算子理论在量子信息和非交换计算中的若干应用
|
摘要
算子矩阵,算子数值域,算子谱理论以及交换子都是近年来算子理论中比较活跃的研究课题,对它们的研究涉及到诸如代数学、矩阵理论、非交换几何理论、非交换计算以及量子计算等多个学科分支.本文主要研究算子理论在量子信息和非交换计算中的若干应用.研究方法上着重使用了算子分块技巧和算子谱理论.研究内容涉及到上三角算子矩阵的值域,算子数值域,广义量子门与自伴算子下确界的谱表示,正压缩交换子的极大范数,斜投影五个方面.全文共分五章：
第一章运用算子分块技巧,对上三角算子矩阵分别就对角线上算子A,B的值域R(A)和R(B)都闭、都不闭、一个闭一个不闭这四种情况,研究了值域R(MC)的闭性.并结合所得结论,给出了上三角算子矩阵是Kato-非奇异的充分必要条件.
第二章将S. Gudder和G. Nagy在研究序贯量子测量理论中所依赖的涉及到自伴算子数值域的一个基本定理,在去掉其自伴性的限制后,推广到了一般有界线性算子中；完全回答了P. J. Psarrakos和M. J. Tsatsomeros提出的,关于矩阵的极小模和内数值域半径的一个公开问题,并证明了相应结论对无限维Hilbert空间上的有界线性算子也成立；借助于算子分块技巧,研究了算子数值域的角点与算子点谱、约化近似点谱之间的关系.
第三章对龙桂鲁在研究对偶量子计算机原理中提出的广义量子门,及S.Gudder在研究量子测量中引入的逻辑序下的自伴算子中的若干问题进行了研究,运用线性算子的谱理论和算子分块的方法,给出了广义量子门的谱刻画及逻辑序下两个自伴算子的下确界的谱表示.
第四章运用线性算子的谱理论,研究了由蔡文端提出的非交换计算中两个正压缩算子的交换子的有关问题,得到了正压缩交换子的极大范数可达的充要条件.
第五章从斜投影算子的几何结构入手,给出了补子空间上斜投影的算子矩阵表示.在无限维Hilbert空间上,得到了斜投影U(VU)(?)V的值域和零空间的若干刻画,给出了一般斜投影的表示.
本文所取得的主要研究成果分为以下7个方面：
(1)利用算子分块的方法,给出了几种不同情形下上三角算子矩阵值域是闭的以及上三角算子矩阵是Kato-非奇异的充分必要条件.
(2)证明了对Hilbert空间H上的有界线性算子A,B,C,若等式(Ax,x)(Bx,x)= (Cx,x)对H中任意的单位向量x都成立,则A和B中至少有一个是恒等算子的倍数,即存在复数c使得A=cI或B=cI.
(3)证明了对Hilbert空间H上的有界线性算子A,若0(?)W(A),则δ(A)≥(?)(A).
(4)利用线性算子的谱理论,给出了压缩算子是广义量子门的充要条件.
(5)得到了逻辑序下两个自伴算子的下确界的谱表示公式.
(6)获得了两个正压缩算子的交换子的极大范数可达的充分必要条件.
(7)给出了斜投影U(VU)(?)V的值域,零空间的若干刻画,得到了一般斜投影的表示.
Operator matrices, numerical ranges of operators, spectral theory of operators and commutaors are some heatly discussed topics in operator theory. The research of these subjects has been related to algebra, matrix analysis, non-commutative geometry, non-commutative computation and quantum computation, etc. The dissertation is related to some applications of operator theory in quantum information and non-commutative computation. The research methods mainly focus on techniques of block operator and spectral theory. The dissertation, which can be divided into five chapters, mainly studies the ranges of upper triangular operator matrices, numerical ranges of operators, spectral characterizations of generalized quantum gate and infimum of two self-adjoint operators, the maximum norm of commutators of positive contractions, and oblique projectors.
