Spectral Theory on the S-Spectrum for Quaternionic Operators
Spectral Theory on the S-Spectrum for Quaternionic Operators
复制标题
DOI：
10.1007/978-3-030-03074-2
复制
发表时间：
2019-01
期刊：
影响因子：
--
通讯作者：
F. Colombo;J. Gantner;D. Kimsey
中科院分区：
文献类型：
--
作者：
F. Colombo;J. Gantner;D. Kimsey
Banach和Hilbert空间中的经典算子理论受到了数学和物理中的几个问题的刺激。此外，全纯函数理论在算子理论中，特别是在算子函数的定义中起着至关重要的作用。上个世纪初，随着量子力学的提出，极大地推动了算子理论的发展，特别是Hilbert空间上无界正规算子的谱定理是最重要的成果之一。1936年，Birkhoff和von Neumann证明了量子力学可以用实数、复数和四元数来表示。因此，一个自然的问题是理解一个人应该在四元数算符理论中使用什么谱概念。直到2006年，随着四元数线性算子的S谱的发现，这个问题才得到解决，从那时起，四元数谱理论得到了迅速的发展。这本书的目的是给出基于S谱的四元数谱理论的系统基础，并提出将用于处理四元数算子论的切片超全纯函数理论。这本书讨论了四个主要主题：S泛函演算，F泛函演算，四元数谱定理，四元数谱算符理论。S泛函演算是RieszDunford泛函演算中四元数情形的自然推广，可用于定义四元数H-∞泛函演算或向量扇形算子。H-∞泛函演算在分数阶扩散过程中有重要的应用，因为它允许人们定义向量算子的分数次方，如梯度或具有非常系数的梯度算子的推广。F-泛函演算基于一个积分变换，称为积分形式的Fueter-SCE映射定理，它定义了四元数算子的Fueter-正则函数。这个演算基于切片超全纯函数和所谓的F-预解算子，它允许我们通过一个积分公式来定义一个四元数算子的函数。我们在S谱的基础上讨论了四元数正规算子的谱定理，该定理于2014年被证明，并于2016年发表。无界反自共轭算子的四元数谱定理是建立四元数的重要工具
Classical operator theory in Banach and Hilbert spaces has been stimulated by several problems in mathematics and physics. Moreover, the theory of holomorphic functions plays a crucial role in operator theory and in particular in the definition of functions of operators. A great impulse was given to the development of operator theory at the beginning of the last century when quantum mechanics was formulated; in particular, the spectral theorem for unbounded normal operators on a Hilbert space was one of the most important achievements. In 1936, Birkhoff and von Neumann showed that quantum mechanics can be formulated on real, complex, and quaternionic numbers. So a natural problem was to understand what notion of spectrum one should use in quaternionic operator theory. This problem was solved only in 2006 with the discovery of the S-spectrum for quaternionic linear operators, and since then the quaternionic spectral theory has grown rapidly. The aim of this book is to give a systematic foundation of quaternionic spectral theory based on the S-spectrum and to present the theory of slice hyperholomorphic functions, which will be used in the treatment of quaternionic operator theory.This book treats four main topics: the S-functional calculus, the F-functional calculus, the quaternionic spectral theorem, and the theory of quaternionic spectral operators. The S-functional calculus is the natural extension to the quaternionic setting of the Riesz–Dunford functional calculus, and it can be used to define the quaternionic H∞-functional calculus for quaternionic or vector sectorial operators. The H∞-functional calculus has important applications in fractional diffusion processes because it allows one to define fractional powers of vector operators such as the gradient or a generalization of the gradient operator with nonconstant coefficients. The F-functional calculus is based on an integral transform, called the Fueter-Sce mapping theorem in integral form, and it defines Fueter-regular functions of quaternionic operators. This calculus is based on slice hyperholomorphic functions and on the so-called F-resolvent operators that allow us to define, via an integral formula, functions of a quaternionic operator. We treat the spectral theorem for quaternionic normal operators based on the S-spectrum, which was proved in 2014 and published in 2016. The quaternionic spectral theorem for unbounded anti-selfadjoint operators is a very important tool for formulating quaternionic