On the boundedness and compactness of extended Cesàro composition operators between weighted Bloch-type spaces

Abstract

Let $\psi \in H(\mathbb{B}_n),$ the space of all holomorphic functions on the unit ball $\mathbb{B}_n$ of $\mathbb{C}^n,$ $\varphi = (\varphi_1, \ldots, \varphi_n) \in S(\mathbb{B}_n)$ the set of holomorphic self-maps of $\mathbb{B}_n.$ Let $C_{\psi, \varphi}: \mathcal B_{\nu}$ (and $ \mathcal B_{\nu,0}$) $\to \mathcal B_{\mu} $ (and $ \mathcal B_{\mu,0}$) be weighted extended Cesàro operators induced by products of the extended Cesàro operator $ C_\varphi $ and integral operator $T_\psi.$ In this paper, we characterize the boundedness and compactness of $ C_{\psi,\varphi} $ via the estimates for either $ |\varphi| $ or $ |\varphi_k| $ for some $ k\in \{1,\ldots,n\}. $ At the same time, we also give the asymptotic estimates of the norms of these operators.

Keywords

• Cesàro operator
• unit ball
• Bloch spaces
• boundedness
• compactness

Supporting Institution

The Science and Technology Planning Project of Quang Ngai Province

References

1. [1] A. Aleman, J. Cima, An integral operator on $H^p$ and Hardys inequality, J. Anal. Math. 85, 157-176, 2001.
2. [2] A. Aleman, A.G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. 46, 337-356, 1997.
3. [3] K. Avetisyanm, S. Stevic, Extended Cesàro operators between different Hardy spaces, Appl. Math. Computation 207(2), 346-350, 2009.
4. [4] H. Hamada, Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space, Sci. China Math. 62(4), 617-628, 2019.
5. [5] Z. Hu, Extended Cesàro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131(7), 2171-2179, 2003.
6. [6] Z.J. Hu, S.S. Wang, Composition operators on Bloch-type spaces, Proc. Roy. Soc. Edinburgh Sect. A 135, 1229-1239, 2005.
7. [7] S. Li, S. Stevic, Compactness of Riemann-Stieltjes operators between $ F(p, q, s) $ spaces and $ \alpha$-Bloch spaces, Publ. Math. Debrecen 72(1-2), 111-128, 2008.
8. [8] S. Li, S. Stevic, Riemann-Stieltjes operators between different weighted Bergman spaces, Bull. Belg. Math. Soc. 15(4), 677-686, 2008.

9. [9] S. Li, S. Stevic, Products of Volterra type operator and composition operator from $ H^\infty $ and Bloch spaces to Zygmund spaces, J. Math. Ana. Appl. 345(1), 40-52, 2008.
10. [10] Yu-X. Liang, Ze-H. Zhou, Product of Extended Cesàro Operator and Composition Operator from Lipschitz Space to $ F(p, q, s) $ Space on the Unit Ball, Hind. Publ. Corp. Abst. Appl. Anal., Article ID 152635, 9 pages, 2011.
11. [11] A. L. Shields, D. L. Williams, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc. 162, 287-302, 1971.
12. [12] X. Tang, Extended Cesàro operators between Bloch-type spaces in the unit ball of $\mathbb{C}^n$, J. Math. Anal. Appl. 326, 1199-1211, 2007.
13. [13] M. Tjani, Compact composition operators on some Möbius invariant Banach spaces, Doctoral Dissertation, Michigan State University, 1996.
14. [14] S.S. Wang, Z.J. Hu, Extended Cesàro operators on Bloch-type spaces, Chinese Ann. Math. Ser. A 26 (5), 613-624 (in Chinese), 2005.
15. [15] J. Xiao, Cesàro operators on Hardy, BMOA and Bloch spaces, Arch. Math. 68, 398- 406, 1997.
16. [16] W. F. Yang, Volterra composition operators from $ F(p, q, s) $ spaces to Bloch-type spaces, Bull. Malay. Math. Sci. Soc. 34(2), 267-277, 2011.
17. [17] K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Graduate Texts in Mathematics, vol. 226, Springer-Verlag, New York, 2005.