作为交织算子的单演演算

Poly slice monogenic functions, Cauchy formulas and the PS-functional calculus

Since 2006 the theory of slice hyperholomorphic functions and the related spectral theory on the S-spectrum have had a very fast development. This new spectral theory based on the S-spectrum has applications, for example, in the formulation of quaternionic quantum mechanics, in Schur analysis and in fractional diffusion problems. In this paper we introduce and study the theory of poly slice monogenic functions, also proving some Cauchy type integral formulas. Then we introduce the associated functional calculus, called PS-functional calculus, which is the polyanalytic version of the S-function…

A polyanalytic functional calculus of order 2 on the S-spectrum

The Fueter theorem provides a two step procedure to build an axially monogenic function, i.e. a null-solutions of the Cauchy-Riemann operator in $ \mathbb{R}^4$, denoted by $ \mathcal{D}$. In the first step a holomorphic function is extended to a slice hyperholomorphic function, by means of the so-called slice operator. In the second step a monogenic function is built by applying the Laplace operator in four real variables ($Δ$) to the slice hyperholomorphic function. In this paper we use the factorization of the Laplace operator, i.e. $Δ= \mathcal{\overline{D}} \mathcal{D}$ to split the previ…

The fine structure of the spectral theory on the $S$-spectrum in dimension five

Holomorphic functions play a crucial role in operator theory and the Cauchy formula is a very important tool to define functions of operators. The Fueter-Sce-Qian extension theorem is a two steps procedure to extend holomorphic functions to the hyperholomorphic setting. The first step gives the class of slice hyperholomorphic functions; their Cauchy formula allows to define the so-called $S$-functional calculus for noncommuting operators based on the $S$-spectrum. In the second step this extension procedure generates monogenic functions; the related monogenic functional calculus, based on the …

Properties of a polyanalytic functional calculus on the $S$-spectrum

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 sl…

谱作为谓词演算的支撑

Spectrum as the Support of Functional Calculus

本文利用SL 2（R）的表示论研究了解析谓词演算的新定义.我们避免了任何使用其代数同态性质，并取代它的要求是一个交织算子。给出了谱的相关概念和谱映射定理。通过一个简单的非正规n × n矩阵的微积分和谱的例子说明了这种构造。关键词：谓词演算，谱，交织算子，谱映射定理，喷流空间。

$S$-谱上多调和与多解析Clifford精细结构的函数与算子

Functions and operators of the polyharmonic and polyanalytic Clifford fine structures on the $S$-spectrum

本文研究$S$-谱上Clifford精细结构中的多调和与多解析函数及其函数演算，并证明单演函数演算与$F$-函数演算等价。