Functional Analysis and Its Applications

, Volume 52, Issue 1, pp 1–8 | Cite as

Duhamel Algebras and Applications

  • M. T. Karaev


We introduce Duhamel algebras and study their properties and applications. We prove that a Banach space of analytic functions on the unit disc that satisfy certain conditions is a Duhamel algebra and describe its closed ideals. These results substantially generalize and improve the main results of Wigley’s papers. Some other related questions are also discussed.

Key words

Duhamel algebra closed ideal Hardy space Banach space of analytic functions invariant subspace 


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  1. [1]
    A. Aleman and B. Korenblum, “Volterra invariant subspaces of H p,” Bull. Sci. Math., 132:6 (2008), 510–528.MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    A. Biswas, A. Lambert, and S. Petrovic, “Extended eigenvalues and the Volterra operator,” Glasgow Math. J., 44:3 (2002), 521–534.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    I. Yu. Domanov and M. M. Malamud, “On the spectral analysis of direct sums of Riemann–Liouville operators in Sobolev spaces of vector functions,” Integral Equations Operator Theory, 63:2 (2009), 181–215.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    H. Guediri, M. T. Garayev, and H. Sadraoui, “The Bergman space as a Banach algebra,” New York J. Math., 21 (2015), 339–350.MathSciNetzbMATHGoogle Scholar
  5. [5]
    B. Hollenbeck and I. E. Verbitsky, “Best constants for the Riesz projection,” J. Funct. Anal., 175:2 (2000), 370–392.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    M. T. Karaev, “Invariant subspaces, cyclic vectors, commutant and extended eigenvectors of some convolution operators,” Methods Funct. Anal. Topology, 11:1 (2005), 48–59.MathSciNetzbMATHGoogle Scholar
  7. [7]
    M. T. Karaev, M. Gürdal, and S. Saltan, “Some applications of Banach algebras techniques,” Math. Nachr., 284:13 (2011), 1678–1689.MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    M. T. Karaev, “On extended eigenvalues and extended eigenvectors of some operator classes,” Proc. Amer. Math. Soc., 134:6 (2006), 2383–2392.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Yu. Linchuk, “On derivation operators with respect to the Duhamel convolution in the space of analytic functions,” Math. Commun., 20:1 (2015), 17–22.MathSciNetzbMATHGoogle Scholar
  10. [10]
    M. M. Malamud, “Invariant and hyperinvariant subspaces of direct sums of simple Volterra operators,” in: Oper. Theory Adv. Appl., vol. 102, BirkhaÜser, Basel, 1998, 143–167.MathSciNetzbMATHGoogle Scholar
  11. [11]
    M. M. Malamud, “Similarity of Volterra operators and related questions of the theory of differential equations of fractional orders,” Trudy Moscov. Mat. Obsch., 55 (1994), 73–148; English transl.: Trans. Moscow Math. Soc., 55 (1994), 57–122.zbMATHGoogle Scholar
  12. [12]
    K. G. Merryfield and S. Watson, “A local algebra structure for H p of the polydisc,” Colloq. Math., 62:1 (1991), 73–79.MathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    N. K. Nikolski, “Selected problems of weighted approximation and spectral analysis [in Russian],” Trudy Mat. Inst. Steklov., 70 (1974), 1–270.Google Scholar
  14. [14]
    N. M. Wigley, “The Duhamel product of analytic functions,” Duke Math. J., 41 (1974), 211–217.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    N. M. Wigley, “A Banach algebra structure for H p,” Canad. Math. Bull., 18 (1975), 597–603.MathSciNetCrossRefzbMATHGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institute of Mathematics and Mechanics National Academy of Sciences of AzerbaijanBakuAzerbaijan
  2. 2.Department of Mathematics, College of ScienceKing Saud UniversityRiyadhSaudi Arabia

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