Compact Toeplitz operators via the Berezin transform on bounded symmetric domains
- Cite this article as:
- Engliš, M. Integr equ oper theory (1999) 33: 426. doi:10.1007/BF01291836
Let Ω be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andAν2 (Ω) the standard weighted Bergman space of holomorphic functions on Ω square-integrable with respect to the measureh(z, z)ν−pdz. Extending the recent result of Axler and Zheng for Ω=D, ν=p=2 (the unweighted Bergman space on the unit disc), we show that ifS is a finite sum of finite products of Toeplitz operators onAν2 (Ω) and ν is sufficiently large, thenS is compact if and only if the Berezin transform\(\bar S\) ofS tends to zero asz approaches ∂Ω. An analogous assertion for the Fock space is also obtained.