Boundedness of the product of some operators from the natural Bloch space into weighted-type space

Xiaoman Liu; Yongmin Liu; Xiaoman Liu; Yongmin Liu

doi:10.3934/math.2024957

AIMS Mathematics

2024, Volume 9, Issue 7: 19626-19644. doi: 10.3934/math.2024957

Previous Article Next Article

Research article

Boundedness of the product of some operators from the natural Bloch space into weighted-type space

Xiaoman Liu ^{1
,
,},
Yongmin Liu ²

1.
College of Sciences, Nanjing Agricultural University, Nanjing 210095, China
2.
School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China

Received: 12 April 2024 Revised: 03 June 2024 Accepted: 03 June 2024 Published: 14 June 2024
MSC : 47B38, 47B33, 47B01, 46E15

Let $ \mathbb{B}_X $ be the unit ball of a complex Banach space $ X $, which may be infinite dimensional. The authors characterize the boundedness of the product of the radial derivative operator and the weighted composition operator from the natural Bloch space (or the little Bloch-type space) into the weighted-type space (or the little weighted-type space) on $ \mathbb{B}_X $.
- weighted composition operator,
- radial derivative operator,
- natural Bloch space,
- weighted-type space,
- boundedness
Citation: Xiaoman Liu, Yongmin Liu. Boundedness of the product of some operators from the natural Bloch space into weighted-type space[J]. AIMS Mathematics, 2024, 9(7): 19626-19644. doi: 10.3934/math.2024957

Related Papers:

Abstract

Let $ \mathbb{B}_X $ be the unit ball of a complex Banach space $ X $, which may be infinite dimensional. The authors characterize the boundedness of the product of the radial derivative operator and the weighted composition operator from the natural Bloch space (or the little Bloch-type space) into the weighted-type space (or the little weighted-type space) on $ \mathbb{B}_X $.

