In this paper we show how the spectral theory based on the notion of S-spectrum allows us to study new classes of fractional diffusion and of fractional evolution processes. We prove new results on the quaternionic version of the \({H^\infty}\) functional calculus and we use it to define the fractional powers of vector operators. The Fourier law for the propagation of the heat in non homogeneous materials is a vector operator of the form$$T = e_{1} a(x)\partial_{x1} + e_{2} b(x)\partial_{x2} + e_{3} c(x)\partial_{x3}$$where \({e_{\ell}, {\ell} = 1, 2, 3}\) are orthogonal unit vectors, a, b, c are suitable real valued functions that depend on the space variables \({x = (x_{1}, x_{2}, x_{3})}\) and possibly also on time. In this paper we develop a general theory to define the fractional powers of quaternionic operators which contain as a particular case the operator T so we can define the non local version \({T^{\alpha}, {\rm for} \alpha \in (0, 1)}\), of the Fourier law defined by T. Our new mathematical tools open the way to a large class of fractional evolution problems that can be defined and studied using our spectral theory based on the S-spectrum for vector operators. This paper is devoted to researchers working on fractional diffusion and fractional evolution problems, partial differential equations, non commutative operator theory, and quaternionic analysis.

中文翻译：

---

S-功能演算在分数扩散过程中的应用

在本文中，我们展示了基于S谱概念的光谱理论如何使我们研究分数扩散和分数演化过程的新类别。我们在\（{H ^ \ infty} \）函数微积分的四元数形式上证明了新结果，并用它来定义矢量算子的分数幂。用于在非均质材料中传播热量的傅立叶定律是形式为$$ T = e_ {1} a（x）\ partial_ {x1} + e_ {2} b（x）\ partial_ {x2 } + e_ {3} c（x）\ partial_ {x3} $$其中\（{e _ {\ ell}，{\ ell} = 1，2，3} \）是正交单位向量a，b，c是取决于空间变量\（{x =（x_ {1}，x_ {2}，x_ {3}）} \）并可能还取决于时间的合适的实值函数。在本文中，我们开发了一种通用理论来定义四元离子算子的分数幂，其中包含作为特殊情况的算子T，因此我们可以定义非局部版本\（{T ^ {\ alpha}，{\ rm for} \ alpha \ in（0，1）} \），由T定义的傅立叶定律。我们新的数学工具为解决一类分数演化问题打开了道路，这些问题可以使用基于S的谱理论来定义和研究向量运算符的频谱。本文致力于研究分数扩散和分数演化问题，偏微分方程，非交换算子理论和四元离子分析。