论文标题
$ s $ spectrum上多芯分析功能积分的属性
Properties of a polyanalytic functional calculus on the $S$-spectrum
论文作者
论文摘要
Fueter映射定理提供了一种建设性的方法，可以将一个复合变量的圆锥形函数扩展到单基因函数，即在$ \ Mathbb {r}^4 $中以$ \ Mathcal {D}表示的$ \ Mathbb {r}^4 $中的广义Cauchy-Riemann运算符的无溶液。该定理分为两个步骤。在第一步中，全体形函数扩展到切片的超晶函数。这些类型的功能的Cauchy公式是$ s $功能演算的起点。在第二步中，通过将拉普拉斯操作员分别以四个真实变量（即$δ$）应用于切片的超酚形态函数来获得单基因函数。我们在本文中研究的多序分析函数演算基于$δ= \ Mathcal {d} \ Mathcal {\ overline {d}} $的分解。与其直接将拉普拉斯操作员应用于切片的超酚形函数，我们首先应用操作员$ \ MATHCAL {\ overline {d}} $，我们获得了订单2的多序分析函数。我们可以以积分形式表示这种类型的函数，然后可以在$ s $ spectrum上定义多序分析函数积分。本文的主要目的是显示该功能演算的主要特性。特别是，我们研究了一个适合证明产品规则并生成Riesz投影仪的分解方程。
The Fueter mapping theorem gives a constructive way to extend holomorphic functions of one complex variable to monogenic functions, i.e., null solutions of the generalized Cauchy-Riemann operator in $\mathbb{R}^4$, denoted by $\mathcal{D}$. This theorem is divided in two steps. In the first step a holomorphic function is extended to a slice hyperholomorphic function. The Cauchy formula for these type of functions is the starting point of the $S$-functional calculus. In the second step a monogenic function is obtained by applying the Laplace operator in four real variables, namely $ Δ$, to a slice hyperholomorphic function. The polyanalytic functional calculus, that we study in this paper, is based on the factorization of $Δ= \mathcal{D} \mathcal{\overline{D}}$. Instead of applying directly the Laplace operator to a slice hyperholomorphic function we apply first the operator $ \mathcal{\overline{D}}$ and we get a polyanalytic function of order 2, i..e, a function that belongs to the kernel of $ \mathcal{D}^2$. We can represent this type of functions in an integral form and then we can define the polyanalytic functional calculus on $S$-spectrum. The main goal of this paper is to show the principal properties of this functional calculus. In particular, we study a resolvent equation suitable for proving a product rule and generate the Riesz projectors.