In Chapter 1, by using the technique of block operator matrices, for the operator matrix in which operators A and B are on the diagonal, under the four cases that R(A) and R(B) are closed or both are not closed, or one of them is closed and the other is not closed, the closedness of R(MC) is studied. Combining the obtained con-clusions, the sufficient and necessary conditions under which the upper triangular operator matrix is Kato non-singular are given.
In Chapter 2, for the fundamental theorem concerning the numerical ranges of self-adjoint operators which was used by S. Gudder and G. Nagy when studying the theory of sequential quantum measurements, the restriction on operators which are self-adjoint is removed and it is generalized to general bounded linear operators. The open problem con-cerning the minimum modulus and inner numerical radius of matrix, which was raised by P. J. Psarrakos and M. J. Tsatsomeros, has been solved completely. The corresponding conclusion is also proved to be correct for operators on an infinite dimensional Hilbert space. Finally, by using the technique of block operator matrices, the research reveals the relations between the corner of numerical range and the point spectrum together with the reduced approximation point spectrum.
In Chapter 3, several problems concerning the generalized quantum gate and self-adjoint operators with respect to the logic order are studied, which were introduced by G. L. Long and S. Gudder when studying the principle of duality quantum computers and the quantum measurements, respectively. By using the spectral theory of linear operators and the technique of block operator matrices, the research puts forward spectral representations of generalized quantum gate and infimum of two self-adjoint operators with respect to the logic order.
In Chapter 4, by using the spectral theory of linear operators, in the non-commutative computation setting, the problem concerning commutators of two positive contract opera-tors which was raised by M. D. Choi is studied. The sufficient and necessary conditions under which the maximum norm attainability of positive contract commutators are estab-lished.
In Chapter 5, starting with the geometry structure of oblique projectors, the oper-ator matrix representation of an oblique projector for complementary subspaces is given. For operators acting on an infinite dimensional Hilbert space, some characterizations of the range and the null space of an oblique projector U(VU)(?)V are obtained, and the representation of a general oblique projector is also given.
The results from the dissertation consist of the following seven statements.
1. By using the block operator matrices technique, the sufficient and necessary con-ditions under which the range closedness and the Kato non-singularity of upper triangular operator matrices under several different conditions are given.
2. We prove that for A, B, C, which are bounded linear operators on a Hilbert space H, if (Ax,x)(Bx,x)= (Cx,x) for all x∈H with‖x‖= 1, then at least one of A and B is a scalar multiple of the identity, i.e. there exists a complex number c∈C such that A=cI or B=cI.
3. For every bounded linear operator A on a Hilbert space H, If 0 (?) W (A),thenδ(A)≥(?)(A).
4. By using the spectral theory of linear operators, a sufficient and necessary condition under which a contract operator is a generalized quantum gate is established.
5. A spectral representation of the infimum of two self-adjoint operators with respect to the logic order is given.
6. Some conditions under which the maximum norm attainability of commutators of two positive contractions are attained.
7. Some representations of the range and the null space of an oblique projector U(VU)(?)V are given, and the representation of a general oblique projector acting on an infinite dimensional Hilbert space is also established.
第一章运用算子分块技巧,对上三角算子矩阵分别就对角线上算子A,B的值域R(A)和R(B)都闭、都不闭、一个闭一个不闭这四种情况,研究了值域R(MC)的闭性.并结合所得结论,给出了上三角算子矩阵是Kato-非奇异的充分必要条件.
第二章将S. Gudder和G. Nagy在研究序贯量子测量理论中所依赖的涉及到自伴算子数值域的一个基本定理,在去掉其自伴性的限制后,推广到了一般有界线性算子中；完全回答了P. J. Psarrakos和M. J. Tsatsomeros提出的,关于矩阵的极小模和内数值域半径的一个公开问题,并证明了相应结论对无限维Hilbert空间上的有界线性算子也成立；借助于算子分块技巧,研究了算子数值域的角点与算子点谱、约化近似点谱之间的关系.
