Study On The Classes Of Irreducible Operators And The Structure Of Operators On G-M Type Spaces

The research of the operator structure on Banach spaces is one of the main issues both for the theory of Banach spaces and for the theory of operator algebras in functional analysis. This paper study the operator structure on the special Banach spaces—G-M type spaces by using the special structure of G-M type spaces and K-theory for operator algebras, with a variety of operators of irreducibility as tools. It reflects the interaction between the research of the structure of Banach spaces and the research of operator structure. Compared to the pure theory of Banach spaces or the pure theory of operator algebras, the idea is unique and innovative.Concerning the research of the operator structure on Banach spaces, Z.J.Jiang thought that the strongly irreducible operators can be considered as the approximate replacement of Jordan blocks on infinite dimensional spaces and hoped that a series of results similar to the Jordan Standard Theorem can be set up with this replacement on the research of the operator structure on infinite dimensional Banach spaces. On separable Hilbert spaces. D.A.Herrero, S.Power and C.L.Jiang had fully confirmed this idea. The work of this paper can be seen to put into practice the idea of Z.J.Jiang on a new class of Banach spaces—G-M type spaces.So far, the domestic and foreign scholars only studied the class of strongly irre-ducible operators or only used the class-of strongly irreducible operators to study the operator structure. This paper expands to a series of operators with irreducibility with the class of strongly irreducible operators as the center. And it uses such classes of operators as tools to study the space structure and the operator structure. This is one of the main feature of this paper.This paper has three main works.Firstly, it gives the concepts of a series of operators with irreducibility, including finite dimensional irreducible operators, infinite dimensional irreducible operators, (NCI) operators, (NFI) operators, Bn operators and B operators. Then it discusses the existence of these classes of operators and discusses in detail the relationships among strongly irreducible operators, Cowen-Douglas operators and them. This paper also discusses some properties of these classes of operators, such as the (quasi)similar invariance and the conjugation invariance. Secondly, this paper discusses the small and compact perturbation problem of operators with irreducibility. It mainly discusses the small and compact perturbation problem of finite dimensional irreducible operators, strongly irreducible operators and (NFI) operators. This paper proves that every operator with a singleton spectrum on separable Banach spaces is a small compact perturbation of a finite dimensional irreducible operator. For the small and compact perturbation problem of strongly irreducible operators, it shows that every operator with a connected spectrum on separable indecomposableΣcdc spaces is a small compact perturbation of a strongly irreducible operator. It further shows that the set{T∈B(X)σa(T) is connected } is the norm-closure of the set of strongly irreducible operators on separable inde-composableΣcdc spaces. For the small and compact perturbation problem of (NFI) operators, this paper shows that an operator with a singleton spectrum{0} can be-come an (NFI) operator by a small and compact perturbation in some cases. Based on these results, this paper establishes the approximate Jordan forms of operators on some kinds of Banach spaces with Schauder bases.Thirdly, this paper studies the similar invariants of operators by using the lan-guage of K-theory. It mainly uses the (order) K-groups to give the sufficient and necessary conditions for the similarity of two strongly irreducible operators and the similarity of two operators in (ΣSI)(X) (where (ESI)(X) denotes the class of op-erators which can be decomposed into the direct sum of finitely strongly irreducible operators on X) onΣcdc spaces.In addition, this paper studies the K0-groups of the operator algebras B(X) on Banach spaces X. It mainly discusses the following problem:does there exist a Banach space X with K0(B(X))=Z2, which is a guess raised by A.Zsak. This paper gives a sufficient condition for K0(B(X))=Z2. Then it discusses the properties of the idempotents on G-M type space XGM4, which provides some ideals and methods to get a Banach space X with K0(B(X))=Z2.