.2 ¡Á 2 Operator Matrices Inverse And The Nature Of The Compression Operator About When The Product Spectrum,

The study of operator theory began in 20th century. Since it is used widely in mathematics and other subjects, it got rapid development at the beginning of the 20th century. In this paper, we devoted to the inverse of 2×2 operator matrices and the spectral properties of Jordan products of positive contractions on separable Hilbert space.The present thesis is divided into four chapters and organized as follows:In Chapter 1, some notations and definitions are introduced and some well-known theorems are given. At first, we give the meanings of some notations. For example, we introduce the definitions of Moore-Penrose inverse, self-adjoint operator, positive operator, positive contractions, spectral, Jordan products, etc. Then we give some well-known theorems, such as, polar decomposition theorem, spectral mapping theorem, etc.In Chapter 2, we discuss the inverse of 2×2 operator matrices. Based on Schur complement, by using the technique of block operator matrices, a sufficient and necessary condition for which 2×2 operator matrixis inverse is given.In Chapter 3, we shall follow Choi's line to extend the work on matrices to operators acting on a separable infinite dimensional Hilbert space. we first discuss the spectral properties of Jordan products of positive contraction operators on separable Hilbert space. For two positive contraction operators A and B, a necessary condition for which -1/4∈σ(AB + BA) holds is given. It is that let A,B∈B(H)₁⁺, if -1/4∈σ(AB + BA), then(1)‖A‖=‖B‖= 1.(2) Both A and B are not inverse.(3) 1/4∈σ(AB).Subsequently, we character the properties of minimum spectral pairs. Through two theorems, we give a complete answer to the question in which condition for a positive contraction operator A there should be a positive contraction operator B such that -1/4∈σ(AB + BA) holds. Meanwhile, we shall see that such an operator B is not unique ifdim H>2.In Chapter 4, we discuss a problem which is introduced by Chapter 3. In Chapter 3, we know that there exist two positive contraction operators A and B such thatσ(AB + BA)∩(-∞,0)≠(?). Then, a natural question appears: whether there exist two positive contraction operators A and B such that when AB + BA≠0, AB + BA≤0 holds? After discussion, we give this question a negative answer. And at the same time, we shall studythat under which conditions for A, B∈B(H)₁⁺,AB + BA≥0 holds.