论文标题
径向操作员在多芯片加权的伯格曼空间上
Radial operators on polyanalytic weighted Bergman spaces
论文作者
论文摘要
令$μ_α$为单位磁盘上的Lebesgue平面度量，带有径向重量$ \ frac {α+1}π（1- | z |^2）^α$。用$ \ MATHCAL {a}^{2} _ {n} $表示$ n $ - 分析函数的空间在设备磁盘上，相对于$μ__α$，可以进行方形 - 积分。扩展了Ramazanov（1999，2002）的结果，我们解释说磁盘多项式（1975年由Koornwinder在1975年研究，2005年的Wünsche）形成了$ \ Mathcal {a} a}^{2} {2} _ {n} $的正常基础。使用此基础，我们将$ \ Mathcal {a}^{2} _ {n} $的傅立叶分解用于与不同频率关联的子空间的正交总和。这导致了径向运算符的von Neumann代数的分解，该代数以$ \ Mathcal {a}^{2} _n $的直接总和为某些矩阵代数的总和。换句话说，所有径向运算符均表示为矩阵序列。特别是，我们以这种形式表示具有有界径向符号的Toeplitz运算符，作用于$ \ Mathcal {a}^{2} _n $。此外，使用Engliš（1996）的想法，我们表明，所有具有有限生成符号的toeplitz运算符的集合在$ \ Mathcal {b}中并不是弱致密的（\ Mathcal {A}^a}^{2} {2} _n）$。
Let $μ_α$ be the Lebesgue plane measure on the unit disk with the radial weight $\frac{α+1}π(1-|z|^2)^α$. Denote by $\mathcal{A}^{2}_{n}$ the space of the $n$-analytic functions on the unit disk, square-integrable with respect to $μ_α$. Extending the results of Ramazanov (1999, 2002), we explain that disk polynomials (studied by Koornwinder in 1975 and Wünsche in 2005) form an orthonormal basis of $\mathcal{A}^{2}_{n}$. Using this basis, we provide the Fourier decomposition of $\mathcal{A}^{2}_{n}$ into the orthogonal sum of the subspaces associated with different frequencies. This leads to the decomposition of the von Neumann algebra of radial operators, acting in $\mathcal{A}^{2}_n$, into the direct sum of some matrix algebras. In other words, all radial operators are represented as matrix sequences. In particular, we represent in this form the Toeplitz operators with bounded radial symbols, acting in $\mathcal{A}^{2}_n$. Moreover, using ideas by Engliš (1996), we show that the set of all Toeplitz operators with bounded generating symbols is not weakly dense in $\mathcal{B}(\mathcal{A}^{2}_n)$.
