Collective Compactness And Fredholmness Of Operator Sequences

Compact operator and Fredholm operator are two kinds of important operators in infinite-dimensional Banach space. They play the core role in the studying of integral equations and mathematical, physical problems and etc. There were well researched in their spectrum analysis and structure. Operator compact sequences and operator Fredholm sequences play important roles in computing approximate solution of equations. In this thesis, we discussed the collective compactness and Fredholmness of operator sequences.The collective compactness of operator sequences on function spaces has roused extensive attention. The collective compactness theory of operator sequences on function spaces plays an important role in computing approximate solution of operator equations and spectrum approximation, which has been studied by foreign scholars at the beginning of the 1970s such as P.M.Anselone. After that a lot of people were interested in it and got many famous results. Li Shaokuan introduced the definition of operator Fredholm sequences in 1986 and discussed some properties of them. Contrary to the collective compactness, there were still in the rough on the Fredholmness of operator sequences, especially about their index. In the thesis, we gave further studies about these two aspects, and got some satisfactory results.Firstly we discussed the collective compactness of weighted composition operator sequences between different Hardy spaces on the unit ball, and we gave the sufficient and necessary condition in terms of the concept of Carleson measure. Secondly we further discussed some properties of operator Fredholm sequences, then obtained equivalent conditions of uniform convergent bounded linear operator sequences to be operator (upper-semi, lower-semi) Fredholm sequences, these are also the generalization of the results of(upper-semi,lower-semi) Fredholm operators. Next we got the other sufficient and necessary condition for the uniform convergent operator sequences to be operator Fredholm sequences. Thus we generalized the Fredholm properties of single Toeplitz and composition operators, and got sufficient and necessary conditions of uniform convergent Toeplitz and composition operator sequences to be operator Fredholm sequences. We also got other sufficient and necessary conditions for Toeplitz operator sequences to be operator Fredholm sequences based on the results of Chen Xiaoman,Li Wenjun, and Zhong Changyong, according to the relationship between operator compact sequences and operator Fredholm sequences, and the property of Toeplitz operator. Last we gave the definition of ascent and descent of operator sequences, we also defined the null space and range of operator sequences properly by means of the construct idea. Then we discussed some properties of operator upper-semi Fredholm sequences and a class of operator sequences which is related to operator compact sequences, and we proved that the collection of operator upper-semi Fredholm sequences with finite ascent (descent) is closed under commuting operator sequences perturbation class associated with them according to the thought of homotopy.