作为经典的p-亚正规算子理论的研究工具，算子乘积共同性质及相关理论一直是国际前沿的热点问题。本项目以算子乘积的共同谱性质、共同空间可补性质等问题的研究为中心，同时结合了广义逆理论与环理论的研究，并将算子乘积共同性质的研究方法应用于复对称算子谱分析的研究中。研究内容主要包括：. (1) 研究算子乘积的零空间、值域、解析核和拟幂零部分的空间可补性问题。. (2) 在有界线性算子范畴内，研究Drazin提出的关于(m,n)-可逆的Jacobson引理问题。. (3) 在有单位的半单的Banach代数上，给出元素乘积的Fredholm指标（或B-Fredholm指标）之间的关系。. (4) 综合运用算子乘积共同性质及其相关理论的研究方法，探讨复对称算子及其对偶算子的各类正则谱之间的关系。由此寻找复对称算子矩阵满足广义Weyl定理的条件。

As the tools of the classical theory of p-hyponormal operators, common properties of operator products and related theory are always the hot and international topics. This project will focus on common properties of operator products, and will combine the research of generalized inverses and ring theory. Some applications in spectral analysis of complex symmetric operators are also given. In this project, we mainly focus on the following aspects: (1) The complement of null, range, analytic core and quasi-nilpotent part of operator products are studied. (2) The open question, posed by Drazin, about the Jacobson lemma for (m,n)-inverses is studied in category of bounded linear operators. (3) In unital semisimple Banach algebra, we study the Fredholm index (or B-Fredholm index) for element products. (4) By utilizing the essential methods of common properties of operator products, we study several regular spectrums of complex symmetric operator and its adjoins. Moreover, we try to provide some conditions so that generalized Weyl type theorem holds for complex symmetric operator matrices.