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Research on Spectra of Operator Matrices and Related Topics
Author: LiYuan
Tutor: DuHongKe
School: Shaanxi Normal University
Course: Basic mathematics
Keywords: Projection Idempotent operator MoorePenrose inverse Drazin inverse Quantum effects Weyl's theorem the a Weyl 's theorem C ~~ * algebra Fredholm operator Essential spectrum
CLC: O177
Type: PhD thesis
Year: 2007
Downloads: 248
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Abstract
Projections, operator spectra, Weyl theorem and quantum effects are heated topics in operator theory and also have important value in both theory and application. The research of these subjects has related to pure and applied mathematics such as algebra, geometry, operator perturbation theory, matrix analysis, approximation theory, multivariate linear modeling, optimality principle and quantum physics etc. The research of this thesis focuses on spectral complements of operator matrices, the MoorePenrose inverse and Drazin inverse of projections in a C^{*}algebra, Weyl theorem and infimum and generalized infimum of quantum effects on a Hilbert space. Using the technique of operator blocks, the internal relations and constructions among operators can be found. This article is divided into five chapters.In chapter 1, by using space decomposition and block operator matrices, the perturbations of left spectra, left essential spectra and essential approximate point spectra of upper triangular operator matrices are treated completely. For a given pair of operators （A, B）, some sufficient and necessary conditions under which M_{c} is left invertible for some left invertible （invertible） operator C are given. As an extension, we also obtain some sufficient and necessary conditions under which M_{c} is left Fredholm and left Weyl for some left invertible （invertible） operator C, respectively.In chapter 2, the sufficient and necessary condition under which the difference of two idempotent operators on infinite dimensional Hilbert spaces is Fredholm （invertible） is discussed. In addition, some equivalent conditions under which products and differences of projections in a C^{*}algebra are MoorePenrose （Drazin） invertible are obtained. At last, some expresses of the MoorePenrose （Drazin） inverse of products and differences of projections in a C^{*}algebra are established.In chapter 3, some conditions under which Weyl’s theorem and Browder’s theorem survive for an operator on the Banach space are considered and explored. As a corollary, we show that Weyl’s theorem holds for analytically Mhyponormal operators and aWeyl’s theorem holds for analytically cohyponormal operators. In addition, we consider some conditions under which Weyl’s and Browder’s theorem hold for operator matrices.In chapter 4, the relationship between the reduced minimum modulus of the left multiplicative operator L_{A} and the reduced minimum modulus of operator A is discussed. In addition, we study the reduced minimum modulus of the generalized Drazin inverse of an operator on a Banach space and give out lower and upper bounds of the reduced minimum modulus of an operator and its generalized Drazin inverse, respectively. In chapter 5, using the method of the spectral theory of operators, we consider condition under which the infimum A∧B exists for two quantum effects A and B. As a corollary, we give an affirmative answer of a conjecture of Gudder. In addition, some properties of generalized infimum A （?） B are considered.The results from the thesis consist of the following statements.1. For a given pair （A, B） of operators, let Mc The perturbations of left spectra, left essential spectra and essential approximate point spectra of an operator matrix Me are considered, then the sets:∩_{c∈} Inv（k, H）σ_{le}（M_{c}）,∪_{c∈} Inv（K, H）σ_{le}（M_{c}）,∩_{c∈} Inv（K, H）σ_{ca}（M_{c}） and∪_{c∈} Inv（K,H）σ（la）（M_{c}） are characterized completely, whereσ_{le}（T） andσ_{ea}（T） denote the left essential spectrum and essential approximate point spectrum of T, respectively.2. Let Mx =（?） be a 2×2 operator matrix acting on the Hilbert space H （?） K. For given operators A, B and C, the sets∪_{x∈B}（H, K）σ_{e}（M_{x}） and∪_{x∈B}（H, K）σ_{w}（M_{x}） are characterized completely, whereσ_{e}（T） andσ_{w}（T） denote the essential spectrum and Weyl spectrum of T, respectively.3. For given A, B and C, when C is a compact operator, we give the sufficient and necessary condition under which Mx is left invertible, for some operator X. And also obtain the sufficient and necessary condition under which Mx is left invertible, for all operators X.4. We give the characterizations that the difference of two idempotent operators is Fredholm （invertible）.5. The Drazin and MoorePenrose inverses of product and difference of orthogonal projections in a C^{*}algebra are established.6. A new sufficient and necessary condition under which aWeyl theorem holds for an operator T on a Banach space is obtained. As a corollary, we obtain that aWeyl’s theorem holds for analytically cohyponormal operators.7. Let A∈B（H）. Define a left （right） multiplicative operator on B（H） by L_{A}（B） = AB（R_{A}（B） = BA）, for all B∈B（H）. We get thatγ（L_{A}） =γ（R_{A}） =γ（A）.8. Using the method of the spectral theory of operators, we give an affirmative answer of a conjecture of Gudder.

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