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

Year 2020, Volume: 49 Issue: 6, 2063 - 2070, 08.12.2020

El Mustapha El Abbassi Lahbib Oubbı

https://doi.org/10.15672/hujms.621628

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.

Keywords

Weighted spaces of holomorphic functions, Vector-valued holomorphic function, Operator-valued weights, Almost isometric embedding

References

• [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.
• [10] W. Lusky, On weighted spaces of harmonic and holomorphic functions, J. London Math. Soc. 51, 309–320, 1995.
• [11] W. Lusky, On the isomorphism classes of weighted spaces of harmonic and holomorphic functions, Studia Math. 175, 19–45, 2006.
• [12] 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.
• [13] W. Rudin, Function Theory in the Unit Ball of $\mathbb{C}^n$, Classics in Mathematics, Springer Berlin, 1980.
• [14] 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.