Glazman, I.M.: Theory of Linear Operators in Hilbert Spaces, vol. I, II. Pitman, Boston (1981)MATHGoogle Scholar
  16. 16.
    Akhiezer, N.I.: The Classical Moment Problem. Oliver and Boyd, Edinburgh (1965)MATHGoogle Scholar
  17. 17.
    Arlinskiĭ, Y., Tsekanovskiĭ, E.: Non-self-adjoint Jacobi matrices with a rank-one imaginary part. J. Funct. Anal. 241, 383–438 (2006)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Cantero, M.J., Moral, L., Velázquez, L.: Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle. Linear Algebra Appl. 408, 40–65 (2005)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Bohner, M., Peterson, A.C.: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser, Boston (2001)CrossRefMATHGoogle Scholar
  20. 20.
    Bohner, M., Peterson, A.C.: Advances in Dynamic Equations on Time Scales. Birkhäuser, Boston (2003)CrossRefMATHGoogle Scholar
  21. 21.
    Akdoǧan, Z., Demirci, M., Sh. Mukhtarov, O.: Green function of discontinuous boundary-value problem with transmission conditions. Math. Methods Appl. Sci. 30(14), 1719–1738 (2007)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Aydemir, K., Sh. Mukhtarov, O.: Green’s function method for self-adjoint realization of boundary-value problems with interior singularities. In: Abstract and Applied Analysis, Article ID 503267, 7 pages (2013). doi: 10.1155/2013/503267
  23. 23.
    Aydemir, K., Sh. Mukhtarov, O.: Spectrum and Green’s function of a many-interval Sturm–Liouville problem. Z. Naturforsch. 70, 301–308 (2015)Google Scholar
  24. 24.
    Sh. Mukhtarov, O., Olǧar, H., Aydemir, K.: Resolvent operator and spectrum of new type boundary value problems. Filomat 29, 1671–1680 (2015)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Sh. Mukhtarov, O., Olǧar, H., Aydemir, K.: Generalized eigenfunctions of one Sturm–Liouville system with symmetric jump conditions. AIP Conf. Proc. 1726, 020086 (2016). doi: 10.1063/1.4945912 CrossRefGoogle Scholar
  26. 26.
    Sh. Mukhtarov, O., Aydemir, K.: Eigenfunction expansion for Sturm–Liouville problems with transmission conditions at one interior point. Acta Math. Sci. 35B(3), 639–649 (2015)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Sh. Mukhtarov, O., Aydemir, K.: Eigenvalues and normalized eigenfunctionsof discontinuous Sturm–Liouville problem with transmission conditions. Rep. Math. Phys. 54, 41–56 (2004)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Shi, Y., Sun, H.: Self-adjoint extensions for second-order symmetric linear difference equations. Linear Algebra Appl. 434, 903–930 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Coddington, E.A.: Extension Theory of Formally Normal and Symmetric Subspaces, vol. 134. Memoirs of the American Mathematical Society, Providence (1973)MATHGoogle Scholar
  30. 30.
    Zhang, C., Zhang, L.: Classification for a class of second-order singular equations on time scales. In: Proceedings of 8th ACIS International Conference on Software Engineering, Artifical Intelligence, Networking, and Parallel/Distributed Computing, Qingdao, 2007Google Scholar
  31. 31.
    Huseynov, A.: Limit point and limit circle cases for dynamic equations on time scales. Hacet. J. Math. Stat. 39, 379–392 (2010)MathSciNetMATHGoogle Scholar
  32. 32.
    Zhang, C., Shi, Y.: Classification and criteria of the limit class for singular second-order linear equations on time scales. Bound. Value Probl. 2012, 103 (2012)CrossRefMATHGoogle Scholar
  33. 33.
    Hilscher, R.S., Zemánek, P.: Overview of Weyl–Titchmarsh theory for second order Sturm–Liouville equations on time scales. Int. J. Differ. Equ. 6, 39–51 (2011)MathSciNetGoogle Scholar
  34. 34.
    Bairamov, E., Ugurlu, E.: The determinants of dissipative Sturm–Liouville operators with transmission conditions. Math. Comput. Model. 53, 805–813 (2011)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Bairamov, E., Ugurlu, E.: Krein’s theorems for a dissipative boundary value transmission problem. Complex Anal. Oper. Theory 7, 831–842 (2013)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    Bairamov, E., Ugurlu, E.: On the characteristic values of the real component of a dissipative boundary value transmission problem. Appl. Math. Comput. 218, 9657–9663 (2012)MathSciNetMATHGoogle Scholar
  37. 37.
    Uğurlu, E., Bairamov, E.: Krein’s theorem for the dissipative operators with finite impulsive effects. Numer. Func. Anal. Optim. 36, 256–270 (2015)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Kreyszig, E.: Introductory Functional Analysis with Applications. Wiley, New York (1978)MATHGoogle Scholar
  39. 39.
    Nikolskiĭ, N: Operators, functions, and systems: an easy reading, vol. 2. In: Model Operators and Systems, Math. Surveys and Monogr., 92. Amer. Math. Soc., Providence (2002)Google Scholar
  40. 40.
    Nikolskiĭ, N.K.: Treatise on the Shift Operator. Springer, Berlin (1986)CrossRefGoogle Scholar
  41. 41.
    Bognar, J.: Indefinite Inner Product Spaces. Springer, Berlin (1974)CrossRefMATHGoogle Scholar
  42. 42.
    Alpay, D., Dijksma, A., Rovnyak, J., de Snoo, H.: Schur Functions, Operator Colligations, and Reproducing Kernel Pontryagin Spaces. Springer, Basel (1997)CrossRefMATHGoogle Scholar
  43. 43.
    Simon, B.: CMV matrices: five years after. J. Comput. Appl. Math. 208, 120–154 (2007)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Cantero, M.J., Moral, L., Velázquez, L.: Measures on the unit circle and unitary truncations of unitary operators. J. Appr. Theory 139, 430–468 (2006)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Gesztesy, F., Zinchenko, M.: Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle. J. Appr. Theory 139, 172–213 (2006)MathSciNetCrossRefMATHGoogle Scholar
  46. 46.
    Simon, B.: Analogs of the \(m-\)function in the theory of orthogonal polynomials on the unit circle. J. Comput. Appl. Math. 171, 411–424 (2004)MathSciNetCrossRefMATHGoogle Scholar
  47. 47.
    Marcellán, F., Shayanfar, N.: OPUC, CMV matrices and perturbations of measures supported on the unit circle. Linear Alg. Appl. 485, 305–344 (2015)MathSciNetCrossRefMATHGoogle Scholar
  48. 48.
    Grenander, U., Szegő, G.: Toeplitz Forms and Their Applications, University of California Press, Berkeley, 1958, 2nd edn. Chelsea, New York (1984)Google Scholar
  49. 49.
    Wang, Z., Wu, H.: Dissipative non-self-adjoint Sturm-Liouville operators and completeness of their eigenfunctions. J. Math. Anal. Appl. 394, 1–12 (2012)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of Arts and SciencesÇankaya UniversityAnkaraTurkey

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