Abstract
Let Ω be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA 2ν (Ω) the standard weighted Bergman space of holomorphic functions on Ω square-integrable with respect to the measureh(z, z) ν−p dz. 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 onA 2ν (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.
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The author's research was supported by GA AV ČR grant A1019701 and GA ČR grant 201/96/0411.