Weighted composition operators on weighted Bergman spaces induced by doubling weights

  • Juntao Du
  • Songxiao Li
  • Yecheng Shi

Abstract

In this paper, we investigate the boundedness, compactness, essential norm and the Schatten class of weighted composition operators $uC_\varphi $ on Bergman type spaces $A_\omega ^p $ induced by a doubling weight ω. Let $X=\{u\in H(\mathbb{D} ): uC_\varphi \colon A_\omega ^p\to A_\omega ^p\ \text {is bounded}\}$. For some regular weights ω, we obtain that $X=H^\infty $ if and only if ϕ is a finite Blaschke product.

References

Aleman, A. and Siskakis, A. G., Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. https://doi.org/10.1512/iumj.1997.46.1373

Contreras, M. D. and Hernández-D\'ıaz, A. G., Weighted composition operators between different Hardy spaces, Integral Equations Operator Theory 46 (2003), no. 2, 165–188. https://doi.org/10.1007/s000200300023

Cowen, C. C. and MacCluer, B. D., Composition operators on spaces of analytic functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1995.

\u Cučković, Ž. and Zhao, R., Weighted composition operators on the Bergman space, J. London Math. Soc. (2) 70 (2004), no. 2, 499–511. https://doi.org/10.1112/S0024610704005605

\u Cučković, Ž. and Zhao, R., Weighted composition operators between different weighted Bergman spaces and different Hardy spaces, Illinois J. Math. 51 (2007), no. 2, 479–498.

Demazeux, R., Essential norms of weighted composition operators between Hardy spaces $H^p$ and $H^q$ for $1\leq p, q\leq \infty $, Studia Math. 206 (2011), no. 3, 191–209. https://doi.org/10.4064/sm206-3-1

Duren, P. and Schuster, A., Bergman spaces, Mathematical Surveys and Monographs, vol. 100, American Mathematical Society, Providence, RI, 2004. https://doi.org/10.1090/surv/100

Peláez, J. A., Small weighted Bergman spaces, in “Proceedings of the Summer School in Complex and Harmonic Analysis, and Related Topics”, Publ. Univ. East. Finl. Rep. Stud. For. Nat. Sci., vol. 22, Univ. East. Finl., Fac. Sci. For., Joensuu, 2016, pp. 29–98.

Peláez, J. A. and Rättyä, J., Weighted Bergman spaces induced by rapidly increasing weights, Mem. Amer. Math. Soc. 227 (2014), no. 1066, vi+124.

Peláez, J. A. and Rättyä, J., Embedding theorems for Bergman spaces via harmonic analysis, Math. Ann. 362 (2015), no. 1-2, 205–239. https://doi.org/10.1007/s00208-014-1108-5

Peláez, J. A. and Rättyä, J., Trace class criteria for Toeplitz and composition operators on small Bergman spaces, Adv. Math. 293 (2016), 606–643. https://doi.org/10.1016/j.aim.2016.02.017

Peláez, J. A. and Rättyä, J., Two weight inequality for Bergman projection, J. Math. Pures Appl. (9) 105 (2016), no. 1, 102–130. https://doi.org/10.1016/j.matpur.2015.10.001

Peláez, J. A., Rättyä, J., and Sierra, K., Embedding Bergman spaces into tent spaces, Math. Z. 281 (2015), no. 3-4, 1215–1237. https://doi.org/10.1007/s00209-015-1528-2

Peláez, J. A., Rättyä, J., and Sierra, K., Berezin transform and Toeplitz operators on Bergman spaces induced by regular weights, J. Geom. Anal. 28 (2018), no. 1, 656–687. https://doi.org/10.1007/s12220-017-9837-9

Rudin, W., Function theory in polydiscs, W. A. Benjamin, Inc., New York-Amsterdam, 1969.

Zhao, L. K. and Hou, S. Z., A note on weighted composition operators on the weighted Bergman space, Acta Math. Sin. (Engl. Ser.) 31 (2015), no. 6, 947–952. https://doi.org/10.1007/s10114-015-4473-4

Zhu, K., Duality of Bloch spaces and norm convergence of Taylor series, Michigan Math. J. 38 (1991), no. 1, 89–101. https://doi.org/10.1307/mmj/1029004264

Zhu, K., Operator theory in function spaces, second ed., Mathematical Surveys and Monographs, vol. 138, American Mathematical Society, Providence, RI, 2007. https://doi.org/10.1090/surv/138
Published
2020-09-03
How to Cite
Du, J., Li, S., & Shi, Y. (2020). Weighted composition operators on weighted Bergman spaces induced by doubling weights. MATHEMATICA SCANDINAVICA, 126(3), 519-539. https://doi.org/10.7146/math.scand.a-119741
Section
Articles