论文标题
矢量运算符的超晶光谱理论和分数的介绍
An introduction to hyperholomorphic spectral theories and fractional powers of vector operators
论文作者
论文摘要
本文的目的是概述与大于一个的尺寸中的圆形概念相关的光谱理论。第一个自然扩展是几个复杂变量的理论，其cauchy公式用于定义$ n $ tubles $ n $ tuplase $（a_1，...，...，a_n）$的全态功能积分。第二种方法是考虑Quaternionic或Paravector变量的超晶函数。在这种情况下，通过Fueter-Sce-Qian映射定理，我们有两个不同的多晶函数概念，称为Slice超酚函数和单基因功能。切片超晶函数基于$ s $ spectrum产生光谱理论，而单基因函数诱导基于单基光谱的光谱理论。通过$ f $功能的演算，两种超晶光谱理论之间也存在一个有趣的关系。两种超晶光谱理论具有不同的和互补的应用。在这里，我们还讨论了如何定义非均匀材料的分数傅立叶定律，此类定义基于$ s $ spectrum的光谱理论。
The aim of this paper is to give an overview of the spectral theories associated with the notions of holomorphicity in dimension greater than one. A first natural extension is the theory of several complex variables whose Cauchy formula is used to define the holomorphic functional calculus for $n$-tuples of operators $(A_1,...,A_n)$. A second way is to consider hyperholomorphic functions of quaternionic or paravector variables. In this case, by the Fueter-Sce-Qian mapping theorem, we have two different notions of hyperholomorphic functions that are called slice hyperholomorphic functions and monogenic functions. Slice hyperholomorphic functions generate the spectral theory based on the $S$-spectrum while monogenic functions induce the spectral theory based on the monogenic spectrum. There is also an interesting relation between the two hyperholomorphic spectral theories via the $F$-functional calculus. The two hyperholomorphic spectral theories have different and complementary applications. Here we also discuss how to define the fractional Fourier's law for nonhomogeneous materials, such definition is based on the spectral theory on the $S$-spectrum.
