加权Dirichlet空间上的Schatten类复合算子

Schatten Class of Composition Operators on Weighted Dirichlet Spaces

本文利用一种积分平均函数给出了加权Ｄｉｒｉｃｈｌｅｔ空间Ｄ_α。（α＞ －１）上的复合算子Ｃ_ψ为Ｓｃｈａｔｔｅｎｐ－类算子的充要条件．此结果包含了过去已有的关于Ｈａｒｄｙ空间及加权Ｂｅｒｇｍａｎ空间Ａ_α（α＞－１）上的复合算子的已有结论．主要定理是：设ｐ＞０，α＞一１，ψεＤ_ａ，则Ｃ_ψ为Ｄ_α上的Ｓｃｈａｔｔｅｎ ｐ－类算子的充要条件是存在δ＞０，使得积分平均函数Φ_δ（ｚ）＝λ（Ｄ（ｚ，δ））＝１ ｉｎｔｅｇｒａｌ ｆｏｒｍ ｎ=Ｄ（ｚ，δ）τψ，α（ω）ｄ－λ（ω）属于Ｌ_２~ｐ（ｄｖ），其中Ｄ（ｚ，δ）为伪双曲圆盘，τψ，α为Ｃψ关于Ｄ_α的确定函数；ｄｖ（ｚ）＝（１－｜ｚ｜~２）~－２ｄλ（ｚ），ｄλ为Ｄ上的就范面积测度．

In this paper, we characterize bounded, compact and Schatten class composition operators on weighted Dirichlet spaces. The method involves integral averages of the determining function of the operator, and the connection between composition operators on Dirichlet spaces and Toeplitz operators on Bergman spaces. The main result is as following: If p>0,α＞-1, and αε D_α(α>-1), then C_ψbelongs to S_p(D_α) if and only if there is a constant δ>0 such that the integral averages Φ_δ of τψ,α defined by belongs to L_2~p (dv). Where D(z,δ) is a pseudohyperbolic disk, τψ,α is the determining function for C_ψon D_α and dv (z) = (1 -|z|~2) ^-2dλ(z).