Schatten(-Herz) class extended Cesàro operators on Bergman spaces in the unit ball of 𝐂ⁿ

Hu, Zhangjian; Tang, Xiaomin

doi:10.1090/S0002-9939-10-10365-7

Schatten(-Herz) class extended Cesàro operators on Bergman spaces in the unit ball of ${\mathbf {C}}^n$
HTML articles powered by AMS MathViewer

by Zhangjian Hu and Xiaomin Tang

Proc. Amer. Math. Soc. 138 (2010), 2803-2814

DOI: https://doi.org/10.1090/S0002-9939-10-10365-7

Published electronically: March 26, 2010

PDF | Request permission

Abstract:

For a large class of weights $\varphi$, we characterize the holomorphic symbols $g$, for which the extended Cesàro operators $T_g$ acting on the weighted Bergman space $A^2_\varphi ({\mathbf B})$ are in the Schatten (or Schatten-Herz) class in the unit ball of $\mathbf {C}^n$.

References

Alexandru Aleman and Joseph A. Cima, An integral operator on $H^p$ and Hardy’s inequality, J. Anal. Math. 85 (2001), 157–176. MR 1869606, DOI 10.1007/BF02788078
Alexandru Aleman and Aristomenis G. Siskakis, An integral operator on $H^p$, Complex Variables Theory Appl. 28 (1995), no. 2, 149–158. MR 1700079, DOI 10.1080/17476939508814844
Alexandru Aleman and Aristomenis G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), no. 2, 337–356. MR 1481594, DOI 10.1512/iumj.1997.46.1373
Boo Rim Choe, Hyungwoon Koo, and Young Joo Lee, Positive Schatten(-Herz) class Toeplitz operators on the half-space, Potential Anal. 27 (2007), no. 1, 73–100. MR 2314190, DOI 10.1007/s11118-007-9044-6
Boorim Choe, Hyungwoon Koo, and Kyunguk Na, Positive Toeplitz operators of Schatten-Herz type, Nagoya Math. J. 185 (2007), 31–62. MR 2301457, DOI 10.1017/S0027763000025733
Zhangjian Hu, Extended Cesàro operators on mixed norm spaces, Proc. Amer. Math. Soc. 131 (2003), no. 7, 2171–2179. MR 1963765, DOI 10.1090/S0002-9939-02-06777-1
Zhangjian Hu, Extended Cesàro operators on Bergman spaces, J. Math. Anal. Appl. 296 (2004), no. 2, 435–454. MR 2075175, DOI 10.1016/j.jmaa.2004.01.045
Eugenio Hernández and Dachun Yang, Interpolation of Herz spaces and applications, Math. Nachr. 205 (1999), 69–87. MR 1709163, DOI 10.1002/mana.3212050104
X. F. Lv, Z. J. Hu, Schatten(-Herz) classes Toeplitz operators on weighted Bergman spaces. (Chinese), Acta Math. Sinica, 53(4) (2010), in press.
Maribel Loaiza, Marcos López-García, and Salvador Pérez-Esteva, Herz classes and Toeplitz operators in the disk, Integral Equations Operator Theory 53 (2005), no. 2, 287–296. MR 2187174, DOI 10.1007/s00020-004-1320-x
Songxiao Li and Stevo Stević, Compactness of Riemann-Stieltjes operators between $F(p,q,s)$ spaces and $\alpha$-Bloch spaces, Publ. Math. Debrecen 72 (2008), no. 1-2, 111–128. MR 2376863, DOI 10.5486/pmd.2008.3803
Daniel H. Luecking, Trace ideal criteria for Toeplitz operators, J. Funct. Anal. 73 (1987), no. 2, 345–368. MR 899655, DOI 10.1016/0022-1236(87)90072-3
Daniel H. Luecking, Finite rank Toeplitz operators on the Bergman space, Proc. Amer. Math. Soc. 136 (2008), no. 5, 1717–1723. MR 2373601, DOI 10.1090/S0002-9939-07-09119-8
Walter Rudin, Function theory in the unit ball of $\textbf {C}^{n}$, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 241, Springer-Verlag, New York-Berlin, 1980. MR 601594
Xiaomin Tang, Extended Cesàro operators between Bloch-type spaces in the unit ball of $\mathbf C^n$, J. Math. Anal. Appl. 326 (2007), no. 2, 1199–1211. MR 2280974, DOI 10.1016/j.jmaa.2006.03.082
Jie Xiao, Riemann-Stieltjes operators on weighted Bloch and Bergman spaces of the unit ball, J. London Math. Soc. (2) 70 (2004), no. 1, 199–214. MR 2064758, DOI 10.1112/S0024610704005484
Ke He Zhu, Positive Toeplitz operators on weighted Bergman spaces of bounded symmetric domains, J. Operator Theory 20 (1988), no. 2, 329–357. MR 1004127
Ke He Zhu, Operator theory in function spaces, Monographs and Textbooks in Pure and Applied Mathematics, vol. 139, Marcel Dekker, Inc., New York, 1990. MR 1074007
Kehe Zhu, Schatten class Toeplitz operators on weighted Bergman spaces of the unit ball, New York J. Math. 13 (2007), 299–316. MR 2357717