關(guān)于算子譜結(jié)構(gòu)和正交投影對(duì)的研究
發(fā)布時(shí)間：2018-08-22 08:19
【摘要】：算子理論是泛函分析重要的研究領(lǐng)域之一,它對(duì)于微分方程,調(diào)和分析及理論物理等學(xué)科都有著深刻應(yīng)用.其中譜結(jié)構(gòu),譜保持問(wèn)題以及正交投影對(duì)一直是眾多學(xué)者研究的熱點(diǎn)問(wèn)題.對(duì)于譜結(jié)構(gòu),Weyl型定理能很好的反映算子譜的分布特點(diǎn),因此對(duì)Weyl型定理及其變形推廣的研究是許多學(xué)者一直關(guān)注的問(wèn)題.同時(shí),譜結(jié)構(gòu)和部分譜子集作為代數(shù)的同構(gòu)不變量研究也引起了學(xué)者們的廣泛關(guān)注,即譜保持問(wèn)題.另一方面,基于Halmos正交投影對(duì)分解,許多學(xué)者運(yùn)用譜理論和Fredholm理論來(lái)研究正交投影對(duì),研究與正交投影對(duì)有關(guān)的范數(shù),譜及正交投影對(duì)的差積等.這些結(jié)果對(duì)算子譜理論有著非常重要的影響,同時(shí)仍有一些問(wèn)題引起學(xué)者們的關(guān)注.本文對(duì)Weyl型定理及其變形在緊攝動(dòng)下的穩(wěn)定性問(wèn)題,保持譜子集的可加映射以及具有固定差的正交投影對(duì)問(wèn)題進(jìn)行了更進(jìn)一步的研究.具體研究?jī)?nèi)容有三方面.在譜結(jié)構(gòu)方面,根據(jù)算子semi-Fredholm域的特點(diǎn)討論了算子Weyl定理在緊攝動(dòng)下有穩(wěn)定性的特征,其次探究了算子T在緊攝動(dòng)下有Weyl定理穩(wěn)定性和T2在緊攝動(dòng)下有Weyl定理穩(wěn)定性的關(guān)系,之后研究了 Weyl定理的一種變形(ω)性質(zhì)在緊攝動(dòng)下有穩(wěn)定性的等價(jià)條件,最后根據(jù)2×2上三角算子矩陣的特點(diǎn),利用對(duì)角線(xiàn)上元素的性質(zhì)來(lái)刻畫(huà)它的單值延拓性質(zhì).對(duì)于譜保持問(wèn)題中,首先由正規(guī)特征值定義了 m-正規(guī)特征值,然后討論m-正規(guī)特征值和m+1-正規(guī)特征值作為B(H)上的一個(gè)同構(gòu)或者反同構(gòu)不變量,之后刻畫(huà)了 B(X)上保持算子譜中semi-Fredholm域的可加映射的結(jié)構(gòu).最后,也討論了保持拓?fù)湟恢陆禈?biāo)集和保持單值延拓性質(zhì)穩(wěn)定性的線(xiàn)性映射.同時(shí),在正交投影對(duì)方面,研究了能表示成兩個(gè)正交投影差的自伴算子.首先討論自伴算子A是純的情形,先給出自伴算子A標(biāo)準(zhǔn)型的定義,探討一個(gè)自伴算子A表示成兩個(gè)正交投影差的充要條件,然后在此基礎(chǔ)上,給出滿(mǎn)足差為A的所有的正交投影對(duì)的一般表示,之后再把得到的結(jié)論推廣到一般自伴算子的情形,最后,從代數(shù)角度考慮,探討了差為A的所有正交投影生成的von Neumann代數(shù)及其換位的形式與結(jié)構(gòu).
