Several Operator Theories On The Indefinite Inner Product Space

Generalized inverse is an important part of modern mathematics.It has important applications in many scientific fields,such as cybernetics,optimization,numerical calcula-tion,etc.Weighted Moore-Penrose inverse(here and after referred to weighted M-P inverse)is an important part of generalized inverse theory.Available references mainly focus on the case of positive definite weight,few references focus on the positive semi-definite weight and fewer references focus on non-positive semi-definite weight.This article based on the adjointalbe operator on a HilbertC~*-module and focus on an adjointable and invertible weight.Then we study the equivalence of the commutator of weighted projection opera-tor(in short weighted projection)and generalized commutator.The problem is that when weight is positive definite,we can induce a new inner product.However,when weight is adjointable and invertible,we can't induce an inner product or a norm.Although we can define a weighted space,it is a indefinite inner product space.Besides,we add a complex number ? to our main result.These condition may cause much trouble during the study,for example,we can't use spectrum theory.Therefore,we need new methods and technics.Assume A and B are two adjointable operator on a Hilbert C~*-module and ? is a complex number.We call A-B the commutator of A and B,A-?B the generalized commutator of A and B.We first introduce the conception of weighted projection' and assume P and Q are two weighted projection.Then we mainly study following two aspect.First one' the equivalence between the generalized commutator of P and Q and the linear combination of P and Q.Second one' the equivalence between the generalized commutator of(PQ)k and(QP)l and the commutator of PQ and QP,k is an positive integer.As far as we know,there are some results on C~*-algebra and Hilbert space concerning the commutator of two projections.However,for generalized commutator our results is new even for matrix cases.In the third chapter of our article,we mainly study the equivalence between the generalized commutator of P and Q and the linear combination of P and Q.Through complicated compute,we gain an mutual expression of them.And then,we use four Penrose equation to verify the accuracy of the expression.Thus,we prove the equivalence of them.In the fourth chapter of our article,we mainly study the equivalence between the generalized commutator of(PQ)k and(QP)k and the commutator of PQ and QP.In the same way,we use four Penrose equation to verify the correctness and prove the equivalence of them.