【作者】 任芳国；

【导师】 杜鸿科；

【作者基本信息】 陕西师范大学 ， 基础数学， 2004， 博士

【摘要】 算子补问题及算子数值域是近年来算子论中最活跃的研究课题，在算子论的研究中有着重要的理论价值和应用价值。对它们的研究涉及到基础数学与应用数学的许多分支，诸如代数学、几何理论、算子扰动理论、Banach代数、C^*-代数、矩阵理论、图论、数值分析、优化理论、组合理论，通过对它们的研究可使得算子结构的内在关系变的更加清晰，为不变子空间问题的研究奠定了坚实的理论基础，特别在控制论、系统论、振动理论及稳定性理论、数值计算、插值理论等学科中得到了广泛的应用。本文的研究内容涉及到算子补问题与广义数值域两大方面。在算子补问题方面，系统地研究了算子的谱配置问题、逆补问题、谱补问题三大问题。在广义数值域方面，深入研究了算子数值域的重要推广：算子n-次数值域、算子多项式数值域及其谱理论两大问题。全文分为两大部分共五章： 第一部分包括前三章，第一章系统阐述了算子补问题的背景、发展概况、所取得的主要结果及其意义，并探讨了算子对可控的一系列等价条件，丰富了算子对可控性的内容，并利用我们的研究成果给出了详细而有清晰的证明，还获得了在一定条件下谱配置问题的深刻结果，并应用空间分解理论、算子分块矩阵技巧、Douglas理论、Read提升理论，给出谱配置定理及其广义谱配置定理的构造性证明。 第二章致力于定义在Hilbert空间上算子逆补问题的研究。利用构造分块初等矩阵、算子广义逆、极分解、谱分解为工具，深入地研究了算子对的逆补M_XY、算子逆配置的一系列等价刻画及一般性结论、缺项算子矩阵存在逆补M_x的等价刻画，并获得了逆补的预解集的性质刻画。 第三章系统地研究了定义在Hilbert空间上缺项算子矩阵M_x的谱补问题。借助于Takahashi理论、算子指标理论及谱理论，讨论了在三元算子对（A，B，C）可控及可容许的条件下，M_x的谱的交集的特性及其谱的分布情况，并利用数学归纳法给出了矩阵迹补的证明。 第二部分包括后两章，第四章论述了数值域的起源、发展及研究成果，深入研究了算子的n-次数值域具有的基本特性、n-数值域与算子谱及二次数值域的关系、n-次数值域与各主子算子矩阵的分块数值域的关系，并讨论了在不同空间分解下，对各算子矩阵的分块数值域之间的关系进行刻画，利用n-次数值域作为研究工具，给出了预解算子范数估计、若当链长度刻画，同时给出了二次数值域的一般性包含区域。 第五章研究了算子多项式数值域的性质、算子多项式数值域与n-次数值域的关系，特别地利用矩阵范数、矩阵的奇异值、非负矩阵的理论、友矩阵为工具，给出了算子多项式数值域及谱的范围全面刻画，并深刻地研究了算子线性束的正则性的等价条件及谱分布. 本文所取得的研究成果可分为以下八个方面: （l）研究了Hilb，艺空间上算子的可控性.通过B叭ach极限理论、构造算子分块矩阵的技巧、正规算子谱理论的恃性，获得了算子可控性的一系列等价刻画. （2）研究了著名的算子谱配置问题.作为研究基础，引入了算子对可控性与可容许性指标及算子卜限制，并利用算子闭值域特性、值域逼近及空间的不同分解，获得了算子谱配置问题及广义的算子谱配置间题一般性证明. （3）通过引入初等分块算子矩阵，研究并获得了算子对可逆的一般性条件. （4）研究了算子逆配置的等价条件，并借助算子的。一限制逆配置的刻画及正算子、紧算子的特性，完全刻画了算子对的逆配置. （5）深刻研究缺项算子矩阵门M丫存在逆补的一般性条件，并获得了补矩阵坷尤的所有预解集交集的特性. （6）借助于极分解、谱分解、算子谱理论，深刻研究缺项算子矩阵.在三元算子对（A，B，c）可控、可容许条件下材x的谱的交集及谱半径的特性. （7）研究了算子的n一次数值域具有的基本特性、并探讨了。此数值域与各主子算子矩阵的分块数值域及其在不同空间分解下分块数值域之间的关系，在此基础上，利用。一次数值域为工具获得了预解算子范数估计、若当链长度刻画，同时给出了二次数值域的一般包含区域. （s）应用矩阵范数理论、矩阵的奇异值理论、非负矩阵的理论、友矩阵、算子的谱理论为工具，获得了算子多项式数值域具有的特性及其包含范围，算子多项式数值域与n一次数值域的关系，最后研究了算子线性束正则性的等价条件及谱分布情况.更多还原

