可分右四元数Hilbert空间上正规算子的伪反自伴性

Pseudoanti-Hermitian operators on complex Hilbert space are closely related to PT-symmetric Hamiltonians of non-Hermitian complex quantum mechanics and have attracted much attention from physical scientist and mathematicians. However, in a recent systematic study of quantum mechanics in quaternionic Hilbert space, the issue of a quaternionic analogue of pseudo-Hermitian Hamiltonians has gained the researchers’ attention naturally. It is called pseudoanti- Hermitian quaternionic operators. Until now, most results about pseudoanti-Hermitian quaternionic operators were obtained under the assumption of diagonalization. As we all know,the class of normal operators is important and contains diagonalized operators. In this project, we will study several problems of pseudoanti-Hermiticity of normal bounded linear operators on separable infinite-dimensional right quaternionic Hilbert space. We will study the pseudoanti-Hermiticity and the spectrum of more general normal operators. The diagonalization of normal pseudoanti-Hermitian quaternionic operators and more general operators will be considered.

复Hilbert 空间上伪自伴算子与非自伴复量子力学中的PT 对称哈密顿量存在着密切联系，并且引起了物理学家和数学家们的广泛关注。近年来，随着人们对四元数Hilbert 空间中量子力学理论的系统研究，越来越关心伪自伴算子在右四元数Hilbert 空间中的类似物，即伪反自伴算子，此类算子在非自伴四元数量子力学中扮演着重要角色。现有关于四元数算子伪反自伴性方面的结果多是在可对角化的条件下得到的，而正规算子是比可对角化算子更大的算子类，且在算子理论中具有十分重要的研究价值。本课题中，我们将研究可分右四元数Hilbert 空间上正规算子的伪反自伴性及相关问题。具体如下：研究比对角算子更一般的正规算子的伪反自伴性及其谱特征；研究正规的伪反自伴算子的对角化条件，并期望以此为基础进一步深入考虑一般算子的对角化问题。

本项目旨在研究四元数Hilbert空间上正规算子的伪反自伴性及相关问题. 我们按照项目计划书展开研究, 基本实现了项目的既定研究目标. 尤其是学习了最新文献关于四元数Hilbert空间算子的球面谱及函数演算等理论, 利用其基本理论和技巧方法对课题展开研究. 取得的成果有：（1）对紧正规算子给出伪反自伴性的条件, 对其球面谱进行了刻画；（2）对算子的自伴性、伪自伴性及伪反自伴性进行了统一的研究；（3）利用可分四元数Hilbert空间与复Hilbert空间之间的同构关系, 找到了一类可块三角化的算子类.