, Volume 33, Issue 4, pp 426-455

Compact Toeplitz operators via the Berezin transform on bounded symmetric domains

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Abstract

Let Ω be an irreducible bounded symmetric domain of genusp, h(x, y) its Jordan triple determinant, andA ν 2 (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 ν 2 (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.

The author's research was supported by GA AV ČR grant A1019701 and GA ČR grant 201/96/0411.