Strongly Irreducible Operators On Banach Spaces

In the matrix theory of finite dimensional spaces, the famous Jordan Standard Theorem sufficiently reveals the internal structure of matrix. From the Jordan Standard Theorem, it is obvious that the Jordan block plays a fundamental and important role in matrix theory. In infinite dimensional separable Hilbert spaces, C.L.Jiang and his cooperators have proved that strongly irreducible operator is a suitable analogue of Jordan block in the set of all bounded linear operators. And they have founded the theorems concerning the unique strongly irreducible decomposition of operators in the sense of similarity, the spectral picture and compact perturbations of strongly irreducible operators. They have also used the K-theory language to find the complete similarity invariant. In fact, they have formed a theoretical system of strongly irreducible operators in infinite dimensional separable Hilbert spaces. Undoubtedly, it is very important and very essential to build the corresponding theory on general Banach spaces. The main purpose of this thesis is to research the important basic properties of strongly irreducible operators on general Banach spaces, which includes the existence of strongly irreducible operators. This thesis consists of six chapters.Chapter 1 presents a survey of the study of strongly irreducible operators.Chapter 2 discusses the existence of strongly irreducible operators, proves that there is a strongly irreducible operator on the Banach space X with the property that conjugate space X^* is separable under the w^* topology; researches the basic properties of strongly irreducible operators, shows that if T is a strongly irreducible operator,then T isn't of finite rank and isn't an algebraic operator. And the relation between strongly irreducible operators and Cowen-Douglas operators is considered, it is showed that if X=c₀ orι_p(1≤p<∞),then there exists T∈B_n(Ω)∩(SI)(X)for every 1≤n≤∞.Chapter 3 gives the result of "filling in holes" of essential spectra of upper triangular operator matrices, that is, for upper triangular operator matrix M_C=(?)∈B(X×Y),σ_Ï„(A)∪σ_Ï„(B)=σ_Ï„(M_C)∪W,where W is the union of certain of the holes inσ_Ï„(M_C) which happen to be subsets ofσ_Ï„(A)∩σ_Ï„(B),σ_Ï„can be equal toσ_b andσ_e,and the result is not true ifσ_T is equal toσ_K andσ_ι,σ_r,σ_ιe,σ_re; presents the corresponding result of "filling in holes" of spectra of upper triangular matrices in M₂(A).And the conditions is discussed such that operator matrices become strongly irreducible operators.Chapter 4 researches the properties of strongly irreducible operators on hereditarily indecomposable space. Shows the special property of strongly irreducible operators on hereditarily indecomposable space according to the special operator structure.For example, if T is a strongly irreducible operator, then ker T (?) (?); and A'(T)/radA'(T)≈C if and only if T∈(SI);moreover,ifσ_p(T)=(?) orσ_γ(T)=(?),then T∈(SI).Gives the definition of finite dimensional strongly irreducible operators and shows some properties of it.Chapter 5 discusses the small and compact perturbations problem of strongly irreducible operators on hereditarily indecomposable space, shows that some kinds of operators with singleton spectra can become strongly irreducible operators by a small and compact perturbation, that is, if the operator T with singleton spectra {0} satisfies one of the following conditions: (1)dim kerT <∞; (2)dim(X/(?))<∞; (3)dim(ker T/(ker T∩(?)))=∞;(4)dim ker T=dim(X/(?))=∞,ker T(?) (?),dim((?)/ ker T)<∞,then T can become strongly irreducible operators by a small and compact perturbation. Proves that diagonal operators with singleton spectra can become strongly irreducible operators by a small and compact perturbation on the space which has a finite dimensional Schauder decomposition.Chapter 6 uses the K-theory language to research the strongly irreducible decomposition,obtains some theorems which are similar to the conclusions on infinite dimensional separable Hilbert spaces, that is, A-∑_i=1^k(?)A_i⁽ⁿⁱ⁾,where A_i∈(SI), i=1,2,…,k,A_i(?)A_j(i≠j) if and only if there exists a isomorphism h from∨(A'(A)) onto N^(k) with h([I])=n₁e₁+n₂e₂+…+n_ke_k,where I is the identityoperator in A'(A),{e_i}_i+1^k are the generators of N^(k),and 0≠n_i∈N.In particular, the K₀-group of commutant of direct sum of two strongly irreducible operators can been characterized the similarity of them in hereditarily indecomposable space.