Solutions Of Operator Equations And Convex Combinations Of Invertible Elements

Since 1990s', the researches on operator equations X + A*X-tA = I(t ≥ 1) in a finite dimensional space which have always played an important role on the subject of functional analysis and have been applied to some fields such as control, dynamic programming and statistic have been considered by many authors. In this paper, we continue to study the operator equation X + A*X~1 A = I ( t ≥ 1) in an infinite dimensional Hilbert space not finite dimensional. We mainly discuss some properties about positive solutions of this equation. Furthermore, we study some relation about idompotent operators. On the other hand, we further investigate the relations between invertible elements and unitary elements and between invertible group and unitary group of von Neumann algebra acting on a Hilbert space. These results play a fundamental part in studying structure of operators.This paper contains four chapters. Chapter 1 mainly introduces some terminologies and notation, definitions and some simpler theorem or more known theorem. Firstly, we introduce some terminologies and notation, and introduce the definitions of numerical range, spectra of operator, radius of numerical range and radius of spectra etc. Secondly, we give some definitions of normal operator, Hermitian operator, positive operator, unitary operator etc. At last, we introduce some known theorems such as spectral theorem, range inclusion theorem, index theorem etc.Chapter 2 discusses some properties about positive solutions of operator equation X + A* X-6 A = I( t ≥ 1) in an infinite dimensional Hilbert space. Firstly, we discuss spectral radius and numerical range radius of A if X + A*X-6A = I has a solution. Secondly, we obtain necessary and sufficient conditions for positive solutions of X + A*X-t A = I and get that if X + A*X-tA =I has a solution X, then ||X|| = 1 if and only if A is not bounded below. When A is normal, we can obtain some necessary and sufficient conditions for this equation having a positive solution and positive solutions of operator equation through iterative methods. We apply iterative method to obtain the maximal and minimal solutions. In the end we study operator equation Xs + A*X-tA = I( s, t ≥ 1) based on the last sections.Chapter 3 is mainly about some properties of idempotent operators. We give necessary and sufficient conditions about sum and difference1 of two idempotent operators tobe idempotent. Furthermore, we study necessary and sufficient conditions about which linear combination of two idempotent operators is still idempotent. Assume P and Q are two idempotent operators, we can get necessary and sufficient conditions which PQ\ QP are invertible operators. In the end we prove that idempotent operator is similar to its adjoint.Chapter 4 deals with the relations between invertible elements and unitary elements and between invertible group and unitary group of von Neumann algebra A acting on an infinite dimensional Hilbert space. We state that if a ∈ [0, r2] and A is in von Neumann algebra, and the spectra of A* A are contained in [1 - 2a, 1], then there exist two unitary element U\ and Vi in von Neumann algebra A such that A = aU1 + (1 - a)U2. Moreover, we get the relation between the closure of the set of invertible elements and linear combinations of unitary elements. Then we prove necessary and sufficient conditions which an elements is in the closure of invertible group which norm is not more than 1 and convex combination of two unitary elements.