算子矩阵的补问题和谱
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摘要
算子矩阵是近年来算子理论中最为活跃的研究课题之一,其研究涉及到基础数学与应用数学的许多分支,如矩阵理论、优化理论和量子物理等等.本学位论文主要考虑算子矩阵的补问题和谱,内容包括算子矩阵的谱补问题、Fredholm补问题和可逆补问题以及非自伴算子矩阵的谱结构等方面.
用H1和H2表示Hilbert空间,召(Hi,Hj)表示从Hi到Hj的所有有界线性算子构成的Banach空间且简记为B(Hi,Hi)＝B(Hi),i,j＝1,2.本文的主要结论如下：
首先,研究了缺项算子矩阵(?)的Moore-Penrose谱补问题.对给定的算子A∈B(H1)和B∈B(H2),刻画了下面的集合其中σM(·)表示Moore-Penrose谱.
其次,讨论了缺项算子矩阵(?)的右(左)Fredholm补问题和右(左)可逆补问题.对给定的算子A∈B(H1)、B∈B(H2)和C∈B(H2,H1),分别得到了存在X∈B(H1,H2)使得算子矩阵为右(左)Fredholm算子和右(左)可逆算子的充分必要条件.
然后,考虑了缺项算子矩阵(?)的Fredholm补问题和可逆补问题.对给定的算子A∈B(H1)和C∈B(H2,H1),分别给出存在X∈B(H1,H2)和Y∈B(H2)使得算子矩阵为右(左)Fredholm算子、右(左)可逆算子、Fredholm算子和可逆算子的充分必要条件.
最后,研究了某类非自伴算子矩阵的谱结构,作为推论得到了无穷维Hamilton算子和J-自伴算子矩阵的相关性质.一般情况下,无穷维Hamilton算子是一类特殊的非自伴算子矩阵,其研究有着重要的理论价值和应用价值.因此,还考虑了斜对角型无穷维Hamilton算子的近似点谱和上三角型无穷维Hamilton算子的本质谱,给出无穷维Hamilton算子的近似点谱(或本质谱)和其元素算子的近似点谱(或本质谱)之间的关系.
Operator matrices are heated topic in operator theory, and the research of operator matrices has related to pure and applied mathematics such as matrix analysis, optimality principle and quantum physics etc. This dissertation mainly studies completion problems and spectra of operator matrices, including spec-tral completion problems, Fredholm completion problems, invertible completion problems of operator matrices and the structure of spectra of nonselfadjoint op-erator matrices.
We use H1 and H2 to denote Hilbert spaces. Let B(Hi,Hj) be the Banach space of all bounded linear operators from Hi to Hj, and write B(Hi, Hi)= B(Hi), i,j= 1,2. The main results of this dissertation are as follows:
Firstly, completion problems of the Moore-Penrose spectrum for an operator partial matrix (?) are discussed. For given operators A∈B(H1) and B∈B(H2), the following sets are characterized, whereσm(·) denotes the Moore-Penrose spectrum.
Secondly, right (left) Fredholm completion problems and right (left) invert-ible completion problems for an operator partial matrix (?) are studied. For given operators A∈B(H1), B∈B(H2) and C∈B(H2,H1), the necessary and sufficient conditions are obtained, respectively, for operator matrix to be a right (left) Fredholm operator and a right (left) invertible operator for some X∈B(H1,H2).
Then, Fredholm completion problems and invertible completion problems for an operator partial matrix (?) are considered. For given operators A∈B(H1) and C∈B(H2,H1), the necessary and sufficient conditions are obtained, respectively, for operator matrix to be a right (left) Fredholm operator, a right (left) invertible operator, a Fred-holm operator and an invertible operator for some X∈B(H1,H2) and Y∈B(H2).
Lastly, the structure of spectra for some class of nonselfadjoint operator matrices is studied. As corollaries, some related properties of infinite dimen-sional Hamiltonian operators and J-selfadjoint operator matrices are obtained. In general, infinite dimensional Hamiltonian operators are a class of special non-selfadjoint operator matrices, its study has important value in both theory and applications. Therefore, the approximate point spectrum of off-diagonal infinite dimensional Hamiltonian operators and the essential spectrum of upper trian-gular infinite dimensional Hamiltonian operators are considered. A relationship between the approximate point spectrum (or the essential spectrum) of infinite dimensional Hamiltonian operators and the approximate point spectrum (or the essential spectrum) of their entries is given.
用H1和H2表示Hilbert空间,召(Hi,Hj)表示从Hi到Hj的所有有界线性算子构成的Banach空间且简记为B(Hi,Hi)＝B(Hi),i,j＝1,2.本文的主要结论如下：
首先,研究了缺项算子矩阵(?)的Moore-Penrose谱补问题.对给定的算子A∈B(H1)和B∈B(H2),刻画了下面的集合其中σM(·)表示Moore-Penrose谱.
其次,讨论了缺项算子矩阵(?)的右(左)Fredholm补问题和右(左)可逆补问题.对给定的算子A∈B(H1)、B∈B(H2)和C∈B(H2,H1),分别得到了存在X∈B(H1,H2)使得算子矩阵为右(左)Fredholm算子和右(左)可逆算子的充分必要条件.
然后,考虑了缺项算子矩阵(?)的Fredholm补问题和可逆补问题.对给定的算子A∈B(H1)和C∈B(H2,H1),分别给出存在X∈B(H1,H2)和Y∈B(H2)使得算子矩阵为右(左)Fredholm算子、右(左)可逆算子、Fredholm算子和可逆算子的充分必要条件.
