Abstract In this paper we investigate when a finite sum of products of two Toeplitz operators with quasihomogeneous symbols is a finite rank perturbation of another Toeplitz operator on the Bergman space. We discover a noncommutative convolution ⋄ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if F j , G j ( 1 ≤ j ≤ N ) are polynomials of z and z ¯ then ∑ j = 1 N T F j T G j − T H is a finite rank operator for some L 1 -function H if and only if ∑ j = 1 N F j ⋄ G j belongs to L 1 and H = ∑ j = 1 N F j ⋄ G j . In the case F j 's are holomorphic and G j 's are conjugate holomorphic, it is shown that H is a solution to a system of first order partial differential equations with a constraint.

中文翻译：

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伯格曼空间上托普利茨乘积的有限秩扰动

摘要 本文研究了两个具有拟齐次符号的Toeplitz算子的乘积的有限和何时是Bergman空间上另一个Toeplitz算子的有限秩摄动。我们在拟齐次函数的空间上发现了一个非对易卷积 ⋄ 并将其用于解决问题。我们的主要结果表明，如果 F j , G j ( 1 ≤ j ≤ N ) 是 z 和 z ¯ 的多项式，那么 ∑ j = 1 NTF j TG j − TH 是某个 L 1 函数 H 的有限秩算子，如果并且仅当 ∑ j = 1 NF j ⋄ G j 属于 L 1 且 H = ∑ j = 1 NF j ⋄ G j 。在 F j 是全纯且 G j 是共轭全纯的情况下，表明 H 是具有约束的一阶偏微分方程组的解。