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 \((A_1,\ldots ,A_n)\). 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. We finally discuss how to define the fractional Fourier’s law for nonhomogeneous materials using the spectral theory on the S-spectrum.

本文的目的是概述与维度大于一的全纯性概念相关的光谱理论。第一个自然扩展是几个复变量的理论，其柯西公式用于定义操作符\((A_1,\ldots ,A_n)\) 的n元组的全纯泛函演算。第二种方法是考虑四元数或副向量变量的超全纯函数。在这种情况下，根据 Fueter-Sce-Qian 映射定理，我们有两种不同的超全纯函数概念，称为切片超全纯函数和单基因函数。切片超全纯函数生成基于S的谱理论-spectrum 而单基因函数诱导基于单基因谱的谱理论。通过F泛函演算，两种超全纯谱理论之间也存在有趣的关系。这两种超全纯光谱理论具有不同的互补应用。我们最后讨论了如何使用S谱上的谱理论来定义非均质材料的分数傅立叶定律。