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Adjoint of generalized Cesáro operators on analytic function spaces

  • Sunanda Naik EMAIL logo and Pankaj K. Nath
Published/Copyright: September 1, 2020

Abstract

In this article, we define a convolution operator and study its boundedness on mixed-norm spaces. In particular, we obtain a well-known result on the boundedness of composition operators given by Avetisyan and Stević in [K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 2009, 2, 304–311]. Also we consider the adjoint 𝒜b,c for b>0 of two parameter families of Cesáro averaging operators and prove the boundedness on Besov mixed-norm spaces Bα+(c-1)p,q for c>1.

Acknowledgements

The authors are thankful to the anonymous referees whose reports have helped in improving the article.

References

[1] M. R. Agrawal, P. G. Howlett, S. K. Lucas, S. Naik and S. Ponnusamy, Boundedness of generalized Cesáro averaging operators on certain function spaces, J. Comput. Appl. Math. 180 (2005), no. 2, 333–344. 10.1016/j.cam.2004.11.004Search in Google Scholar

[2] A. Aleman and J. A. Cima, An integral operator on Hp and Hardy’s inequality, J. Anal. Math. 85 (2001), 157–176. 10.1007/BF02788078Search in Google Scholar

[3] G. E. Andrews, R. Askey and R. Roy, Special Functions, Encyclopedia Math. Appl. 71, Cambridge University, Cambridge, 1999. 10.1017/CBO9781107325937Search in Google Scholar

[4] K. Avetisyan and S. Stević, The generalized Libera transform is bounded on the Besov mixed-norm, BMOA and VMOA spaces on the unit disc, Appl. Math. Comput. 213 (2009), no. 2, 304–311. 10.1016/j.amc.2009.03.022Search in Google Scholar

[5] D. Borgohain and S. Naik, Generalized Cesàro operators on the spaces of Cauchy transforms, Acta Sci. Math. (Szeged) 83 (2017), no. 1–2, 143–154. 10.14232/actasm-016-542-6Search in Google Scholar

[6] P. L. Duren, Theory of Hp Spaces, Pure Appl. Math. 38, Academic Press, New York, 1970. Search in Google Scholar

[7] H. Hedenmalm, B. Korenblum and K. Zhu, Theory of Bergman Spaces, Grad. Texts in Math. 199, Springer, New York, 2000. 10.1007/978-1-4612-0497-8Search in Google Scholar

[8] Z. Hu, Extended Cesàro operators on Bergman spaces, J. Math. Anal. Appl. 296 (2004), no. 2, 435–454. 10.1016/j.jmaa.2004.01.045Search in Google Scholar

[9] O. S. Kwon and N. E. Cho, A class of nonlinear integral operators preserving double subordinations, Abstr. Appl. Anal. 2008 (2008), Article ID 792160. 10.1155/2008/792160Search in Google Scholar

[10] S. Li and S. Stević, Integral type operators from mixed-norm spaces to α-Bloch spaces, Integral Transforms Spec. Funct. 18 (2007), no. 7–8, 485–493. 10.1080/10652460701320703Search in Google Scholar

[11] S. Li and S. Stević, Generalized composition operators on Zygmund spaces and Bloch type spaces, J. Math. Anal. Appl. 338 (2008), no. 2, 1282–1295. 10.1016/j.jmaa.2007.06.013Search in Google Scholar

[12] S. Li and S. Stević, Products of composition and integral type operators from H to the Bloch space, Complex Var. Elliptic Equ. 53 (2008), no. 5, 463–474. 10.1080/17476930701754118Search in Google Scholar

[13] S. Li and S. Stević, Products of Volterra type operator and composition operator from H and Bloch spaces to Zygmund spaces, J. Math. Anal. Appl. 345 (2008), no. 1, 40–52. 10.1016/j.jmaa.2008.03.063Search in Google Scholar

[14] S. Li and S. Stević, Riemann–Stieltjes operators between different weighted Bergman spaces, Bull. Belg. Math. Soc. Simon Stevin 15 (2008), no. 4, 677–686. 10.36045/bbms/1225893947Search in Google Scholar

[15] S. Li and S. Stević, Riemann-Stieltjes operators between mixed norm spaces, Indian J. Math. 50 (2008), no. 1, 177–188. Search in Google Scholar

[16] S. Li and S. Stević, Products of integral-type operators and composition operators between Bloch-type spaces, J. Math. Anal. Appl. 349 (2009), no. 2, 596–610. 10.1016/j.jmaa.2008.09.014Search in Google Scholar

[17] S. Naik, Generalized Cesáro operators on mixed norm spaces, J. Indian Acad. Math. 31 (2009), no. 1, 295–306. Search in Google Scholar

[18] S. Naik, Generalized Cesàro operators on certain function spaces, Ann. Polon. Math. 98 (2010), no. 2, 189–199. 10.4064/ap98-2-6Search in Google Scholar

[19] S. Naik, Cesáro type operators on spaces of analytic functions, Filomat 25 (2011), no. 4, 85–97. 10.2298/FIL1104085NSearch in Google Scholar

[20] A. G. Siskakis, Semigroups of composition operators in Bergman spaces, Bull. Austral. Math. Soc. 35 (1987), no. 3, 397–406. 10.1017/S0004972700013381Search in Google Scholar

[21] K. Stempak, Cesàro averaging operators, Proc. Roy. Soc. Edinburgh Sect. A 124 (1994), no. 1, 121–126. 10.1017/S030821050002922XSearch in Google Scholar

[22] S. Stević, Cesàro averaging operators, Math. Nachr. 248/249 (2003), 185–189. 10.1002/mana.200310013Search in Google Scholar

[23] N. M. Temme, Special Functions, John Wiley & Sons, New York, 1996. 10.1002/9781118032572Search in Google Scholar

[24] J. Xiao, Cesàro-type operators on Hardy, BMOA and Bloch spaces, Arch. Math. (Basel) 68 (1997), no. 5, 398–406. 10.1007/s000130050072Search in Google Scholar

Received: 2018-10-22
Accepted: 2019-11-12
Published Online: 2020-09-01
Published in Print: 2020-12-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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