Properties Of Drazin Invertible Operators And Linear Combinations Of Hypergeneralized Projectors

Abstract: Drazin inverse is a main concept of generalized inverse theory of operators. It has unexpected relations and infiltrations with the theory of finite Markov chains, cryptograph, iterative methods in numerical analysis, indeed multi-body system dynamics as well as some other important branches of mathematics. Recently, due to the development of science and technology and need of practical problems, many scholars both here and abroad are very interested in and have focused on the study of generalized inverse of linear operators in an abstract space. The paper researches some properties of Drazin invertible operators. In addition, Hypergeneralized projectors have also become a hot topic in recent years. The paper studies the linear combinations of Hypergeneralized projectors. The paper is divided into three chapter in all. The details as follows:In chapter 1, some notations, definitions are introduced and some well-known theorems are given. In section I, we give some technologies and notations, and introduce the definitions of Moore-Penrose inverse, Drazin inverse, acent and decent of an operator and B-Fredholm operators and so on. In section II, we give some well-known theorems, such as spectral mapping theorem.In chapter 2, we first discuss the characterization of eigenprojections at zero of Drazin invertible operators on a Hilbert space. By using the technique of block operator matrices, we extend the results that the characterization of eigenprojections at zero of matrices, obtained by J. J. Koliha, I. Straskraba, N. Castro-Gonzalez and Yimin Wei through to operators in B(H). Furthermore, we discuss the Drazinspectrum of 2 x 2 upper triangular operator matrices on H (?) K. Suppose that M_C =is an operator which acts on H (?) K, where A ∈ B(H),B ∈ B(K),C ∈ B(K,H). Point out that if σ_D(M_C) = σ_D(A) ∪ σ_D(B), then σ(M_C) = σ(A) ∪ σ(B). Subsequently, we give conditions of whence σ_D(T+N) = (σ__D(T), where T ∈ B(H), T is nilpotent. We also give conditions of whence the perturbation of Drazin invertible operators are also Drazin invertible. Finally, we study the properties of invertible operators, Moore-Penrose invertible operators and Drazin invertible operators.In chapter 3, we first discuss the relation about star-orthogonal, star partialordering and Hypergeneralized projector and prove that for A, B G B(H)HGP, the following three statements hold:(1) UAL* B, then A + B <= B{H)HGP,(2) If A <* B, then B - A e B{H)HGP,(3) For nonzero A and a G C, the condition a A <* B implies either a = 0 or a3 = 1. Subsequently, we discuss the linear combinations of Hypergeneralized projectors. In [37], J. K. Baksalary, O. M. Baksalary and J. Grofi give out the result of when a linear combination a A + PB of two hypergeneralized projectors A, B is also a hypergeneralized projector under certain commutativity property imposed on matrices A and B. In the paper, we not only extend the result to infinite dimentional Hilbert space but also weaken the conditions.