论文标题
伯格曼空间上Toeplitz产品的有限等级扰动
Finite Rank Perturbations of Toeplitz Products on the Bergman Space
论文作者
论文摘要
在本文中，我们调查了两个带有准偶氮符号的Toeplitz操作员的产品的有限总和是对伯格曼空间上另一家Toeplitz运营商的有限级扰动。我们在准综合功能的空间上发现了一个非交通性卷积$ \钻石$，并将其用于解决问题。我们的主要结果表明，如果$ f_j，g_j $（$ 1 \ leq j \ leq n $）是$ z $的多项式和$ \ bar {z} $，则$ \ sum_ {j = 1}^{n}^{n} t_ { $ h $ if，仅当$ \ sum_ {j = 1}^{n} f_j \ diamond g_j $属于$ l^1 $和$ h = \ sum_ {j = 1}^{n} f_j \ diamond g_j $。在$ f_j $的情况下，$ g_j $是共轭全体形状，这表明$ h $是对具有约束的一阶偏微分方程系统的解决方案。
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 $\diamond$ on the space of quasihomogeneous functions and use it in solving the problem. Our main results show that if $F_j, G_j$ ($1\leq j\leq N$) are polynomials of $z$ and $\bar{z}$ then $\sum_{j=1}^{N}T_{F_j}T_{G_j}-T_{H}$ is a finite rank operator for some $L^{1}$-function $H$ if and only if $\sum_{j=1}^{N}F_j\diamond G_j$ belongs to $L^1$ and $H=\sum_{j=1}^{N}F_j\diamond G_j$. 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.
