Some Theories Of Projection Operators And Idempotent Operators On Hilbert C~*-module

Projection and idempotent element are two well-known mathematical objects,which correspond to the orthogonal decomposition and the direct sum decomposition of a space respectively,and have various applications in analysis,algebra and geometry.Although projections and idempotent elements have been intensely studied in the literature,there are still some fundamental issues that remain to be unknown,one of which is the approx-imation to an idempotent element by projections.Taking inspiration from the research on the approximation of projections to an idempotent element,we introduce firstly the matched projection for each idempotent element,which has a high degree of correlation with the given idempotent element.Based on the characterizations of the relationship be-tween an idempotent element and its matched projection,we introduce secondly a new term called the quasi-projection pair.This dissertation contains many original results by focusing on the investigations of these two brand new mathematical objects.In the general setting of adjointable operators on Hilbert C^*-modules,this disserta-tion is focused on the study of the matched projections and the quasi-projection pairs.It contains the following four topics:the fundamental issues about the quasi-projection pairs,the matched projections of idempotent elements,the semi-harmonious quasi-projection pairs and the harmonious quasi-projection pairs by using such mathematical tools as the generalized inverses of operators or matrices,the polar decompositions of operators,and the weak majorizations of vectors,which are developed from the theory of C^*-algebras,Hilbert C^*-modules and numerical algebras.Firstly,some equivalent characterizations of a quasi-projection pair are provided,and six operators as well as six closed submod-ules are defined for each quasi-projection pair,based on which the new terms called the semi-harmonious quasi-projection pair and the harmonious quasi-projection pair are in-troduced.Secondly,some basic properties of the matched projection are clarified and the application of the matched projection in the theory of the generalized inverse is investi-gated.Specifically,it is shown thatm（Q）,Qis always a quasi-projection pair for every idempotent element Q.Meanwhile,the consistent compressibility of the matched projec-tion is dealt with.It is proved that m（Q₁）-m（Q₂）≤Q₁-Q₂ whenever Q₁and Q₂are idempotent elements and at least one of them is a projection.Much efforts have been paid in the study of the distances from projections to a given idempotent elementQ via the matched projection m（Q）.A full characterization of the Frobenius distances between projections and a given idempotent element,including the minimum value,the maximum value,the intermediate value and the uniqueness of the projections that take the minimum value and the maximum value respectively,is provided in the matrix case when all the matrices are endued with the Frobenius norm.In the case of operators endowed with operator norm,among other things it is shown that for every idempotent element Q,m（Q）-Q≤P-Q for any quasi-projection pair（P,Q）.Thirdly,some detailed de-scriptions and applications of the semi-harmonious quasi-projection pair are employed.In particular,many equivalent conditions are provided for a quasi-projection pair to be semi-harmonious,and the solvability of certain operator equations concerning the com-mon similarity and unitary equivalence are dealt with for some operators closely related with a quasi-projection pair.It is proved that these operator equations are solvable when-ever the underlying quasi-projection pair is semi-harmonious,and a counterexample is constructed to show the necessity of the semi-harmony.Finally,some equivalent condi-tions for a quasi-projection pair to be harmonious are also figured out.As applications,the 2×2 standard decomposition and the 6×6 Halmos decomposition are derived for every harmonious quasi-projection pair.The study of these contents can deepen the under-standing of projection and idempotent element,add new materials to the fields of operator algebra and operator theory,as well as numerical algebra,and promote the development of related disciplines.