【Abstract】 Operator completions and numerical range of operator are very active topics in operator thoery 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,perturbation theory,Banach-algebra,C*-algebra, matrix analysis,graph,numerical analysis, optimation theory,combinatorial theory ect.Through research for them, the interior relation and construction among operators can be found and a substantial basis can be provided for the study of the invariant subspaces problems.At the same time,they can be extendly applied in other sciences such as control theory, system theory,vibration theory,stability theory,numerical computer,interpolation theory ect. The research of this thesis is on both operator completion and generalized numerical ranges of operator.The research on operator completion comes the following topics:operator spectrum as-signment,invertible completions,spectral completions. The research on generalized numerical ranges of operator contains n-numerical ranges and numerical ranges of operator polynomial.This article is divided into five chapters.In chapter 1, the background ,the developing general situation,the main results and significance of operator completion problems are described. The equivalent characterizations on controllable operator pairs are explored. By the results that we ob-tained,we supply a carefull and clear proof in controllable oprators.Moreover,we give construction proofs on spectrum assignment problem and generalized spectrum assignment problem by the techniques in block operator matrix, Douglas theory,Read lift theory.In chapter 2, the invertible completions on operator pairs are discussed firstly. Secondly,the invertible assignment problem are studied.At last,we explored the conditions on the invertible completion of part matrix MX and obtained the characterizations of the relovent operator of invertible completion.In chapter 3, the spectral completion problem over operator space are treated completely. We studied the intersection of spectra of operator completions under the conditions that the triple (A, B, C) is controllabe and admissible respectively and the part results on distribution of spectrum are solved.In chapter 4, the origin,development and researing results of numerical range are introducd. The basic properties on n-numerical range are presented and the relationships between n-numerical range and operator spectrum.Moreover, the block numerical ranges of main submatrix are established.The estimate of norm of resolvent operator and the length of Jordan are characterized in the terms of n-numerical range.Lastly,the general locations of quadratic numerical range are obtained.In chapters, the serial properties of numerical range of operator polynomial areput firstly.Secondly,the relation between numerical range of operator polynomial and n-numerical range are considered. Thirdly,in the light of matrix norm.matrix singular value.the regions and bounds of numerical range and spectrum of operator polynomial are dicussed carefully.In the end, the regularity and spectral distribution of operator linear pencils are studied.The researching results on the thesis consist of the following statements.1. The equivalent conditions of controllability of operators are obtained.2. The well-known spectrum assignment problem are discuussed and a general proofs for it and its generalization are gived.3. The general conditions of invertible completions of operator pairs are acquired.4. By introducing the e-restriction of operator and the character on positive and compact operators,we completely solved the invertible assignment of operator pairs.5. The general conditions of invertible completions of operator matrix MX are supplied and the intersective characterization of the resolvent set of operator of MX are been desribed.6. By using the polar decomposition,spectral decomposition,the spectral theory of operators,the intersection of the spectra of operator completi更多还原