Idempotent Operator And Combination Technique Approximation

Opeator theory is one of the most important fields in functional analysis. In recent years, Idempotent operator and combination technique 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 geometry characterizations between, two subspaces, combination technique and the closeness of the range of the linear combinations. 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 investigated the geometry structure between two subspaces, New characterizations of the gap and the angle between two subspaces are established. Particularly, we have pointed out that pass the tight calculation and reason logically, we have further charactered the sizes of the minimum gap.In chapter 2, We have mainly discussed the characterizitions of the idempotent operators on an infinite dimensional Hilbert space. In section 2, we using the technique of block operator matrices redepicted the relation between the norm of the idempotent operator and the norm of the two orthogonal projections which speak of in [4], i. e At last, by using block operator matrices representation of idempotent operators, we proved that when c₁(c₂+c₃)≠0, c₂(c₁+c₃)≠0, c₁+c₂+c₃≠0 and c₁／(c₂+c₃)(?)[-1, 0] or c₂／(c₁+c₃)(?)[-1, 0], the closness of R(c₁P+c₂Q+c₃PQ) is independent of the choice of (c₁, c₂, c₃).In chapter 3, We have mainly discussed combination technique approximation. The combination technique has repeatedly been shown to be an effective tool for the approximation with sparse grid spaces. In this section we discussion two subspaces V₁, V₂(?)H, satified V=V₁+V₂ and P_V1, P_V2 are orthogonal projections onto V₁, V₂, respectively, The approximations are either additive, where T^a=c₁P_V1+c₂P_V2+c₁₂P_V1∩V2or multiplicative, where T^mc=c₁P_V1+c₂P_V2+c₁₂P_V1P_V2. The theorem 3.2.2 extends the result in [40].