| Let H be a complex and separable Hilbert space, B(H) denote the set of all bounded linear operators on H. If P∈B(H) satisfies P~2=p=p~*, we call P an orthogonal projection on H; if P∈B(H) only satisfies P~2=P, we call P an idempotent operator on H; if P∈B(H) only satisfies P~2=P~*, we call P is a generalized projection on H; if there exists a positive integer k≥2 such that P∈B(H) satisfies P~2=P, we call P is a k-idempotent operator on H. if there exists a positive integer k≥2 such that P∈B(H) satisfies P~2=P~*, we call P is a k-generalized projection on H. The results obtained from the research on problems about idempotent operators always still hold for k-idempotent operators. The results obtained from the research on problems about generalized projections always still hold for k-generalized projections. We will mainly study two aspects of problems in this paper: the Drazin invertibility of the linear combinations of orthogonal projections, path connectivity of generalized projections and k-generalized projections.The research on orthogonal projections and idempotent operators have begun a long time ago(see [1-16]). Hong-Ke Du and Chun-Yuan Deng obtained an important conclusion that the invertibility of the linear combinations is independent of the coefficients(see [13]), some fairly complete theroems about the Drazin inverse and Moore-Penrose inverse of the products and difference of orthogonal projections were established (see [14-16]). In this paper, we will study the Drazin inverse and MoorePenrose inverse of the linear combinations of orthogonal projections and show the equivalence between the existence of the Drazin inverse and Moore-Penrose inverse of the linear combinations of orthogonal projections. In recent ten years, the research problems on generalized projections have obsorbed many scholars, such as H. K. Du, Y. Li, X. J. Liu, J. Groin, G. Trenkler, O. M. Baksalary, J. K. Baksalary, G. W. Stewart, J. Benitez, L. Lebtahi, N. Thome and so on. They have a further research on problems about generalized projections(see [17-26]). The concept of generalized projections was introduced by H. K. Du and Y. Li in [21]. In 1997 J. Groβand G. Trenkler published an adjoint work, Generalized and hypergeneralized projectors(see [22]). The authors introduced generalized and hypergeneralized projectors on finite dimensional Hilbert spaces. H. K. Du and Y. Li extended the concept of generalized projectors to infinite dimensional Hilbert spaces in [21] and hence introduced the concept of generalized projections. An important result in [21] is the spectral characterization of generalized projections. In the published papers, the research on path connectivity of generalized projections never was investigated. In this paper, by the spectral characterization of generalized projections, we establish a complete solution for the problem on path connectivity of generalized projections.There are three chapters in this article, and the main content as follows:In the first chaper, we mainly introduce the preliminaries of orthogonal projections and generalized projections. This chapter consists of two sections. In the first section, published results on orhtgonal projections are introduced. In the second section, the concept of generalized projections, characterizations of generalized projections(including the original characterizations by J. Groβand G. Trenkler, the alternative characterizations by J. K. Baksalary and X. J. Liu, the spectral characterizations by H. K. Du and Y. Li) and published results on generalized projections are introduced.In the second chapter, after study the explicit block matrix representation for orthogonal projections, we establish a relation between the Drazin invertibility and Moore-Penrose invertibility of the linear combinations of orthogonal projections and the coefficients. This chapter consists of two sections. In the first section, we simply state the published results on the Drazin inverse and Moore-Penrose inverse of the products and difference of orthogonal projections. In the second section, we study the Drazin invertibility of the linear combinations of orthogonal projections and the coefficients, where the coefficients are nonzero.And in the chapter three, we introduce the problem on path connectivity of generalized projections. This chapter consists of three sections. In the first section, by the spectral characterization of generalized projections, we establish a complete solution for the problem on path connectivity of generalized projections. In the second section, we extend the path connectivity problem to k-generalized projections and mainly explain that there is a little difference between the study of path connectivity of k-generalized projections and that of generalized projections. In the third section, we investigate some problems on generalized projections which remained unknown. |
