The paper gives some characterization theorems for the compact composition operator on some function spaces over the unit ball Bn in C n . Especially, it gives a characterization for compact composition operators on BMOA(Bn), which generalizes a result proved by Bourdon, Cima and Matheson for the case n = 1.
<abstract><p>Let $ R_{{\cal A}} $ be the Cartan classical domains of type Ⅲ and Ⅳ, and $ \Delta_g $ is assumed to be the Laplace-Beltrami operator associated to the Bergman metric $ g $ on $ R_{{\cal A}} $. In this paper, we derive an estimate for $ \lambda_1(\Delta_g) $, which is the bottom of the spectrum of $ \Delta_g $ on $ R_{{\cal A}} $.</p></abstract>
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