第三章对龙桂鲁在研究对偶量子计算机原理中提出的广义量子门,及S.Gudder在研究量子测量中引入的逻辑序下的自伴算子中的若干问题进行了研究,运用线性算子的谱理论和算子分块的方法,给出了广义量子门的谱刻画及逻辑序下两个自伴算子的下确界的谱表示.
第四章运用线性算子的谱理论,研究了由蔡文端提出的非交换计算中两个正压缩算子的交换子的有关问题,得到了正压缩交换子的极大范数可达的充要条件.
第五章从斜投影算子的几何结构入手,给出了补子空间上斜投影的算子矩阵表示.在无限维Hilbert空间上,得到了斜投影U(VU)(?)V的值域和零空间的若干刻画,给出了一般斜投影的表示.
本文所取得的主要研究成果分为以下7个方面：
(1)利用算子分块的方法,给出了几种不同情形下上三角算子矩阵值域是闭的以及上三角算子矩阵是Kato-非奇异的充分必要条件.
(2)证明了对Hilbert空间H上的有界线性算子A,B,C,若等式(Ax,x)(Bx,x)= (Cx,x)对H中任意的单位向量x都成立,则A和B中至少有一个是恒等算子的倍数,即存在复数c使得A=cI或B=cI.
(3)证明了对Hilbert空间H上的有界线性算子A,若0(?)W(A),则δ(A)≥(?)(A).
(4)利用线性算子的谱理论,给出了压缩算子是广义量子门的充要条件.
(5)得到了逻辑序下两个自伴算子的下确界的谱表示公式.
(6)获得了两个正压缩算子的交换子的极大范数可达的充分必要条件.
(7)给出了斜投影U(VU)(?)V的值域,零空间的若干刻画,得到了一般斜投影的表示.
Operator matrices, numerical ranges of operators, spectral theory of operators and commutaors are some heatly discussed topics in operator theory. The research of these subjects has been related to algebra, matrix analysis, non-commutative geometry, non-commutative computation and quantum computation, etc. The dissertation is related to some applications of operator theory in quantum information and non-commutative computation. The research methods mainly focus on techniques of block operator and spectral theory. The dissertation, which can be divided into five chapters, mainly studies the ranges of upper triangular operator matrices, numerical ranges of operators, spectral characterizations of generalized quantum gate and infimum of two self-adjoint operators, the maximum norm of commutators of positive contractions, and oblique projectors.
In Chapter 1, by using the technique of block operator matrices, for the operator matrix in which operators A and B are on the diagonal, under the four cases that R(A) and R(B) are closed or both are not closed, or one of them is closed and the other is not closed, the closedness of R(MC) is studied. Combining the obtained con-clusions, the sufficient and necessary conditions under which the upper triangular operator matrix is Kato non-singular are given.
In Chapter 2, for the fundamental theorem concerning the numerical ranges of self-adjoint operators which was used by S. Gudder and G. Nagy when studying the theory of sequential quantum measurements, the restriction on operators which are self-adjoint is removed and it is generalized to general bounded linear operators. The open problem con-cerning the minimum modulus and inner numerical radius of matrix, which was raised by P. J. Psarrakos and M. J. Tsatsomeros, has been solved completely. The corresponding conclusion is also proved to be correct for operators on an infinite dimensional Hilbert space. Finally, by using the technique of block operator matrices, the research reveals the relations between the corner of numerical range and the point spectrum together with the reduced approximation point spectrum.
In Chapter 3, several problems concerning the generalized quantum gate and self-adjoint operators with respect to the logic order are studied, which were introduced by G. L. Long and S. Gudder when studying the principle of duality quantum computers and the quantum measurements, respectively. By using the spectral theory of linear operators and the technique of block operator matrices, the research puts forward spectral representations of generalized quantum gate and infimum of two self-adjoint operators with respect to the logic order.
