Strongly Irreducible Operators on Banach Spaces

This paper firstly discusses the existence of strongly irreducible operators on Banach spaces. It shows that there exist strongly irreducible operators on Banach spaces with w＊-separable dual. It also gives some properties of strongly irreducible operators on Banach spaces. In particular, if T is a strongly irreducible operator on an infinite-dimensional Banach space, then T is not of finite rank and T is not an algebraic operator. On Banach spaces with subsymmetric bases, including infinite-dimensional separable Hilbert spaces, it shows that quasisimilarity does not preserve strong irreducibility. In addition, we show that the strong irreducibility of an operator does not imply the strong irreducibility of its conjugate operator, which is not the same as the property in Hilbert spaces.

THE (ｕ＋κ)-ORBIT OF ESSENTIALLY NORMAL OPERATORS AND COMPACT PERTURBATIONS OF STRONGLY IRREDUCIBLE OPERATORS

Let н be a complex, separable, infinite dimensional Hilbert space, T ε（Ｈ）. （ｕ＋Ｋ）（T） denotes the （ｕ＋ｋ）-orbit of T, i.e., （ｕ＋ｋ）（T） = {R-1TR: R is invertible and of the form unitary plus compact}. Let be an analytic and simply connected Cauchy domain in C and n ε N. A（, n） denotes the class of operators, each of which satisfies （i） T is essentially normal; （ii） σ（T） =, ρF（T） ∩ σ（T） = ; （iii） ind （λ-T） = -n, nul （λ-T） = 0 （λ∈Ω ）. It is proved that given T1, T2 ε A（, n） and ε > 0, there exists a compact operator K with K <ε such that T1 +Kε （ｕ＋ｋ）（T2）. This result generalizes a result of P. S. Guinand and L. Marcoux [6,15]. Furthermore, the authors give a character of the norm closure of （ｕ＋Ｋ）（T）, and prove that for each T ε A（, n）, there exists a compact （SI） perturbation of T whose norm can be arbitrarily small.

The approximate Jordan forms of operators on Σ_e^1 type Banach spaces

This paper studies the structure of operators on Σ1e type Banach spaces.It solves the problem of the small compact perturbations of operators with connected spectra.Namely,it shows that every operator with a connected spectrum on separable Σ1e type Banach spaces is a small compact perturbation of a strongly irreducible operator.Based on this result,this paper establishes the approximate Jordan forms of operators on Σ1e type Banach spaces with Schauder bases.

Similarity invariants of operators on a class of Gowers-Maurey spaces

This paper studies the similarity invariants of operators on a class of Gowers-Maurey spaces, ∑dc spaces, where an infinite dimensional Banach space X is called a ∑dc space if for every bounded linear operator on X the spectrum is disconnected unless it is a singleton. It shows that two strongly irreducible operators T1 and T2 on a ∑dc space are similar if and only if theK0-group of the commutant algebra of the direct sum T1 GT2 is isomorphic to the group of integers Z. On a ∑dc space X, it uses the semigroups of the commutant algebras of operators to give a condition that an operator is similar to some operator in （∑SI）（X）, it further gives a necessary and sufficient condition that two operators in （∑SI）（X） are similar by using the ordered K0-groups. It also proves that every operator in （∑SI）（X） has a unique （SI） decomposition up to similarity on a ∑dc space X, where （∑SI）（X） denotes the class of operators which can be written as a direct sum of finitely many strongly irreducible operators.

Two Classes of Operators with Irreducibility and the Small and Compact Perturbations of Them

This paper gives the concepts of finite dimensional irreducible operators（（FDI） operators）and infinite dimensional irreducible operators（（IDI） operators）. Discusses the relationships of（FDI）operators,（IDI） operators and strongly irreducible operators（（SI） operators） and illustrates some properties of the three classes of operators. Some sufficient conditions for the finite-dimensional irreducibility of operators which have the forms of upper triangular operator matrices are given. This paper proves that every operator with a singleton spectrum is a small compact perturbation of an（FDI） operator on separable Banach spaces and shows that every bounded linear operator T can be approximated by operators in（Σ FDI）（X） with respect to the strong-operator topology and every compact operator K can be approximated by operators in（Σ FDI）（X） with respect to the norm topology on a Banach space X with a Schauder basis, where（ΣFDI）（X） ：= {T∈B（X） ： T=Σki=1Ti, Ti ∈（FDI）, k ∈ N}.

THE(U+K)-ORBITS OF ESSENTIALLY NORMAL OPERATORS AND THEIR STRONG IRREDUCIBILITY

The authors characterize the (U+K) orbits of a class essentially normal operators and prove that some essentially normal operators with connected spectrum are strongly irreducible after a small compact perturbation. This partially answers a question of Domigo A. Herrero.

Unitary Equivalence,Similarity and Calculation of K_0-Group

In this paper, we are concerned with the classification of operators on complex separable Hilbert spaces, in the unitary equivalence sense and the similarity sense, respectively. We show that two strongly irreducible operators A and B are unitary equivalent if and only if W＊（A＋B）′≈M2（C）, and two operators A and B in B1（Ω） are similar if and only if A′（AGB）/J≈M2（C）. Moreover, we obtain V（H^∞（Ω,μ）≈N and Ko（H^∞（Ω,μ）≈Z by the technique of complex geometry, where Ω is a bounded connected open set in C, and μ is a completely non-reducing measure on Ω.

K_0-group and similarity of operator weighted shifts

By using simultaneous triangularization technique, similarity for operator weighted shifts with finite multiplicity is characterized in terms of K0-group of commutant algebra. The result supports the conjecture posed by Cao et al. in 1999. Moreover, we discuss the relations between similarity and quasi-similarity for operator weighted shifts.