Commuting Toeplitz Operators with Harmonic Symbols on A^2(overline{mathbb{D}},dmu)$
首发时间:2009-01-06
Abstract:For any given symmetric measure u on the closed unit disk $overline{mathbb{D}}$, by characterizingthe relationship between the Berezin transform and harmonic functions, we obtain that if both $f$ and $g$ are bounded harmonic on $mathbb{D}$, then the Toeplitz operator $T_f$ commutes with $T_g$ on $A^2(overline{mathbb{D}},dmu)$ if and only if at least one of the following conditions holds: (1) both $f$ and $g$ are analytic on $mathbb{D}$; (2) both $overline{f}$ and $overline{g}$ are analytic on $mathbb{D}$;(3) there exist constants $a,binmathbb{C}$, not both 0, such that $af+bg$ is constant on $mathbb{D}$.
keywords: symmetric measure Toeplitz operator Berezin transform
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A^2(overline{mathbb{D}},dmu)上的调和符号的Toeplitz算子的交换性
摘要:对闭单位圆盘$overline{mathbb{D}}$上的任一对称测度 $mu$,通过刻画Berezin 变换和调和函数之间的关系, 我们得到了下面的结果.若$f$和$g$ 在开单位圆盘 $mathbb{D}$上都是有界调和的, 则$A^2overline{mathbb{D}},dmu)$上的 Toeplitz 算子$T_f$ 和$T_g$ 交换当且仅当下列条件至少有一个成立:(1)$f$和$g$ 在$mathbb{D}$上都解析;(2)$f$和$g$在$mathbb{D}$上都余解析;(3)存在不全为零的常数$a,b in mathbb{C}$使得$af+bg$在$ mathbb{D}$上是常值函数.
关键词: 对称测度 Toeplitz算子 Berzin变换
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