In Chapter 4, by using the spectral theory of linear operators, in the non-commutative computation setting, the problem concerning commutators of two positive contract opera-tors which was raised by M. D. Choi is studied. The sufficient and necessary conditions under which the maximum norm attainability of positive contract commutators are estab-lished.
In Chapter 5, starting with the geometry structure of oblique projectors, the oper-ator matrix representation of an oblique projector for complementary subspaces is given. For operators acting on an infinite dimensional Hilbert space, some characterizations of the range and the null space of an oblique projector U(VU)(?)V are obtained, and the representation of a general oblique projector is also given.
The results from the dissertation consist of the following seven statements.
1. By using the block operator matrices technique, the sufficient and necessary con-ditions under which the range closedness and the Kato non-singularity of upper triangular operator matrices under several different conditions are given.
2. We prove that for A, B, C, which are bounded linear operators on a Hilbert space H, if (Ax,x)(Bx,x)= (Cx,x) for all x∈H with‖x‖= 1, then at least one of A and B is a scalar multiple of the identity, i.e. there exists a complex number c∈C such that A=cI or B=cI.
3. For every bounded linear operator A on a Hilbert space H, If 0 (?) W (A),thenδ(A)≥(?)(A).
4. By using the spectral theory of linear operators, a sufficient and necessary condition under which a contract operator is a generalized quantum gate is established.
5. A spectral representation of the infimum of two self-adjoint operators with respect to the logic order is given.
6. Some conditions under which the maximum norm attainability of commutators of two positive contractions are attained.
7. Some representations of the range and the null space of an oblique projector U(VU)(?)V are given, and the representation of a general oblique projector acting on an infinite dimensional Hilbert space is also established.
引文
[1]H. K. Du, J. Pan. Perturbation of spectrums of 2×2 operator matrices[J]. Proc. Amer. Math. Soc.,1994,121:761-776.
[2]M. Barraa, M. Boumazgour. A note on the spectrum of an upper triangular operator matrices[J]. Proc. Amer. Math. Soc.,2003,131:3083-3088.
[3]X. H. Cao. Browder essential approximate point spectra and hypercyclicity for operator matrices[J]. Linear Algebra Appl.,2007,426(2-3):317-324.
[4]X. H. Cao, M. Z. Guo, B. Meng. Semi-Fredholm spectrum and Weyl's theorem for operator matrices[J]. Acta Math. Sinica,2006,22(1):169-178.
[5]D. S. Djordjevic. Perturbations of spectra of operator matrices[J]. J. Operator Theory, 2002,48:467-486.
[6]S. V. Djordjevic, Y. M. Han. A note on Weyl's theorem for operator matrices[J]. Proc. Amer. Math. Soc.,2003,131:2543-2547.
[7]B. A. Barnes. Riesz points of upper triangular operator matrices[J]. Proc. Amer. Math. Soc.,2005,133:1343-1347.
[8]X. H. Cao, B. Meng. Essential approximate point spectra and Weyl's theorem for oper-ator matrices[J]. J. Math. Anal. Appl.,2005,304:759-771.
[9]W. Y. Lee. Weyl's theorem for operator matrices[J]. Int. Equ. Oper. Theory,1998,32: 319-331.
[10]W. Y. Lee. Weyl's spectra for operator matrices[J]. Proc. Amer. Math. Soc.,2000,129: 131-138.
[11]Y. Li, X. H. Sun, H. K. Du. The intersection of left (right) weyl spectrum of 2 x 2 upper triangular operator matrices [J]. Acta Math. Sinica,2005,48(4):653-660.
[12]候国林,阿拉坦仓.2×2阶上三角算子矩阵的谱扰动[J].系统科学与数学.2006,26(3)：257-263.
