Abstract
In this paper, we investigate the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a class of operators that we call radial operators, an oscillation criterion and diagonal are sufficient conditions under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit sphere. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L 1(B n ) symbol.
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References
Ahern P.: On the range of the Berezin transform. J. Funct. Anal. 215, 206–216 (2004)
Ahern P., Čučković Ž.: A theorem of Brown-Halmos type for Bergman space Toeplitz operators. J. Funct. Anal. 187, 200–210 (2001)
Ahern P., Zheng D.: Compact operators via the Berezin transform. Indiana Univ. Math. J. 47, 387–400 (1998)
Ahern P., Zheng D.: The Berezin transform on the Toeplitz algebra. Stud. Math. 127, 113–130 (1998)
Čučković Ž., Rao N.V.: Mellin transform, monomial symbols and commuting Toeplitz operators. J. Funct. Anal. 154(1), 195–214 (1998)
Dong, X.T., Zhou, Z.H.: Algebraic properties of Toeplitz operators with separately quasihomogeneous symbols on the Bergman space of the unit ball. J. Oper. Theory (in press)
Dong X.T., Zhou Z.H.: Products of Toeplitz operators on the harmonic Bergman space. Proc. Am. Math. Soc. 138(5), 1765–1773 (2010)
Englis M.: Function invariant under the Berezin transform. J. Funct. Anal. 121, 223–254 (1994)
Hardy G.H.: Divergent Series. Clarendon Press, Oxford (1949)
Korenblum B., Zhu K.H.: An application of Tauberian theorems to Toeplitz operators. J. Oper. Theory 33, 353–361 (1995)
Louhichi I., Strouse E., Zakariasy L.: Products of Toeplitz operators on the Bergman space. Integ. Eq. Oper. Theory 54, 525–539 (2006)
Louhichi I., Zakariasy L.: On Toeplitz operators with quasihomogeneous symbols. Arch. Math. 85, 248–257 (2005)
Luecking D.: Trace ideal criteria for Toeplitz operators. J. Funct. Anal. 73, 345–368 (1987)
Rudin, W.: Function Theory in the Unit Ball of \({\mathbb{C}^{n}}\) . Springer, Berlin (1980)
Stroethoff K.: The Berezin transform and operators on spaces of analytic functions. Banach Center Publ. 38, 361–380 (1997)
Stroethoff K.: Compact Toeplitz operators on Bergman space. Math. Proc. Cambridge Philos Soc. 124, 151–160 (1998)
Suarez D.: The essential norm of operators in the toeplitz algebra on A p(B n ). Indiana Univ. Math. J. 56(5), 2185–2232 (2007)
Vasilevski N.: Toeplitz operator on the Bergman spaces: inside-the-dommain effects. Contemp. Math. 289, 79–146 (2001)
Vasilevski N.: Bergman space structure, commutative algebras of Toeplitz operators and hyperbolic geometry. Integ. Eq. Oper. Theory 46, 235–251 (2003)
Zhou Z.H., Dong X.T.: Algebraic properties of Toeplitz operators with radial symbols on the Bergman space of the unit ball. Integ. Eq. Oper. Theory 64, 137–154 (2009)
Zhu K.H.: Positive Toeplitz operators on weighted Bergman space of bounded dymmetric domains. J. Oper. Theory 20, 329–357 (1988)
Zhu K.H.: Operator Theory in Function Space. Marcel Dekker, New York (1990)
Zhu, K.H.: Spaces of Holomorphic Functions in the Unit Ball, GTM 226. Springer, Berlin (2004)
Zorboska N.: The Berezin transform and radial operators. Proc. Am. Math. Soc. 131, 793–800 (2002)
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Communicated by H. Turgay Kaptanoglu.
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Zhou, ZH., Chen, WL. & Dong, XT. The Berezin Transform and Radial Operators on the Bergman Space of the Unit Ball. Complex Anal. Oper. Theory 7, 313–329 (2013). https://doi.org/10.1007/s11785-011-0145-2
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DOI: https://doi.org/10.1007/s11785-011-0145-2