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Solutions of Operator Equations and Convex Combinations of Invertible Elements

Author: ZhangHui
Tutor: DuHongKe
School: Shaanxi Normal University
Course: Basic mathematics
Keywords: Operator equation Idempotent operator von Neumann algebra Convex combination
CLC: O177
Type: Master's thesis
Year: 2004
Downloads: 33
Quote: 3
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Abstract


Operator equation is an important branch of functional analysis. Operator the equation XA * X -t A = (t ≥ 1) positive operator sub-solution study has already begun in the 1990s, and in cybernetics, dynamic programming and statistical aspects of the application. But the majority of this equation is finite dimensional, this paper is mainly to study the characteristics of the solution of this equation in the infinite-dimensional case. Linear combination of still another type of special operator equation further research, that the idempotent operator operator equation, explore the power of the operator is idempotent operator some necessary and sufficient conditions. Attention has been the relationship between the the invertible operator algebra and unitary elements, this paper will further study the von Neumann algebra invertible and unitary elements. The following describes the structure and main contents of this article. The first chapter introduces some of the symbols to be used in this article, the definition and some of the more famous or known theorem. First, we introduce some symbolic representation, followed by the introduction of a numerical range and spectrum of the operator as well as the numerical range radius and the definition of the spectral radius given special operator, such as normal operator, self-adjoint operators, positive operators unitary operator definition. At the same time we are given the preparatory theorem is used, such as spectrum decomposition theorem, range contains Theorem indicators for Theorem. The second chapter of the infinite dimensional Hilbert space operator equation XA * X -t A = I (t ≥ 1) some properties. First study the equation is operator solution spectral radius of A, the radius of the range of values ??domain restrictions. Second, given the operator equation XA * X -t A = I (t ≥ 1) positive operators necessary and sufficient conditions of sub-solutions, and get more special: the equation is the norm operator solution for a necessary and sufficient condition is that A is not bounded below. And thus A the formal operator neutrons equation positive operators the necessary and sufficient conditions of sub-solutions and the use of the iterative method equation operator solution. Next, A norm in a certain range restrictions equations the positive operator sub-range solution, great solution and minimum solution to prove certain interval and by iterative methods. On the basis of the last in the operator equation XA * X -t A = I (t ≥ 1) is operator solution studies, further study the operator equation X S A * X -t A = I (s, t ≥ 1) the positive operator solution. Chapter of idempotent operators on a Hilbert space, the sum and difference of two idempotent operator is still necessary and sufficient condition for idempotent operator. And further study of the convex combination of two idempotent operators still Idempotent the necessary and sufficient conditions for the operator, but also gives a linear combination of the two idempotent operators meet their difference is self-adjoint idempotent operator equivalent proposition. Let P and Q are two idempotent operator, has been PQ ± QP is reversible operator equivalent proposition. Finally proved idempotent operator and its adjoint is similar. The fourth chapter focuses on the research role von Neumann algebra on the Hilbert space in the special relationship between invertible and unitary elements. Prove that a real number α ∈ [0,1 / 2] and elements in the von Neumann algebra A, if A- * A spectrum of [1-2α 1] in, there are two unitary elements U- to 1 and the U- 2 such that A = of αU 1 (1-α) U 2 was founded by reversible elements of all von Neumann algebra closure with a linear combination of the two unitary elements of the relationship between followed proved the von Ncumam, the algebra norm not greater than l elements is the closure of the reversible group necessary and sufficient condition. Finally, the overall relationship between the unit ball of reversible convex combination of the elements of the closure of the entire two unitary elements. households,

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