SavedAdd your journal

Journals / Hacettepe Journal of Mathematics and Statistics / 2020 / Cilt: 49 - Sayı: 6

Embedding the weighted space $Hv_0(G, E)$ of holomorphic functions into the sequence space $c_0(E)$

El Mustapha El Abbassi
Lahbib Oubbi
Pages
2063–2070
Publication Date
2022-10-18
DOI
—
🏛️Bibliographic metadata and full text access for this article are provided via DergiPark.
DergiPark Record ↗
✨

AI Summary & Key Takeaways

AI Insight
⚡ TL;DR(Too Long; Didn't Read)

We then deduce the non 1-Hyers-Rassias stability of the isometry functional equation in the framework of Banach spaces.

📌 3 Key Takeaways

🎯
Objective & Hypothesis:We embed almost isometrically the generalized weighted space $Hv_0(G, E)$ of holomorphic functions on an open subset $G$ of $\mathbb{C}^N$ with values in a Banach space $E$, into $c_0(E)$, the space of all null sequences in $E$, where $v$ is an operator-valued continuous function on $G$ vanishing nowhere.
📊
Key Findings & Data:This extends and generalizes some known results in the literature.
💡
Conclusion & Impact:We then deduce the non 1-Hyers-Rassias stability of the isometry functional equation in the framework of Banach spaces.
🔬Design:Research Article
📍Scope:Comprehensive Sample
⚙️Variables:Weighted spaces of holomorphic functions · Vector-valued holomorphic function · Operator-valued weights

Abstract

We embed almost isometrically the generalized weighted space $Hv_0(G, E)$ of holomorphic functions on an open subset $G$ of $\mathbb{C}^N$ with values in a Banach space $E$, into $c_0(E)$, the space of all null sequences in $E$, where $v$ is an operator-valued continuous function on $G$ vanishing nowhere. This extends and generalizes some known results in the literature. We then deduce the non 1-Hyers-Rassias stability of the isometry functional equation in the framework of Banach spaces.

🔖

Cite This Article

Abbassi, E. M. E., Oubbi, L. (2022). Embedding the weighted space $Hv_0(G, E)$ of holomorphic functions into the sequence space $c_0(E)$. Hacettepe Journal of Mathematics and Statistics, Cilt: 49 - Sayı: 6, 2063–2070. https://ajindex.com/articles/529603

📄 Full Text PDF Viewer

📄

PDF görüntüleyici hazırlanıyor...

References (13)

  1. J. Bonet and J.A. Conejero, The sets of monomorphisms and of almost open operators between locally convex spaces, Proc. Amer. Math. Soc. 129, 3683–3690, 2001
  2. J. Bonet and E. Wolf, A note on weighted Banach spaces of holomorphic functions, Arch. Math. (Basel) 81, 650–654, 2003
  3. C. Boyd and P. Rueda, The v-boundary of weighted spaces of holomorphic functions, Ann. Acad. Sci. Fenn. Math. 30, 337–352, 2005
  4. Z. Gajda, On stability of additive mappings, Int. J. Math. Math. Sci. 14, 431–434, 1991
  5. H.A. Gindler and A.E. Taylor, The minimum modulus of a linear operator and its use in spectral theory, Studia Math. 22, 15–41, 1962
  6. L. Hörmander, An Introduction to Complex Analysis in Several Variables, North- Holland Mathematical Library 7, North-Holland, 1990
  7. M. Klilou and L. Oubbi, Multiplication operators on generalized weighted spaces of continuous functions, Mediterr. J. Math. 13, 3265–3280, 2016
  8. M. Klilou and L. Oubbi, Weighted composition operators on Nachbin spaces with operator-valued weights, Commun. Korean Math. Soc. 33, 1125–1140, 2018 [9] W. Lusky, On the structure of $H_{v_0}(D)$ and $h_{v_0}(D)$, Math. Nachr. 159, 279–289, 1992
  9. W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. 51, 309–320, 1995
  10. W. Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. 175, 19–45, 2006
  11. V. Müller, Spectral theory of linear operators and spectral systems in Banach algebras, in: Operator Theory: Advances and Applications, second ed. 139, Birkhäuser Verlag, Basel, 2007
  12. W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$, Classics in Mathematics, Springer Berlin, 1980
  13. C. Shekhar and B.S. Komal, Multiplication operators on weighted spaces of continuous functions with operator-valued weights, Int. J. Contemp. Math. Sci. 7 (38), 1889–1894, 2012.