[13]海国君,阿拉坦仓.2×2阶上三角型算子矩阵的可能谱.中国科技论文在线.http://www.paper.edu.cn/paper.php?serialn umber=200805-625.2008年5月22.
[14]J. C. Hou, H. K. Du. Norm inequalities of positive operator matrices[J]. Int. Equ. Oper. Theory.,1995,22(3):281-294.
[15]H. Y. Zhang, H. K. Du. Browder spectra of upper-triangular operator matrices [J]. J. Math. Anal. Appl.,2006,323(1):700-707.
[16]C. Benhida, E. H. Zerouali, F. Zguittl. Spectra of upper triangular operator matrices [J]. Proc. Amer. Math. Soc.,2005,133(10):3013-3020.
[17]张海燕,张希花,杜鸿科.2×2阶上三角算子矩阵的Drazin谱[J].数学物理学报.2009,29A(2)：272-282.
[18]邓春源,杜鸿科.Hilbert空间上线性算子的Drazin可逆性[J].数学学报,2007,50(6)：1262-1270.
[19]J. K. Han, H. Y. Lee. Invertible completions of 2×2 upper triangular operator matrices[J]. Proc. Amer. Math. Soc.,2000,128:119-123.
[20]K. Takahashi. Invertible completions of operator matrices[J]. Int. Equ. Oper. Theory, 1995,21:355-361.
[21]M. Barraa, M. Boumazgour. On the perturbations of spectra of upper triangular operator matrices[J]. J. Math. Anal. Appl.,2008,347(1):315-322.
[22]I. S. Hwang, W. Y. Lee. The boundedness below of 2 x 2 upper triangular operator matrices[J]. Int. Equ. Oper. Theory,2001,39:267-276.
[23]Y. N. Dou, G. C. Du, C. F. Shao, H. K. Du. Closedness of ranges of upper-triangular operators[J]. J. Math. Anal. Appl.,2009,356:13-20.
[24]S. Gudder, G. Nagy. Sequential quantum measurements [J]. J. Math. Phys.,2001,42: 5212-5222.
[25]H. K. Du, Y. N. Dou, H. Y. Zhang, L. M. Shen. A generalization of Gudder-Nagy Theorem with numerical ranges of operators[J]. submitted.
[26]M. D. Choi, D. W. Kribs, K. Zyczkowskid. Higher-rank numerical ranges and compres-sion problems[J]. Linear Algebra Appl.,2006,418(2-3):828-839.
[27]C. K. Lia, Y. T. Poonb, N. S. Sze. Higher rank numerical ranges and low rank pertur-bations of quantum channels[J]. J. Math. Anal. Appl.,2008,348(2):843-855.
[28]T. S. herbruggen, G. Dirr, U. Helmke, S J. Glaser. The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information [J]., Linear and Multilinear Algebra,2008,56(1-2):3-26.
[29]A. Y. Kazakov. An elementary constructive approach to the higher-rank numerical ranges of unitary matrices and quantum error-correcting codes[J]. Journal of Physics A:Mathematical and Theoretical,2008,41:255306(1-9).
[30]M. D. Choi, J. A. Holbrook, D.W. Kribs, K. zyczkowski. Higher-rank numerical ranges of unitary and normal matrices[J]. Operators and Matrices,2007,1(3):409-426
[31]R. A. Horn, C. R. Johnson, Topics in Matrix Analysis[M]. Cambridge, England:Univer-sity of Cambridge.1991.
[32]K. E. Gustafson, D. K. M.RaO. Numerical Range, The Field of Values of Linear Operators and Matriees[M], New York:Springer-Verlag,1997.
[33]P. J. Psarrakos, M. J. Tsatsomeros. Numerical range:(in) a matrix nutshell, part 1[M]. Mathematical Notes from Washington State University,2002,45(2).
[34]A. W. Wafula, J. M. Khalagai. Invariance of numerical range under isometric transfor-mation[J]. Afr. J. Math. Comput. Sci. Res.,2010,3(2):036-037.
