L^2(X,A,μ)上的Schatten类复合算子的特征

Characterization of Schatten class of composition operators on L^2(X,A,μ)

设 ( X,A,μ)是σ-有限测度空间 ,Cφ 是由 X上的非奇异可测函数φ导出的 Hilbert空间 L2 ( X,A,μ)上的复合算子 ,hφ 表示测度 μ φ- 1关于 μ的 Radon-Nikodym导数 .证明了 :Cφ∈ Sp( p≥ 1 )的充要条件是 X为纯原子测度空间且 ∑∞n=1hp2φ ( n) <∞ .

Let (X,A,μ) be a σ finite measure space and C\-φ a composition operator on Hilbert space L\+2(X,A,μ) induced by the holomorghic self mapping φ from D into itself, and let h\-φ be Radon Nikodym derivative of the measure μφ\+\{-1\} on the measure μ .In the paper,it was showed that C\-φ∈S\-p(p≥1) ,the Schatten p class,if and only if X is a pure atomic measure space and ∑∞n=1h\+\+\{p2\}\-φ(n)<∞.