References

[1]	J. Anderson, J. Clunie, C. Pommerenke, On Bloch functions and normal functions, J. Reine Angew. Math., 270 (1974), 12–37. https://doi.org/10.1515/crll.1974.270.12 doi: 10.1515/crll.1974.270.12
[2]	R. M. Aron, P. Galindo, M. Lindström, Compact homomorphisms between algebras of analytic functions, Studia Math., 123 (1997), 235–247.
[3]	O. Blasco, P. Galindo, M. Lindström, A. Miralles, Composition operators on the Bloch space of the unit ball of a Hilbert space, Banach J. Math. Anal., 11 (2017), 311–334. https://doi.org/10.1215/17358787-0000005X doi: 10.1215/17358787-0000005X
[4]	O. Blasco, P. Galindo, A. Miralles, Bloch functions on the unit ball of an infinite dimensional Hilbert space, J. Funct. Anal., 267 (2014), 1188–1204. https://doi.org/10.1016/j.jfa.2014.04.018 doi: 10.1016/j.jfa.2014.04.018
[5]	C. H. Chu, H. Hamada, T. Honda, G. Kohr, Bloch functions on bounded symmetric domains, J. Funct. Anal., 272 (2017), 2412–2441. https://doi.org/10.1016/j.jfa.2016.11.005 doi: 10.1016/j.jfa.2016.11.005
[6]	F. Colonna, New criteria for boundedness and compactness of weighted composition operators mapping into the Bloch space, Centr. Eur. J. Math., 11 (2013), 55–73. https://doi.org/10.2478/s11533-012-0097-4 doi: 10.2478/s11533-012-0097-4
[7]	F. Colonna, M. Tjani, Operator norms and essential norms of weighted composition operators between Banach spaces of analytic functions, J. Math. Anal. Appl., 434 (2016), 93–124. https://doi.org/10.1016/j.jmaa.2015.08.073 doi: 10.1016/j.jmaa.2015.08.073
[8]	C. C. Cowen, B. D. MacCluer, Composition operators on spaces of analytic functions, Boca Raton: CRC Press, 1995.
[9]	F. Deng, C. Ouyang, Bloch spaces on bounded symmetric domains in complex Banach spaces, Sci. China Ser. A, 49 (2006), 1625–1632. https://doi.org/10.1007/s11425-006-2050-0 doi: 10.1007/s11425-006-2050-0
[10]	Z. S. Fang, Z. H. Zhou, New characterizations of the weighted composition operators between Bloch type spaces in the polydisk, Can. Math. Bull., 57 (2014), 794–802. https://doi.org/10.4153/CMB-2013-043-4 doi: 10.4153/CMB-2013-043-4
[11]	D. García, M. Maestre, P. Rueda, Weighted spaces of holomorphic functions on Banach spaces, Studia Math., 138 (2000), 1–24.
[12]	D. García, M. Maestre, P. Sevilla-Peris, Composition operators between weighted spaces of holomorphic functions on Banach spaces, Ann. Acad. Sci. Fenn. Math., 29 (2004), 81–98.
[13]	H. Hamada, Weighted composition operators from $H^\infty$ to the Bloch space of infinite dimensional bounded symmetric domains, Complex Anal. Oper. Theory, 12 (2018), 207–216. https://doi.org/10.1007/s11785-016-0624-6 doi: 10.1007/s11785-016-0624-6
[14]	H. Hamada, Bloch-type spaces and extended Cesàro operators in the unit ball of a complex Banach space, Sci. China Math., 62 (2019), 617–628. https://doi.org/10.1007/s11425-017-9183-5 doi: 10.1007/s11425-017-9183-5
[15]	R. A. Hibschweiler, Products of composition, differentiation and multiplication from the Cauchy spaces to the Zygmund space, Bull. Korean Math. Soc., 60 (2023), 1061–1070. https://doi.org/10.4134/BKMS.b220471 doi: 10.4134/BKMS.b220471
[16]	R. A. Hibschweiler, N. Portnoy, Composition followed by differentiation between Bergman and Hardy spaces, Rocky Mountain J. Math., 35 (2005), 843–855. https://doi.org/10.1216/rmjm/1181069709 doi: 10.1216/rmjm/1181069709
[17]	C. S. Huang, Z. J. Jiang, Product-type operators from weighted Bergman-Orlicz spaces to weighted-type spaces on the unit ball, J. Math. Anal. Appl., 519 (2023), 126739. https://doi.org/10.1016/j.jmaa.2022.126739 doi: 10.1016/j.jmaa.2022.126739
[18]	C. S. Huang, Z. J. Jiang, On a sum of more complex product-type operators from Bloch-type spaces to the weighted-type spaces, Axioms, 12 (2023), 566. https://doi.org/10.3390/axioms12060566 doi: 10.3390/axioms12060566
