Several Theories Of Hilbert C~*-modular Upprojection Operators

The(orthogonal)projection is one of the most important kinds of operators,which has a wide range of applications in many disciplines.The complexity of the theory for projections lies in the interaction of multiple operators.It is well-known that every closed linear subspace of a Hilbert space is orthogonally complemented.This leads much conve-nience to the study of representations of two projections on a Hilbert space and the norm estimation of their linear combinations.The Hilbert C*-module is a natural generalization of the Hilbert space,a closed submodule of it may fail to be orthogonally complemented.Therefore,some new phenomena may happen when we study the properties of projection-s in the framework of Hilbert C*-modules,so the innovation in methods is required in dealing with these new situations.In the framework of adjointable operators on Hilbert C*-modules,this dissertation mainly studies the harmonious pair of projections and its block matrix representations,the product of finite projections,norm equalities and norm inequalities related to a pair of projections,and the weighted Moore-Penrose invertibility associated with a pair of weighted projections.The second chapter deals mainly with the harmonious pair of projections and its block matrix representations.Firstly,we study the orthogonal complementarity of some operator range closures related to two projections.As an application,some necessary and sufficient conditions are given such that the two projections are harmonious.Then we study the 2 × 2 block matrix representations.Specifically,we show that the two projections have the standard 2 × 2 block matrix representation if and only if they are harmonious.Based on this standard block representations,we give a full content of the Halmos,two projections theorem.Then we show that for the given pair of projections,this theorem is valid if and only if they are harmonious.Finally,more equivalent conditions are provided by using the Halmos,two projections theorem to ensure that the given pair of projections are harmonious.The third chapter deals mainly with products of finite projections on Hilbert C*-modules.Firstly,we consider such kind of finite products of projections,whose left side and right side are changeable,whereas those in the middle are fixed.In the case that an adjointable operator T has the polar decomposition,some characterizations are given such that T can be expressed as the product mentioned above.Secondly,we focus on the case that there is only one fixed projection,which is the identity operator.Note that an adjointable operator may have no polar decomposition,so the notation of(?)is introduced as in the Hilbert space case for the products of two projections,and(?)for those operators in(?)that have polar decompositions.It is shown that(?)is a proper subset of(?),and some properties derived for operators taken in(?)in the Hilbert space case are also valid for operators taken in(?)in the Hilbert C*-module case.In virtue of some counterexamples,we show that the same is not true for certain operators taken in(?)in the Hilbert C*-module case.Some characterizations are given for the partial isometry factor of the polar decomposition of an operator in(?).Finally,we study the PQP representation of the powers of positive contractive operators.Let A be a positive contractive linear operator on Hilbert C*-module H such that the closure of the range of operator A-A2 is orthogonally complemented in ?,? is any positive number,and P is any projection on H.We prove that the solvability of the operator equation A?=PQP with variable Q is independent of the parameter ?,where Q is a projection on H such that(P,Q)is harmonious.The fourth chapter deals mainly with norm equalities and norm inequalities related to a pair of projections.Since the norm of two positive elements in a C*-algebra is generally not equal to the sum of their norms,the Pythagorean theorem is generally not applicable to two mutually orthogonal elements in a Hilbert C*-module.This makes it difficult in the Hilbert C*-module case to give a direct proof of certain norm equalities by just using the definition of the operator norm.Firstly,we study the feasibility of constructing unitary operators to make the unitary equivalence between operators on both sides of the equation.We consider the differences between two groups of projections related to a pair of projections,and construct a common unitary operator by using the polar decomposition for adjointable operators,so that the differences between the two groups of projections are both unitarily equivalent via this unitary operator.Some operator norm equalities are then derived.Secondly,we obtain some norm equalities and norm inequalities of certain linear combinations of the given two projections.Then,for every pair of projections P and Q,we give the full characterizations of ?P-Q?=1 and ?P-Q?<1,respectively Finally,for the given pair of projections,we introduce three invariant closed submodules and obtain some operator norm equalities and norm inequalities related to these three closed submodulesThe fifth chapter deals mainly with the weighted Moore-Penrose invertibility of op-erators related to a pair of weighted projections.The indefinite inner-product spaces considered in this chapter are induced by those weights that are self-adjoint and invertible operators on Hilbert C*-modules.We mainly study the weighted Moore-Penrose invertibil-ity of PQ-QP,where P and Q are weighted projections.Through a series of lemmas and calculations,we finally prove that PQ-QP is weighted Moore-Penrose invertible if and only if for any natural number ? and any complex number ?,the corresponding operator(PQ)?-?(QP)? is weighted Moore-Penrose invertible.