Linear Combinations, Path Connectivity Of Idempotents And Drazin Inverse In B(H)

Opeator theory is one of the most important fields in functional analysis. In recent years, idempotents and Drazin Inverse have been live topics in operator theory. The research of these subjects have related to pure and applied mathematics such as algebra, geometry, perturbation theory, matrix analysis, approximation theory, multivariate linear modeling, Banach-algebra, numerical analysis, optimality principle and quantum physics etc. It is from the study in this field that we have a clear impression on the internal relations and the constructions among operators. In this essey, we will deal with the linear combinations, path connectivity of two idempotents and Drazin Inverse on a Hilbert space. This article is divided into three chapters.In chapter 1, using the way of space decomposition and the technique of block operator matrices, we have given the matrix representations of idempotents and orthogonal projecitons on a Hilbert space. Subsequently, We have established the sufficient and necessary condition that the linear combinations of two idempotents P and Q are also idempotent, which extends the result in [1] to an infinite dimensional Hilbert space. Particularly, we have pointed out that the condition P₁P₂ ≠ P₂P₁ of the statement (b) of the theorem in [1] can be deleted.In chapter 2, We have mainly discussed that two homotopic idempotents belong to the same connected idempotent-valued components on an infinite dimensional Hilbert space. About this problem, The Kovarik formula which Z. V. Kovarik brought up in 1977 Plays an important role, so in the second section, the forms, properties of Kovarik formula and generalized Kovarik formula have been investigated in depth. In the sequel, the path connectivity of two homtopic idempotents have been studied. In fact, we have established the sufficient and necessary condition that (s|⁾(P, Q) ≤ 2 holds, where (s|⁾(P, Q) ≤ 2 denotes that two homotopic idempotents can be connected by two pairs of idmpotent-valued path.In chapter 3, the representation and characterization of Drazin Inverse of operators on a Hilbert space are treated completely. According to the the Index theory of operators, spectrum theory and the representation of Drazin Inverse of operatorsin [35], we have investigated the Drazin Inverse of the sum of two idempotents, andhave given out the concrete representation of the Drazin Inverse.