[19]	C. S. Huang, Z. J. Jiang, Y. F. Xue, Sum of some product-type operators from mixed-norm spaces to weighted-type spaces on the unit ball, AIMS Math., 7 (2022), 18194–18217. https://doi.org/10.3934/math.20221001 doi: 10.3934/math.20221001
[20]	O. Hyvärinen, I. Nieminen, Weighted composition followed by differentiation between Bloch-type spaces, Rev. Mat. Complut., 27 (2014), 641–656. https://doi.org/10.1007/s13163-013-0138-y doi: 10.1007/s13163-013-0138-y
[21]	S. Li, S. Stević, Composition followed by differentiation between $H^\infty$ and $\alpha$-Bloch spaces, Houston J. Math., 35 (2009), 327–340.
[22]	S. Li, S. Stević, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl., 349 (2009), 596–610. https://doi.org/10.1016/j.jmaa.2008.09.014 doi: 10.1016/j.jmaa.2008.09.014
[23]	S. Li, S. Stević, Products of composition and differentiation operators from Zygmund spaces to Bloch spaces and Bers spaces, Appl. Math. Comput., 217 (2010), 3144–3154. https://doi.org/10.1016/j.amc.2010.08.047 doi: 10.1016/j.amc.2010.08.047
[24]	S. Li, J. Zhou, A note of weighted composition operators on Bloch-type spaces, Bull. Korean Math. Soc., 52 (2015), 1711–1719. https://doi.org/10.4134/BKMS.2015.52.5.1711 doi: 10.4134/BKMS.2015.52.5.1711
[25]	X. Liu, Y. Liu, Weighted composition operators between the Bloch type space and the Hardy space on the unit ball of complex Banach spaces, Numer. Funct. Anal. Optim., 43 (2022), 1578–1590. https://doi.org/10.1080/01630563.2022.2112599 doi: 10.1080/01630563.2022.2112599
[26]	Y. Liu, Y. Yu, Weighted composition operators between the Bloch type space and $H^\infty(\mathbb{B}_X)$ of infinite dimensional bounded symmetric domains, Complex Anal. Oper. Theory, 13 (2019), 1595–1608. https://doi.org/10.1007/s11785-018-00884-w doi: 10.1007/s11785-018-00884-w
[27]	X. Liu, S. Li, Norm and essential norm of a weighted composition operator on the Bloch space, Integr. Equ. Oper. Theory, 87 (2017), 309–325. https://doi.org/10.1007/s00020-017-2349-y doi: 10.1007/s00020-017-2349-y
[28]	J. S. Manhas, R. Zhao, New estimates of essential norms of weighted composition operators between Bloch type spaces, J. Math. Anal. Appl., 389 (2012), 32–47. https://doi.org/10.1016/j.jmaa.2011.11.039 doi: 10.1016/j.jmaa.2011.11.039
[29]	A. Miralles, Bloch functions on the unit ball on a Banach space, Proc. Amer. Math. Soc., 149 (2021), 1459–1470. https://doi.org/10.1090/proc/14966 doi: 10.1090/proc/14966
[30]	S. Ohno, K. Stroethoff, R. Zhao, Weighted composition operators between Bloch-type spaces, Rocky Mountain J. Math., 33 (2003), 191–215. https://doi.org/10.1216/rmjm/1181069993 doi: 10.1216/rmjm/1181069993
[31]	S. Ohno, R. Zhao, Weighted composition operators on the Bloch space, Bull. Aust. Math. Soc., 63 (2001), 177–185. https://doi.org/10.1017/S0004972700019250 doi: 10.1017/S0004972700019250
[32]	C. Pommerenke, On Bloch functions, J. Lond. Math. Soc., 2 (1970), 689–695. https://doi.org/10.1112/jlms/2.Part_4.689 doi: 10.1112/jlms/2.Part_4.689
[33]	J. S. Manhas, Weighted composition operators and dynamical systems on weighted spaces of holomorphic functions on Banach spaces, Ann. Funct. Anal., 4 (2013), 58–71. https://doi.org/10.15352/afa/1399899525 doi: 10.15352/afa/1399899525
[34]	B. Sehba, S. Stević, On some product-type operators from Hardy-Orlicz and Bergman-Orlicz spaces to weighted-type spaces, Appl. Math. Comput., 233 (2014), 565–581. https://doi.org/10.1016/j.amc.2014.01.002 doi: 10.1016/j.amc.2014.01.002
[35]	A. L. Shields, D. L. Williams, Bounded projections, duality, and multipliers in spaces of analytic functions, Trans. Amer. Math. Soc., 162 (1971), 287–302. https://doi.org/10.2307/1995754 doi: 10.2307/1995754
[36]	S. Stević, Products of composition and differentiation operators on the weighted Bergman space, Bull. Belg. Math. Soc. Simon Stevin, 16 (2009), 623–635. https://doi.org/10.36045/bbms/1257776238 doi: 10.36045/bbms/1257776238
