Perturbation Theory And Applications Of Conjugable Operators On Hilbert C~*-modules

As an extremely important part of operator theory,the perturbation theory of oper-ators originating from the practical needs of numerical calculation,signal processing and quantum physics etc,has a wide range of applications.The perturbation analysis of oper-ators is closely related to the underlying space,which usually turns out to be the Hilbert space or the Banach space.Up to now little is done in the case of the Hilbert C~*-module.In the setting of the multiplicative perturbation of adjointable operators on Hilbert C~*-modules,this dissertation focuses on the study of such topics as the Moore-Penrose inverse(briefly M-P inverse),the partial isometry factor in the polar decomposition,the general-ized parallel sum.In addition,the generalized Douglas range inclusion theorem associated with certain indefinite inner product spaces is also studied.The second chapter deals mainly with two kinds of multiplicative perturbations,name-ly,the strong perturbation and the weak perturbation introduced in recent years.Based on block matrix representations of M-P inverses associated with the weak perturbations of operators,together with some C~*-algebraic techniques,some key norm equations relat-ed to M-P inverses of the weak perturbation are obtained.As a result,some new norm upper bounds of M-P inverses of operators are derived in the cases of the strong perturba-tion and the weak perturbation,respectively.The sharpness of the newly obtained upper bounds is illustrated by numerical examples and the numerical values of the solutions of certain least squares problems.The relationship between several kinds of multiplicative perturbations,as well as the internal relationship between multiplicative perturbations and additive perturbations,are clarified.Specifically,it is shown that in the matrix case,a multiplicative perturbation is a strong perturbation if and only if it is a rank preserving weak perturbation.The third chapter is concerned with the perturbation estimation of the partial isometry factor in the polar decomposition both for operators and matrices.The singular value decomposition,acts as the main tool in the matrix case,is apparently not feasible for general operators.A formula in block matrix form is derived for the partial isometry factor under concern conditions,which gives a new way to study the perturbation estimation of the partial isometry factor in the polar decomposition.Based on this newly obtained formula,some necessary and sufficient conditions are given under with U_M=U_T,where M and T are two matrices of same order and same rank,U_M=U_T are the partial isometries in the polar decompositions ofand,respectively.A new upper bound of the Frobenius norm of U_M=U_T is provided in the case that M=ETF,where E and F are both positive definite matrices.The sharpness of this upper bound is illustrated by numerical tests.The parallel sum of operators is a natural generalization of the parallel sum of positive semi-definite matrices.Most literatures dealing with the parallel sum of two operators A and B are focused either on the case that the range of A+B is closed,or on the case that both A and B are positive.As far as we know,little is done in the exceptional case,which is the concern of the fourth chapter.The term of tractable pair of operators is introduced,and a new kind of parallel sum,called the generalized sum,is also introduced for a tractable pair of operators.Some basic properties of this generalized parallel sum are given,a factorization theorem concerning the positive perturbation of the generalized parallel sum is provided.Based on this factorization theorem and the construction of a common upper bound for a tractable pair of positive operators,some norm upper bounds of the generalized parallel sum are derived.Consequently,some results known in the matrix case are generalized.In the last part of this chapter,a gap of a known result on the harmonic mean of two operators is removed.Some new phenomena may happen if an operator acts on a weighted inner product space induced by a weight that is not positive definite.The weights considered in the fifth chapter are self-adjoint and invertible operators that act on Hilbert C~*-modules.A generalized Douglas range inclusion theorem is derived in this setting of weighted inner product spaces.It is helpful to study the perturbation estimates of solutions of linear operator equations on indefinite inner product spaces.