多解析Fock核的水平Fourier变换
Horizontal Fourier transform of the polyanalytic Fock kernel
AI总结：
本文研究多解析Fock空间中与水平Weyl平移可交换的算子代数，通过计算水平Fourier变换将空间分解为Hermite函数乘积，并证明其等距同构于向量函数空间，算子代数同构于矩阵函数代数。
摘要：
设$n,m\ge 1$且$\alpha>0$。我们用$\mathcal{F}_{\alpha,m}$表示$m$-解析Bargmann--Segal--Fock空间，即定义在$\mathbb{C}^n$上、关于Gauss权$\exp(-\alpha|z|^2)$平方可积的所有$m$-解析函数构成的Hilbert空间。我们研究作用在$\mathcal{F}_{\alpha,m}$中且与所有“水平”Weyl平移（即与$\mathbb{R}^n$中元素相关联的Weyl酉算子）可交换的有界线性算子构成的von Neumann代数$\mathcal{A}$。$\mathcal{F}_{1,m}$的再生核由Youssfi [Polyanalytic reproducing kernels in $\mathbb{C}^n$, Complex Anal. Synerg., 2021, 7, 28]计算得出。将$\mathcal{F}_{\alpha,m}$的元素乘以适当的权函数，我们将该空间变换为另一个再生核Hilbert空间，其核$K$在水平平移下不变。利用Laguerre函数与Hermite函数之间众所周知的Fourier联系，我们计算$K$在“水平方向”上的Fourier变换，并将其分解为$d$个Hermite函数乘积之和，其中$d=\binom{n+m-1}{n}$。最后，应用Herrera-Yañez、Maximenko、Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31]提出的方案，我们证明$\mathcal{F}_{\alpha,m}$等距同构于向量函数空间$L^2(\mathbb{R}^n)^d$，且$\mathcal{A}$等距同构于矩阵函数代数$L^\infty(\mathbb{R}^n)^{d\times d}$。
英文摘要：
Let $n,m\ge 1$ and $α>0$. We denote by $\mathcal{F}_{α,m}$ the $m$-analytic Bargmann--Segal--Fock space, i.e., the Hilbert space of all $m$-analytic functions defined on $\mathbb{C}^n$ and square integrables with respect to the Gaussian weight $\exp(-α|z|^2)$. We study the von Neumann algebra $\mathcal{A}$ of bounded linear operators acting in $\mathcal{F}_{α,m}$ and commuting with all ``horizontal'' Weyl translations, i.e., Weyl unitary operators associated to the elements of $\mathbb{R}^n$. The reproducing kernel of $\mathcal{F}_{1,m}$ was computed by Youssfi [Polyanalytic reproducing kernels in $\mathbb{C}^n$, Complex Anal. Synerg., 2021, 7, 28]. Multiplying the elements of $\mathcal{F}_{α,m}$ by an appropriate weight, we transform this space into another reproducing kernel Hilbert space whose kernel $K$ is invariant under horizontal translations. Using the well-known Fourier connection between Laguerre and Hermite functions, we compute the Fourier transform of $K$ in the ``horizontal direction'' and decompose it into the sum of $d$ products of Hermite functions, with $d=\binom{n+m-1}{n}$. Finally, applying the scheme proposed by Herrera-Yañez, Maximenko, Ramos-Vazquez [Translation-invariant operators in reproducing kernel Hilbert spaces, Integr. Equ. Oper. Theory, 2022, 94, 31], we show that $\mathcal{F}_{α,m}$ is isometrically isomorphic to the space of vector-functions $L^2(\mathbb{R}^n)^d$, and $\mathcal{A}$ is isometrically isomorphic to the algebra of matrix-functions $L^\infty(\mathbb{R}^n)^{d\times d}$.
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