Approximation To Maintain The Orthogonal Mapping And Orthogonal Equations And Hilbert K (h) - Mode Nature

In this dissertation, we frist consider the stability of orthogonality preserving mapping from a inner product space to a Hilbert space and the orthogonality preserving mapping (approximate orthogonality preserving mapping) on the direct sum of Hilbert spaces. Then we consider the stability of the orthogonality equation on finite dimensional Hilbert spaces. In the end we give some results about Hilbert K(H)-module.This dissertation consists of four sections.The first section is the introduction.In the second section, we consider the stability of approximate orthogonality preserving mapping. When T∈B(H), T^-1exists and is continuous and T = U|T| is its polar decomposition, we give the precise value of ||T－U|| using of functional calculus. We also give the stability of approximate orthogonality preserving mapping from a inner product space to a Hilbert space. Suppose H is a inner product space, K is a Hilbert space, T : H�'K is a nonzero linearε-OP mapping, e∈[0,1). Then there existsa isometry U : H�'K such that ||T-||T||||U||≤(1-((1-ε)/(1+ε))^1/2)||T||. In the end of the section, we study the orthogonality preserving mapping (approximate orthogonality preserving mapping) on the direct sum of Hilbert spaces.In the third section, we consider the stability of orthogonality equation on finite dimensional Hilbert spaces. We give a new definition of approximate solution of the orthogonality equation which is more precise than that given before. We show the stability of the orthogonality equation on finite dimensional Hilbert spaces.In the fourth section, we consider the Hilbert K(H)-module. It is known that whenV and W are Hilbert K(H)-modules, T:V�'W is a bounded K(H)-linear mapping, if these are m, M >0 such that (?)x∈V, m||T||||x||≤||T_x||≤M||T||||x||, then there exists a K(H)-linear isometry U:V�'W such that ||T-||T||U||≤max{|M-1|, |m-1|}||T||. In this section we use this result and the properties of Hilbert K(H)-module to study the Hilbert K(H)-module. We show that if A is C^*-algebra and K(H) (?) A (?) B(H),V and W is Hilbert A-module,ε∈[0,1), T:V�'W is A-linearε-OP mapping,δ=(1-((1-ε)/(1+ε))^1/2), then...