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Convolution operators on Banach–Orlicz algebras


For a locally compact group G, let \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G) be Orlicz and weighted Orlicz spaces, respectively, where Φ is a Young function and ω is a weight on G. We study the harmonic and convolution operators on Orlicz and weighted Orlicz spaces. We prove that under some conditions the harmonic operators on \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G) are compact. We characterize convolution operators on Orlicz and weighted Orlicz spaces \(\mathcal{L}^{\Phi}\)(G) and \(\mathcal{L}_\omega^{\Phi}\)(G).

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  1. [1]

    I. Akbarbaglu and S. Maghsoudi, Banach-Orlicz algebras on a locally compact group, Mediterr. J. Math., 10 (2013), 1937–1934.

  2. [2]

    J. Alexopoulos, Weakly compact sets in Banach spaces, PhD Thesis, Kent State University Graduate College (1992).

    Google Scholar 

  3. [3]

    C-H. Chu, Harmonic function spaces on groups, J. London Math. Soc., 70 (2004), 182–198.

    MathSciNet  Article  Google Scholar 

  4. [4]

    C-H. Chu, Matrix Convolution Operators on Groups, Lecture Notes in Math., Springer-Verlag (Berlin, Heidelberg, 2008).

    Book  Google Scholar 

  5. [5]

    A. Derighetti, Convolution Operators on Groups, Springer-Verlag ( Berlin, 2011).

    Book  Google Scholar 

  6. [6]

    J. Diestel, Sequences and Series in Banach Spaces, Springer-Verlag ( Berlin, 1984).

    Book  Google Scholar 

  7. [7]

    F. Ghahramini, Compact elements of weighted group algebras, Pacific J. Math., 113 (1984), 77–84.

    MathSciNet  Article  Google Scholar 

  8. [8]

    F. Ghahramini, Weighted group algebra as an ideal in its second dual space, Proc. Amer. Math. Soc., 90 (1984), 71–76.

    MathSciNet  Article  Google Scholar 

  9. [9]

    M. A. Krasnosel’skiĭ and Ya. B. Rutickiĭ, Convex functions and Orlicz spaces, translated from the first Russian edition by L. F. Boron, P. Noordhoff Ltd. ( Groningen, The Netherlands, (1961).

  10. [10]

    A. Osançhol and S. Öztop, Weighted Orlicz algebras on locally compact groups, J. Aust. Math. Soc., 99 (2015), 399–414.

    MathSciNet  Article  Google Scholar 

  11. [11]

    S. Öztop and E. Samei, Twisted Orlicz algebras. I, Studia Math., 236 (2017), 271–296.

    MathSciNet  Article  Google Scholar 

  12. [12]

    S. Öztop and E. Samei, Twisted Orlicz algebras. II, Math. Nachr., 292 (2019), 1122–1136.

    MathSciNet  Article  Google Scholar 

  13. [13]

    S. Öztop, E. Samei and V. Shepelska, Weak amenability of weighted Orlicz algebras, Arch. Math. (Basel), 110 (2018), 363–376.

    MathSciNet  Article  Google Scholar 

  14. [14]

    M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Marcel Dekker (1991).

    MATH  Google Scholar 

  15. [15]

    M. M. Rao and Z. D. Ren, Applications of Orlicz Spaces, Marcel Dekker (2002).

    Book  Google Scholar 

  16. [16]

    S. Sakai, Weakly compact operators on operator algebras, Pacific J. Math., 14 (1964), 659–664.

    MathSciNet  Article  Google Scholar 

  17. [17]

    J. J. Uhl, Compact operators on Orlicz spaces, Rend. Semin. Math. Univ. Padova, 42 (1969), 209–219.

    MathSciNet  MATH  Google Scholar 

  18. [18]

    J. J. Uhl, On a class of operators on Orlicz spaces, Studia Math., 40 (1971), 17–22.

    MathSciNet  Article  Google Scholar 

Download references


The authors would like to thank the referee for careful reading of the paper and for his/her useful suggestions, which greatly improved the presentation of the paper. We revised the whole of the subsection 4.2, in light of his/her comments. In particular, we corrected the proof of Theorem 5.3 and added Theorem 5.4 by his/her suggestions.

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Correspondence to A. Jabbari.

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Ebadian, A., Jabbari, A. Convolution operators on Banach–Orlicz algebras. Anal Math 46, 243–264 (2020).

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Key words and phrases

  • convolution operator
  • harmonic operator
  • locally compact group
  • Orlicz space
  • weighted Orlicz space

Mathematics Subject Classification

  • 47B48
  • 46E25
  • 46E30