Research On The Generalized Aluthge Transformation And Orthogonal Projections In B(H)

Aluthge transformation, numerical range, projections and Drazininverses are heated topics in operator theory and also have important value in boththeory and application. The research of these subjects has related to pure and ap-plied mathematics such as geometry, operator perturbation theory, matrix analysis,C~*-algebra, numerical analysis, system theory, quantum physics etc. Through re-search, the internal relations and constructions among operators can be found anda substantial basis can be provided for the study of operators.The research of this thesis focuses on numerical range, essential numerical rangeand different kinds of spectrum between an operator and its generalized Aluthgetransform, Drazin invertibility of product and difference of orthogonal projections,commutativity of projections on a complex Hilbert space. The research on thenumerical range of the generalized Aluthge transform brings out the numerical rangeof inclusion relationships between an operator and its generalized Aluthge transform,which extend two results which are obtained by Peiyuan Wu in[1]. The research onprojections on a complex Hilbert space contains the representation of Drazin inversesof product and difference of projections and the commutativity of projections. Thearticle is divided into four chapters.In chapter 1, some notations, definitions are introduced and some well-knowntheorems are given. In sectionⅠ, we give some technologies and notations, andintroduce the definitions of Moore-Penrose inverse, Drazin inverse, ascent and decentof operators and B-Fredholm operators and so on. In sectionⅡ, we give some well-known theorems, such as spectral mapping theorem.In chapter 2, we discuss numerical range, essential numerical range, differentkinds of spectrum among an operator itself, its generalized Aluthge transform, itsgeneralized ~*-Aluthge transform, we prove that(1)(?), (?)t∈(0, 1), which extends the conclusions about numericalranges between T and Aluthge transform (?) in [1].(2)W_e(?)W_e(T), (?)t∈(0,1).In chapter 3, through the study of the spectral properties of the generalizedAluthge transform, we prove that r-Weyl's theorem holds for T if and only if r- Weyl's theorem holds for (?) if and only if r-Weyl's theorem holds for (?)(*). Andalso we show that a-Weyl's theorem holds for T if and only if a-Weyl's theoremholds for (?), (?)λ∈(0, 1).In chapter 4, let P and Q be orthogonal projections on a complex Hilbert space.By using block operator matrices, the characterization of the Drazin invertibility ofproduct and difference of projections P and Q are established. We also give therepresentation of Drazin inverses of product and difference of two projections. Inaddition, we obtain a very interesting result: PQ (res. P-Q) is Drazin invertibleif and only if PQ (res. P-Q) is Moore-Penrose invertible. At last, we do someresearch on the commutativity between two projections. We prove that ifσ(PQ)={0, 1}, then PQ=QP. Meanwhile, we also obtain that if PQ is EP operator, thenPQ=QP.