DOI QR코드

DOI QR Code

SOME RESULTS FOR THE FRACTIONAL INTEGRAL OPERATOR DEFINED ON THE SOBOLEV SPACES

  • Received : 2021.04.15
  • Accepted : 2021.05.25
  • Published : 2022.01.30

Abstract

We investigated the invariant subspaces of the fractional integral operator in the Sobolev space Wkp[0, 1] and prove unicellularity of the operator Jα by using the Duhamel product.

Keywords

References

  1. N. Ahmad, S.K. Sharma and S.A. Mohiuddine, Generalized entire sequence spaces defined by fractional difference operator and sequence of modulus functions, TWMS J. App. and Eng. Math. 10 (2020), 63-72.
  2. L. Brickman and P.A. Fillmore, The invariant subspace lattice of a linear transformation, Can. J. Math. 19 (1967), 810-822. https://doi.org/10.4153/CJM-1967-075-4
  3. M.S. Brodskii, Triangular and Jordan Representations of Linear Operators, Translational Mathematical Monographs, 32, AMS Providence, RI, 1971.
  4. I.H. Dimovski, Convolutional Calculus, Kluwer Academic Publisher, London, 1990.
  5. I.Y. Domanov and M.M. Malamud, Invariant and hyperinvariant subspaces of an operator Jαand related operator algebras in Sobolev spaces, Linear Algebra and its Appl. 348 (2002), 209-230. https://doi.org/10.1016/S0024-3795(01)00581-X
  6. N. Dunford and L. Schwartz, Linear Operators, P.I: General Theory, Springer-Verlag, New York, 1958.
  7. I.C. Gohberg and M.G. Krein, Theory and Applications of Volterra Operators in Hilbert Space, Translational Mathematical Monographs 24, AMS Providence, RI, 1970.
  8. I.C. Gohberg, P. Lancaster and L. Rodman, Invariant Subspaces of Matrices with Applications, Wiley-Interscience, New York, 1986.
  9. M. Gurdal, Description of extended eigenvalues and extended eigenvectors of integration operator on the Wiener algebra, Expo. Math. 27 (2009), 153-160. https://doi.org/10.1016/j.exmath.2008.10.006
  10. M. Gurdal, On the extended eigenvalues and extended eigenvectors of shift operator on the Wiener algebra, Appl. Math. Lett. 22 (2009), 1727-1729. https://doi.org/10.1016/j.aml.2009.06.008
  11. M. Gurdal, M.T. Garayev and S. Saltan, Some concrete operators and their properties, Turkish J. Math. 39 (2015), 970-989. https://doi.org/10.3906/mat-1502-48
  12. B.B. Jena, S.K. Paikray, S.A. Mohiuddine and V.N. Mishra, Relatively equi-statistical convergence via deferred Norlund mean based on difference operator of fractional-order and related approximation theorems, AIMS Mathematics 5 (2020), 650-672. https://doi.org/10.3934/math.2020044
  13. M.T. Karaev, Usage of convolution for the proof of unicellularity, Zap.Nauchn.Sem. LOMI 135 (1984), 66-68.
  14. M.T. Karaev, Invariant subspaces, cyclic vectors, commutant and extended eigenvectors of some convolution operators, Methods Funct. Anal. Topology 11 (2005), 45-59.
  15. M.T. Karaev and M. Gurdal, Strongly splitting weighted shift operators on Banach spaces and unicellularity, Oper. Matrices 5 (2011), 157-171.
  16. M.T. Karaev, M. Gurdal and S. Saltan, Some applications of Banach algebra techniques, Math. Nachr. 284 (2011), 1678-1689. https://doi.org/10.1002/mana.200910129
  17. M.M. Malamud, Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators, Oper. Theory: Adv. Appl. Integral Differential Oper. 102 (1988), 143-167.
  18. S.A. Mohiuddine, K. Raj, M. Mursaleen and A. Alotaibi, Linear isomorphic spaces of fractional-order difference operators, Alexandria Eng. J. 60 (2021), 1155-1164. https://doi.org/10.1016/j.aej.2020.10.039
  19. N.K. Nikolskii, Invariant subspaces in operator theory and function theory, Itogi Nauki i Tekniki, Ser. Mat. Analiz, Moscow, 12 (1974), 199-412.
  20. N.K. Nikolskii, Treatise on the Shift Operator, Springer, Berlin, 1986.
  21. P.V. Ostapenko and V.G. Tarasov, Unicellularity of the integration operator in certain function spaces, Teor. Funkcii, Funkcional Anal. i Prilojen 27 (1977), 121-128.
  22. S. Saltan and M. Gurdal, Spectral multiplicities of some operators, Complex Var. Elliptic Equ. 56 (2011), 513-520. https://doi.org/10.1080/17476933.2010.487207
  23. R. Tapdigoglu, Invariant subspaces of Volterra integration operator: Axiomatical approach, Bull. Sci. Math. 136 (2012), 574-578. https://doi.org/10.1016/j.bulsci.2011.12.006
  24. R. Tapdigoglu, On the description of invariant subspaces in the space C(n)[0, 1], Houston J. Math. 39 (2013), 169-176.
  25. E.R. Tsekanovskii, About description of invariant subspaces and unicellularity of the integration operator in the space W2(p), Uspehi Mat. Nauk. 6 (1965), 169-172.
  26. N.M. Wigley, The Duhamel product of analytic functions, Duke Math. J. 41 (1974), 211-217. https://doi.org/10.1215/S0012-7094-74-04123-4
  27. T. Yaying, B. Hazarika and S.A. Mohiuddine, On difference sequence spaces of fractional order involving Padovan numbers, Asian-European J. Math. 14 (2020), doi: 10.1142/S1793557121500959.