Theory Of Topological Uniform Descent And Two Classes Of Operators

In this thesis we pay attention to the interplay between space structures and operator structures. We mainly study the theory of topological uniform descent and two classes of operators-left decomposably regular operators and (n, k)-quasi-*-paranormal operators. The main results are as follows:One is for the theory of topological uniform descent. First of all, two appli-cations of Grabiner theory are given. We obtain some characterizations of Banach spaces which are not isomorphic to any of their proper subspaces. To improve and generalize the corresponding results of Fang X., we study the Samuel multiplicity and the structure of essentially semi-regular operators. Secondly, we consider small essential spectral radius perturbations of operators with topological uniform descent. Since small essential spectral radius perturbations cover compact, quasinilpotent and Riesz perturbations, its research meaning is self-evident. The results we obtained not only generalize Grabiner’s perturbational results, but also become powerful tools to slove some open questions posed by Berkani et al. As far as we know, most of our perturbational results seem to be new even for Fredholm operators. Thirdly, we give two applications of small essential spectral radius perturbational results. Property (gb), which is a variant of Browder theorem, is discussed in detail. We not only give a counterexample to show that property (gb) in general is not preserved under commuting quasi-nilpotent perturbations, but also revise some new results of Rashid in2011by using small essential spectral radius perturbational results. A result of Burgos et al. is extended to various spectra originated from seni-B-Fredholm theory. As immediate consequences, we give affirmative answers to several questions posed by Berkani et al. Lastly, relating to operators with topological uniform descent, we investigate common properties of RS and SR. This work should be viewed as a continuation of researches of Barnes and Lin et al. In particular, we show that I-SR and I-RS share common property of topological uniform descent and com-plementability of kernels. Three-space theorems in operator theory are discussed. We obtain a widely-used three-space theorem for semi-Fredholmness, and it is also generalize to Banach space complexes.The other is for the discussion of two new classes of operators. We introduce and study a new class of left decomposably regular operators and the corresponding holomorphic version. By using R. Harte’s techniques, we obtain various characteri-zations of these classes of operators. As the applications of these characterizations, we can compute the topological interiors and closures of them. We also introduce and study another new class of (n,k)-quasi-*-paranormal operators. We discuss some inclusion relations and examples related to (n,k)-quasi-*-paranormal opera-tors. We prove that, for every (n,k)-quasi-*-paranormal operator T, the nonzero points of its point spectrum and joint point spectrum are identical, the nonzero points of its approximate point spectrum and joint approximate point spectrum are identical. One question posed by Mecheri [Studia Math.208(2012),87-96] and another question posed by Mecheri and Braha [Oper. Matrices6(2012),725-734] are answered.