四元数算子理论中的伪反自伴算子

The quaternionic quantum mechanics (QQM) was first suggested by Birkhoff and von Neumann in 1936. After that in 1962, Finkelstein et al. constructed the fundamental work of QQM which links the states and operator with quaternionic Hilbert space. In fact, hamiltonians of time-reversal invariant system are pseudoanti-Hermitian operators on right quaternionic Hilbert space in QQM. Hence,pseudoanti-Hermitian operators are important both in the study of non-Hermitian quaternionic quantum mechanics and operator theory. Most study of pseudoanti-Hermitian operators are under the condition of diagonalization. So there are still a lot of problems to be considered. we will study several problems of pseudoanti-Hermiticity of operators on quaternionic Hilbert space. In this project, we will study several problems of pseudoanti-Hermiticity of operators by spectral analysis and space decomposition. (1) Study the structure of pseudoanti-Hermitian operators by their spherical spectrum sphere pseudoanti-Hermiticity of quaternionic bounded linear operators starting. (2) Characterize the anticommutant algebras of pseudoanti-Hermitian operators, and establish the connection between pseudoanti-Hermitian operators and their anticommutant algebras.(3) The diagonalization of pseudoanti-Hermitian operators and their applications on pseudoanti-Hermitian operators.

继1936年Birkhoff和von Neumann提出四元数框架下的量子力学之后，Finkelstein等构建了四元数量子力学的数学结构，将量子力学中的态对应四元数Hilbert空间中的向量。在量子力学中, 时间反演不变系统的哈密顿量是四元数Hilbert空间中的伪反自伴算子，因此伪反自伴算子既有深刻的物理背景，又是四元数算子理论中值得研究的一类非自伴算子。目前为止，关于伪反自伴算子的结果多有可对角化的条件，此类算子仍有大量值得研究的问题。 本课题中，我们将利用四元数算子理论的谱分析和空间分解等方面的方法和技巧，围绕伪反自伴算子的性质研究以下内容：(1) 从四元数Hilbert空间上线性算子的球面谱入手，研究伪反自伴算子的结构; (2) 从特殊算子类出发, 描述伪反自伴算子的反换位代数，建立算子伪反自伴性与其反换位代数之间的关系; (3)伪反自伴算子的对角化及其在四元数量子力学中的应用。

我们考虑了四元数Hilbert空间上紧正规算子的伪反自伴性问题, 利用算子的切片函数演算及球面谱理论对紧正规伪反自伴算子的球面谱进行刻画, 并给出算子的自伴性与伪自伴性之间的关系. 在考虑算子的换位代数研究伪反自伴算子的性质过程中，我们发现了四种Weyl型定理在非交换紧扰动下的稳定性，并且得到了Drazin可逆性在紧扰动下不变的充要条件. 我们讨论了Hilbert空间中一种谱的性质，即谱的（R）性质,得到谱的（R）性质在紧扰动下不变的充要条件. 关于对角化相关问题，我们考虑算子的伪谱半径，得到了Hilbert空间上的有界线性算子谱半径和伪谱半径之间的关系.