[37]	S. Stević, Composition followed by differentiation from $H^\infty$ and the Bloch space to $n$th weighted-type spaces on the unit disk, Appl. Math. Comput., 216 (2010), 3450–3458. https://doi.org/10.1016/j.amc.2010.03.117 doi: 10.1016/j.amc.2010.03.117
[38]	S. Stević, On a new product-type operator on the unit ball, J. Math. Inequal., 16 (2022), 1675–1692. https://doi.org/10.7153/jmi-2022-16-109 doi: 10.7153/jmi-2022-16-109
[39]	S. Stević, Note on a new class of operators between some spaces of holomorphic functions, AIMS Math., 8 (2023), 4153–4167. http://doi.org/10.3934/math.2023207 doi: 10.3934/math.2023207
[40]	S. Stević, Z. J. Jiang, Weighted iterated radial composition operators from weighted Bergman-Orlicz spaces to weighted-type spaces on the unit ball, Math. Methods Appl. Sci., 44 (2021), 8684–8696. https://doi.org/10.1002/mma.7298 doi: 10.1002/mma.7298
[41]	S. Stević, Z. J. Jiang, Weighted iterated radial composition operators from logarithmic Bloch spaces to weighted-type spaces on the unit ball, Math. Methods Appl. Sci., 45 (2022), 3083–3097. https://doi.org/10.1002/mma.7978 doi: 10.1002/mma.7978
[42]	S. Stević, C. S. Huang, Z. J. Jiang, Sum of some product-type operators from Hardy spaces to weighted-type spaces on the unit ball, Math. Methods Appl. Sci., 45 (2022), 11581–11600. https://doi.org/10.1002/mma.8467 doi: 10.1002/mma.8467
[43]	S. Stević, A. K. Sharma, On a product-type operator between Hardy and $\alpha$-Bloch spaces of the upper half-plane, J. Inequal. Appl., 2018 (2018), 273. https://doi.org/10.1186/s13660-018-1867-8 doi: 10.1186/s13660-018-1867-8
[44]	S. Stević, A. K. Sharma, A. Bhat, Essential norm of products of multiplication composition and differentiation operators on weighted Bergman spaces, Appl. Math. Comput., 218 (2011), 2386–2397. https://doi.org/10.1016/j.amc.2011.06.055 doi: 10.1016/j.amc.2011.06.055
[45]	S. Stević, A. K. Sharma, A. Bhat, Products of multiplication, composition and differentiation operators on weighted Bergman space, Appl. Math. Comput., 217 (2011), 8115–8125. https://doi.org/10.1016/j.amc.2011.03.014 doi: 10.1016/j.amc.2011.03.014
[46]	R. M. Timoney, Bloch functions in several complex variables, I, Bull. Lond. Math. Soc., 12 (1980), 241–267. https://doi.org/10.1112/blms/12.4.241 doi: 10.1112/blms/12.4.241
[47]	Z. Tu, L. Xiong, Weighted space and Bloch-type space on the unit ball of an infinite dimensional complex Banach space, Bull. Iran. Math. Soc., 45 (2019), 1389–1406. https://doi.org/10.1007/s41980-019-00204-8 doi: 10.1007/s41980-019-00204-8
[48]	S. Wang, M. Wang, X. Guo, Products of composition, multiplication and iterated differentiation operators between Banach spaces of holomorphic functions, Taiwanese J. Math., 24 (2020), 355–376. https://doi.org/10.11650/tjm/190405 doi: 10.11650/tjm/190405
[49]	L. X. Zhang, Product of composition and differentiation operators and closures of weighted Bergman spaces in Bloch type spaces, J. Inequal. Appl., 2019 (2019), 310. https://doi.org/10.1186/s13660-019-2259-4 doi: 10.1186/s13660-019-2259-4
[50]	J. Zhou, X. Zhu, Product of differentiation and composition operators on the logarithmic Bloch space, J. Inequal. Appl., 2014 (2014), 453. https://doi.org/10.1186/1029-242X-2014-453 doi: 10.1186/1029-242X-2014-453
[51]	K. Zhu, Spaces of holomorphic functions in the unit ball, New York: Springer, 2005. https://doi.org/10.1007/0-387-27539-8
[52]	X. Zhu, Products of differentiation, composition and multiplication from Bergman type spaces to Bers type spaces, Integral Transforms Spec. Funct., 18 (2007), 223–231. https://doi.org/10.1080/10652460701210250 doi: 10.1080/10652460701210250

Reader Comments

Your name:*

Email:*
© 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)