Solutions Of Operator Equations And The Drazin Inverse Of The Operators

Operator theory has always played an important role on the subject of functional analysis and has been considered by many authors. Operator equation is one of the hotest topics in operator theory. The researches on the positive solutions to operator equations began in 1990s' and have been applied to some fields such as control, dynamic programming and statistic. In resent years, operator equation got a great developtment and many scholors devoted to studying different kinds of operator equations. In this article, we discuss three kinds of operator equations X + A^*X^-2A = Q,X + A^*X^-tA = Q (0 < t ≤ 1) and X - A^*X^-tA = I(t > l)in an infinite dimensional space. We mainly discuss some properties of the positive operator solutions to these equations and characterise these three kinds of equations. Furthermore, we study the Drazin invertibility of the perturbed operator in an infinite dimensional Hilbert space and give the bound of the relative error.This paper contains three chapters.Chapter 1 mainly introduces some terminologies, notations and some well-known theorems which are used late in this paper. Firstly, we introduce some terminologies and notation, and introduce the definitions of numerical range, spectrum , radius of numerical range and spectral radius of operator etc. Secondly, we give some definitions of normal operator, self-adjoint operator, positive operator, the Drazin inverse of operator and the index of Drazin invertible operator etc. At last, we introduce some properties of positive operator and some well-known theorem such as spectral theorem, spectral mapping theorem, range inclusion theorem ect.Chapter 2 discusses three kinds of operator equations: X + A^*X^-2A = Q, X + A^*X^-tA = Q and X — A^*X^-tA = I in infinite dimensional Hilbert space. Firstly, we study the properties of the positive operator solutions to the operator equation X + A^*X^-2A = Q. Also, we dicuss the relations of operator A, Q and X in the form of norm, spectral radius and radius of numerical range when this equation has positive solutions. Moreover, we obtain some necessary conditions under whichthe operator equation X + A*X 2A = Q has positive operator solutions. Sen-condly, we obtain some conditions for positive operator solutions to the operator equation X + A*X^tA = Q and discuss the norm, the spectral radius of A and Q when the equation has positive operator solutions. We also get that if the equation X + A*X^lA = Q has positive operator solutions, then it has the maximal solution Xl- Moreover, we apply iterative method to obtain the maximal positive solution of the equation X + A*X^tA = Q. Thirdly, we obtain some necessary conditions for positive operator solution of the operator equation X — A*X^*A = /. We also get that the equation X + A*X^lA = J(t > 1) has a positive solution with norm of 1 if and only if A is not bounded below. When A is normal, t = 2m(m is an positive interge), we apply effective iterative method to obtain the positive solution of the equation X - A*X^lA = I.Chapter 3 is mainly about the Drazin inverse of the perturbed operator. We obtain some sufficient conditions for a Drazin invertible operator to be still Drazin invertible under a perturbation and the expression for the Drazin inverse of the perturbed operator is derived. Moreover, based on the expressing, we obtain the bound of the relative error, which extends the conclusions of Y.M.Wei.