Complete Quasi-wandering Sets and Kernels of Functional Operators

  • Victor D. Didenko
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)


Kernels of functional operators generated by mapping that possess complete quasi-wandering sets are studied. It is shown that the kernels of the operators under consideration either consist of a zero element or contain a subset isomorphic to a space \( L_\infty \left( \mathbb{S} \right) \)), where \( \mathbb{S} \subset \mathbb{R}^n \) has a positive Lebesgue measure. Consequently, such operators are Fredholm if and only if they are invertible.

Mathematics Subject Classification (2000)

Primary 47B33, 39A33, 45E10 Secondary 42C40, 39B42. 


Quasi-wandering set homogeneous equation solution. 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    A. Antonevich, Linear Functional Equations. Operator Approach. Operator Theory. Advances and Applications, Vol. 83, Birkhäuser, Basel-Boston-Berlin, 1996.Google Scholar
  2. [2]
    A. Antonevich, A. Lebedev, Functional Differential Equations: C *-Theory. Pitman Monographs and Surveys in Pure and Applied Mathematics, Vol. 70. Longman, Harlow, 1994.Google Scholar
  3. [3]
    N.V. Azbelev, G.G. Islamov, A certain class of functional-differential equations. Differencial’nye Uravnenija, 12 (1976), 417–427.zbMATHMathSciNetGoogle Scholar
  4. [4]
    G. Belitskii, V. Tkachenko, One-dimensional functional equations. Operator Theory: Advances and Applications, Vol. 144, Birkhäuser, Basel, 2003.Google Scholar
  5. [5]
    E. Castillo, A. Iglesias, R. Ruíz-Cobo, Functional equations in applied sciences. Mathematics in Science and Engineering, Vol. 199, Elsevier, Amsterdam, 2005.Google Scholar
  6. [6]
    A.V. Chistyakov, A pathological counterexample to the non-Fredholmness conjecture in algebras of weighted shift operators, Izv. Vyssh. Uchebn. Zaved. Mat., No. 10 (1995), 76–86.Google Scholar
  7. [7]
    V.D. Didenko, Fredholm properties of multivariate refinement equations. Math. Meth. Appl. Sci., 30 (2007), 1639–1644zbMATHCrossRefMathSciNetGoogle Scholar
  8. [8]
    Yu.I. Karlovich, V.G. Kravchenko, G.S. Litvinchuk, Invertibility of functional operators in Banach spaces. In: Functional-differential equations, Perm. Politekh. Inst., Perm’, 1990, 18–58.Google Scholar
  9. [9]
    V.G. Kravchenko, G.S. Litvinchuk, Introduction to the theory of singular integral operators with shift. Mathematics and its Applications, Vol. 289, Kluwer, Dordrecht, 1994.Google Scholar
  10. [10]
    M. Kuczma, Functional equations in single variable. Monografie Matematyczne, Vol. 46, PWN, Warszawa, 1968.Google Scholar
  11. [11]
    V.G. Kurbatov, A conjecture in the theory of functional-differential equations, Differentsial’nye Uravnenija, 14 (1978), 2074–2075.zbMATHMathSciNetGoogle Scholar
  12. [12]
    G.S. Litvinchuk, Solvability theory of boundary value problems and singular integral equations with shift. Mathematics and its Applications, Vol. 523, Kluwer, Dordrecht, 2000.Google Scholar
  13. [13]
    P.J. Nicholls, The Ergodic Theory of Discrete Groups. London Mathematical Society Lecture Note Series, Vol. 143, Cambridge University press, Cambridge, 1989.Google Scholar

Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Victor D. Didenko
    • 1
  1. 1.Department of MathematicsUniversity Brunei DarussalamBandar Seri BegawanBrunei

Personalised recommendations