The Theory Of Linear Operators On A Class Of Indecomposable Banach Spaces And Related Problems

There are mainly three aspects in this paper. Firstly, on the basis of the new type of Banach spaces ∑_e¹ introduced by Gonzalez and Herrera, the theory of linear operators , strongly continuous (semi-)groups and the properties of cosine family are studied on the ∑_e¹-type Banach spaces, they are main contents in this paper. On such spaces: it is shown that the spectrum σ(T) of a bounded linear operator T has some special properties, for instance, there is an unique point λ_T in σ(T) such that T - λtI is an inessential operator; some properties of the spectrum of linear operators(not necessarily bounded) are obtained; it is shown that every well-bounded operator T can be represented as T = λI + K , here A is scalar, K is a compact operator; it is shown that the family of all Riesz operators is just the ideal of all inessential operators; it is shown that the generator of every C₀-group is necessarily a bounded operator; the generators of C₀-semigroups need not be bounded by an example; it is shown that the generators of certain classes of C₀-semigroups of the operators such as Hermitian operators and isometries are always bounded ; some sufficient conditions are obtained under which the generators of unanimously bounded C₀-semigroups are bounded (in particularly on reflexive spaces); the constitution of the spectrum of the generator of any C₀-semigroup is obtained; it is shown that the generator of any C₀-semigroup satisfies the spectral mapping theorem; a spectral characterization of stability for unanimously bounded Co-semigroups is obtained, and an application to the stability theorem is given; it is shown that the generator of any strongly continuous, non-quasianalytic cosine family is necessarily a bounded operator.On general Banach spaces , a constitutive representation for a well-bounded Riesz operator is obtained , so it is shown that every well-bounded Riesz operator is necessarily a compact operator .Secondly,the concepts of the left(right)Browder spectrum of operators are introduced on general Banach spaces, the intersection relations between the left (right) Browder spectrum of the operator S and that of the operator T are studied when 5 and T are quasisimilar, it is shown that every component of the right Browder spectrum of S intersects with the left Browder spectrum of T.Thirdly, on general Banach spaces, some sufficient and necessary conditions are obtained under which the operators with completely finite ascent satisfy the Weyl theorem, and the sufficient conditions are obtained under which two quasisimilar operators both satisfy the Weyl theorem; it is shown that every operator with completely finite ascent satisfies the Browder theorem.