最后,研究了某类非自伴算子矩阵的谱结构,作为推论得到了无穷维Hamilton算子和J-自伴算子矩阵的相关性质.一般情况下,无穷维Hamilton算子是一类特殊的非自伴算子矩阵,其研究有着重要的理论价值和应用价值.因此,还考虑了斜对角型无穷维Hamilton算子的近似点谱和上三角型无穷维Hamilton算子的本质谱,给出无穷维Hamilton算子的近似点谱(或本质谱)和其元素算子的近似点谱(或本质谱)之间的关系.
Operator matrices are heated topic in operator theory, and the research of operator matrices has related to pure and applied mathematics such as matrix analysis, optimality principle and quantum physics etc. This dissertation mainly studies completion problems and spectra of operator matrices, including spec-tral completion problems, Fredholm completion problems, invertible completion problems of operator matrices and the structure of spectra of nonselfadjoint op-erator matrices.
We use H1 and H2 to denote Hilbert spaces. Let B(Hi,Hj) be the Banach space of all bounded linear operators from Hi to Hj, and write B(Hi, Hi)= B(Hi), i,j= 1,2. The main results of this dissertation are as follows:
Firstly, completion problems of the Moore-Penrose spectrum for an operator partial matrix (?) are discussed. For given operators A∈B(H1) and B∈B(H2), the following sets are characterized, whereσm(·) denotes the Moore-Penrose spectrum.
Secondly, right (left) Fredholm completion problems and right (left) invert-ible completion problems for an operator partial matrix (?) are studied. For given operators A∈B(H1), B∈B(H2) and C∈B(H2,H1), the necessary and sufficient conditions are obtained, respectively, for operator matrix to be a right (left) Fredholm operator and a right (left) invertible operator for some X∈B(H1,H2).
Then, Fredholm completion problems and invertible completion problems for an operator partial matrix (?) are considered. For given operators A∈B(H1) and C∈B(H2,H1), the necessary and sufficient conditions are obtained, respectively, for operator matrix to be a right (left) Fredholm operator, a right (left) invertible operator, a Fred-holm operator and an invertible operator for some X∈B(H1,H2) and Y∈B(H2).
Lastly, the structure of spectra for some class of nonselfadjoint operator matrices is studied. As corollaries, some related properties of infinite dimen-sional Hamiltonian operators and J-selfadjoint operator matrices are obtained. In general, infinite dimensional Hamiltonian operators are a class of special non-selfadjoint operator matrices, its study has important value in both theory and applications. Therefore, the approximate point spectrum of off-diagonal infinite dimensional Hamiltonian operators and the essential spectrum of upper trian-gular infinite dimensional Hamiltonian operators are considered. A relationship between the approximate point spectrum (or the essential spectrum) of infinite dimensional Hamiltonian operators and the approximate point spectrum (or the essential spectrum) of their entries is given.
引文
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[3]阿拉坦仓,黄俊杰,范小英,L2×L2中的一类无穷维Hamilton算子的剩余谱,数学物理学报,2005,25(7)：1040-1045.
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[5]Alatancang, D. Wu, Spectrums of 2×2 upper triangular Hamiltonian oper-ators, Mong. Math. J.,2007,11:13-22.
[6]Alatancang, D. Wu, Completeness in the sense of Cauchy principal value of the eigenfunction systems of infinite dimensional Hamiltonian operator, Sci. China Ser. A,2009,52(1):173-180.
[7]阿拉坦仓,张鸿庆,钟万勰,一类偏微分方程的无穷维Hamilton正则表示,力学学报,1999,31(3)：347-357.
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[9]T. Y. Azizov, I. S. Iokhvidov. Linear operators in spaces with an indefinite metric, John Wiley & Sons, Chichester,1989.
[10]T. Y. Azizov, V. K. Kiriakidi, G. A. Kurina, An indefinite approach to the problem of the reducibility of a nonnegatively Hamiltonian operator function to block diagonal form, Funct. Anal. Appl.,2001,35(3):220-221.
[11]T. Y. Azizov, A. Dijksma, and I. V. Gridneva, On the boundedness of Hamiltonian operators, Pro. Amer. Math. Soc.,2003,131(2):563-576.
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[16]G. D. Birkhoff, Note on the expansion problems of ordinary linear differential equations, Rendiconti del Circolo Matematico di Palermo,1913,36(1):115-126,
[17]X. Cao, Browder essential approximate point spectra and hypercyclicity for operator matrices, Linear Algebra Appl.,2007,426(2-3):317-324.
[18]X. Cao, M. Guo, B. Meng, Semi-Fredholm spectrum and Weyl's theorem for operator matrices, Acta Math. Sin. (Engl. Ser.),2006,22(1):169-178.
[19]X. Cao, M. Guo, B. Meng, Drazin spectrum and Weyl's theorem for operator matrices, J. Math. Res. Exposition,2006,26(3):413-422.
[20]X. Cao, B. Meng, Essential approximate point spectra and Weyl's theorem for operator matrices, J. Math. Anal. Appl.,2005,304(2):759-771.
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[27]S. V. Djordjevic, H. Zguitti, Essential point spectra of operator matrices through local spectral theory, J. Math. Anal. Appl.,2008,338(1):285-291.
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