Some New Classes Of Operators On Banach Spaces And The Perturbation Classes Problem

Semi-Fredholm operators and Riesz operators are two basic and important classes of operators in operator theory. Researching into them can exert important signifi-cance for the other aspects of operator theory. This paper makes a deep study of the perturbation classes problem and some new classes of operators—infinitely cosingular operators, polynomially Riesz operators, generalized Riesz operators—that are closely related to semi-Fredholm operators and Riesz operators.Firstly, the characterizations of infinity cosingular operators are given(see Theorem 1.2.10).The characterizations and properties of polynomially Riesz operators are obtained (see Theorem 2.2.8,The-orem 2.2.14—Theorem 2.2.16),and some new characterizations of Browder operators are given from the angle of polynomially Riesz operators (see Theorem 2.3.6).We ob-tain the characterizations and properties of generalized Riesz operators (see Theorem 3.2.5,Theorem3.2.6), and we also discuss the relations amongΩK+(X),Ωk-(X) and generalized Riesz operators,Ωk+(X),Ωk-(X) and inessential operators (see Theorem 3.3.4—Theorem3.3.8). Secondly, we give a negative answer to the perturbation classes problem onΣen (n≥2) type Banach space(see Theorem4.2.1).A feature of this article is that we define some new classes of operators and make a systematically discussion on them.