[Abstract]:Operator theory is one of the important research fields of functional analysis. It has profound applications in the fields of differential equation harmonic analysis and theoretical physics. Among them, spectral structure, spectral preservation and orthogonal projection pairs have been the hot topics of many scholars. Weyl type theorem of spectral structure can well reflect the distribution characteristics of operator spectrum, so the study of Weyl type theorem and its extension is a problem that many scholars have been paying close attention to. At the same time, the study of spectral structure and partial spectral subsets as isomorphic invariants of algebras has also attracted extensive attention of scholars, that is, the problem of spectral preservation. On the other hand, based on the decomposition of Halmos orthogonal projection pairs, many scholars use spectral theory and Fredholm theory to study orthogonal projection pairs, and study the norm related to orthogonal projection pairs, the difference product of spectrum and orthogonal projection pairs, etc. These results have a very important influence on the theory of operator spectrum, and there are still some problems that attract the attention of scholars. In this paper, the problem of the stability of Weyl type theorem and its deformation under compact perturbation, the additive mapping of preserving spectral subsets and the orthogonal projection pair problem with fixed difference are studied further. The concrete research content has three aspects. In terms of spectral structure, according to the characteristics of operator semi-Fredholm domain, the stability of operator Weyl theorem under compact perturbation is discussed. Secondly, the relation between Weyl theorem stability of operator T under compact perturbation and Weyl theorem stability under compact perturbation is discussed. Then we study the equivalent condition that a kind of deformation (蠅) property of Weyl theorem has stability under compact perturbation. Finally, according to the characteristics of 2 脳 2 upper triangular operator matrix, we use the properties of elements on diagonal line to characterize its single-valued continuation property. For spectral preserving problem, m- normal eigenvalues are defined by normal eigenvalues, and then m- normal eigenvalues and m1-normal eigenvalues are discussed as an isomorphism or anti-isomorphism invariant on B (H). Then we characterize the structure of additive mappings in the semi-Fredholm domain in the spectrum of preserving operators on B (X). Finally, we also discuss the linear mapping that preserves the topological uniform scale-reducing set and the stability of the single-valued continuation property. At the same time, the self-adjoint operators which can be expressed as two orthogonal projection difference are studied on the orthogonal projection surface. In this paper, the definition of the canonical form of self-adjoint operator A is given, and the necessary and sufficient conditions for a self-adjoint operator A to be expressed as two orthogonal projection differences are discussed. In this paper, we give the general representation of all orthogonal projection pairs satisfying difference A, then generalize the result to the case of general self-adjoint operator. Finally, from the algebraic point of view, The form and structure of von Neumann algebras generated by all orthogonal projections with difference A and their commutations are discussed.
【學(xué)位授予單位】：陜西師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O177
本文編號(hào)：2196530
[Abstract]:Operator theory is one of the important research fields of functional analysis. It has profound applications in the fields of differential equation harmonic analysis and theoretical physics. Among them, spectral structure, spectral preservation and orthogonal projection pairs have been the hot topics of many scholars. Weyl type theorem of spectral structure can well reflect the distribution characteristics of operator spectrum, so the study of Weyl type theorem and its extension is a problem that many scholars have been paying close attention to. At the same time, the study of spectral structure and partial spectral subsets as isomorphic invariants of algebras has also attracted extensive attention of scholars, that is, the problem of spectral preservation. On the other hand, based on the decomposition of Halmos orthogonal projection pairs, many scholars use spectral theory and Fredholm theory to study orthogonal projection pairs, and study the norm related to orthogonal projection pairs, the difference product of spectrum and orthogonal projection pairs, etc. These results have a very important influence on the theory of operator spectrum, and there are still some problems that attract the attention of scholars. In this paper, the problem of the stability of Weyl type theorem and its deformation under compact perturbation, the additive mapping of preserving spectral subsets and the orthogonal projection pair problem with fixed difference are studied further. The concrete research content has three aspects. In terms of spectral structure, according to the characteristics of operator semi-Fredholm domain, the stability of operator Weyl theorem under compact perturbation is discussed. Secondly, the relation between Weyl theorem stability of operator T under compact perturbation and Weyl theorem stability under compact perturbation is discussed. Then we study the equivalent condition that a kind of deformation (蠅) property of Weyl theorem has stability under compact perturbation. Finally, according to the characteristics of 2 脳 2 upper triangular operator matrix, we use the properties of elements on diagonal line to characterize its single-valued continuation property. For spectral preserving problem, m- normal eigenvalues are defined by normal eigenvalues, and then m- normal eigenvalues and m1-normal eigenvalues are discussed as an isomorphism or anti-isomorphism invariant on B (H). Then we characterize the structure of additive mappings in the semi-Fredholm domain in the spectrum of preserving operators on B (X). Finally, we also discuss the linear mapping that preserves the topological uniform scale-reducing set and the stability of the single-valued continuation property. At the same time, the self-adjoint operators which can be expressed as two orthogonal projection difference are studied on the orthogonal projection surface. In this paper, the definition of the canonical form of self-adjoint operator A is given, and the necessary and sufficient conditions for a self-adjoint operator A to be expressed as two orthogonal projection differences are discussed. In this paper, we give the general representation of all orthogonal projection pairs satisfying difference A, then generalize the result to the case of general self-adjoint operator. Finally, from the algebraic point of view, The form and structure of von Neumann algebras generated by all orthogonal projections with difference A and their commutations are discussed.
【學(xué)位授予單位】：陜西師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2016
【分類(lèi)號(hào)】：O177
【參考文獻(xiàn)】
相關(guān)期刊論文 前2條
1 M.BERKANI;;On the Equivalence of Weyl Theorem and Generalized Weyl Theorem[J];Acta Mathematica Sinica(English Series);2007年01期
2 ;SPECTRUM-PRESERVING ELEMENTARY OPERATORS ON B(X)[J];Chinese Annals of Mathematics;1998年04期
，本文編號(hào)：2196530
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