Some G-M-type Banach spaces and K-groups of operator algebras on them

By providing several new varieties of G-M-type Banachspaces according to decomposable and compoundable properties, this paper discusses the operator structures of thesespaces and the K-theory of the algebra of the operators on these G-M-type Banach spaces throughcalculation of the K-groups of the operator ideals contained in the class of Riesz operators.

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.

Samuel multiplicity and the structure of essentially semi-regular operators: A note on a paper of Fang

Motivated by a paper of Fang (2009), we study the Samuel multiplicity and the structure of essentially semi-regular operators on an infinite-dimensional complex Banach space. First, we generalize Fang's results concerning Samuel multiplicity from semi-Fredholm operators to essentially semi-regular operators by elementary methods in operator theory. Second, we study the structure of essentially semi-regular operators. More precisely, we present a revised version of Fang's 4 × 4 upper triangular model with a little modification, and prove it in detail after providing numerous preliminary results, some of which are inspired by Fang's paper. At last, as some applications, we get the structure of semi-Fredholm operators which revised Fang's 4 × 4 upper triangular model, from a different viewpoint, and characterize a semi-regular point λ∈ C in an essentially semi-regular domain.