Compact Toeplitz Operators for Weighted Bergman Spaces on Bounded Symmetric Domains

摘要

Let Ω ? ?_d be an irreducible bounded symmetric domain of type (r, a, b) in its Harish-Chandra realization. We study Toeplitz operators T~ν_g with symbol g acting on the standard weighted Bergman space H~2_ν over Ω with weight ν. Under some conditions on the weights ν and ν_0 we show that there exists C(ν, ν_0) > 0, such that the Berezin transform ?_(ν0) of g with respect to H~2_(ν0) satisfies:for all g in a suitable class of symbols containing L~∞(Ω). As a consequence we apply a result in Engli? (Integr Equ Oper theory 33:426-455, 1999), to prove that the compactness of T~ν_g is independent of the weight ν, whenever g ∈ L~∞ (Ω) and ν > C where C is a constant depending on (r, a, b).