| 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... |
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