[35]Y. N. Dou, H. K. Du. An alternative proof of a theorem about the corner of numerical ranges[J]. Submitted.
[36]Y. N. Dou, H. K. Du. An answer of an open problem concerning numerical ranges [J]. Submitted.
[37]H. J. Woerdeman. The higher rank numerical range is convex[J]. Linear and Multilinear Algebra,2008,56(1-2):65-67.
[38]D. Farenick. Matricial extensions of the numerical range:A brief survey [J]. Linear and Multilinear Algebra[J].1993,34(3-4),197-211.
[39]M. D. Choi, C. K. Li. Numerical ranges of the powers of an operator [J]. J. Math. Anal. Appl.,2010,365(2):458-466.
[40]P. Y. Wu. Numerical range of Aluthge transform of operator[J]. Linear Algebra Appl., 2002,357(1-3):295-298.
[41]O. Axelsson, B. Polman, On the numerical radius of matrices and its application for iterative solution methods[J]. Linear and Multilinear Algebra.1994,37:225-238.
[42]B. Kuzma, G. Lesnjak, C. K. Li, T. Pete, L.Rodman. Conditions for linear dependence of two operators[J]. Operator Theory:Advances and Applications,25 pages, accepted for publications.
[43]C. K. Fong. On the essential maximal numerical range[J]. Acta Sci. Math.,1979,41: 307-315.
[44]K. C. Das, B. D. Craven. Linearity and weak convergence on the boundary of numerical range[J]. J. Austral. Math. Soc. (Series A),1983,35:221-226.
[45]C. R. Crawford. A stable generalized eigenvalue problem[J]. SIAM J. Numer. Anal. 1976,13(6):854-860.
[46]G. W. Stewart. Perturbation bounds for the definete generalized eigenvalue problem[J]. Linear Algebra Appl.,1979,23:69-85.
[47]G. L. Long. The general quantum interference principle and the duality computer [J]. Commun. Theor. Phys.(Beijing, China),2006,45(5):825-844.
[48]S. Gudder. Duality quantum computers and quantum operations[J]. Int. J. Theor. Phys., 2008,47(1):268-279.
[49]S. Gudder. Mathematical theory of duality quantum computers [J]. Quantum Information Processing,2007,6(1):37-48.
[50]Y. Q. Wang, H. K. Du, Y. N. Dou. Note on generalized quantum gates and quantum operations[J]. Inter. J. Theor. Phys.,2008,47:2268-2278.
[51]H. K. Du, Y. Q. Wang, J. L. Xu. Applications of the generalized Luders theorem [J]. J. Math. Phys.,2008,49(1):013507.
[52]H. K. Du, Y. N. Dou. A spectral characterization for generalized quantum gates [J]. J. Math. Phys.2009,50:032101(1-6).
[53]G. L. Long, Y. F. Zhou, J. Q. Jin, Y. Sun, H. W. Lee. Density matrix in quantum mechanics and distinctness of ensembles having the same compressed density matrix[J]. Foundations of Physics,2006,36(8):1217-1243.
[54]G. L. Long, Y. Liu. Duality and recycling computing in quantum computers[J]. Commun. Theor. Phys.(Beijing, China),2008,50(6):1303-1306.
[55]P. Busch, J. Singh. Luders theorem for unsharp quantum measurements[J]. Phys. Letters A,1998,249:10-12.
[56]Z. Li, H. X. Zhu, H. Zhang, H. K. Du. Two combinations of unitary operators and frame representations [J]. Northeastern Math. J.,2005,21:212-220.
[57]G. L. Long. Mathematical theory of the duality computer in the density matrix formal-ism, Quantum Information Processing[J].2007,6:49-54.
[58]M. Nielsen, J. Chuang. Quantum Computation and Quantum Information[M]. Cam-bridge, England:University of Cambridge.2000.
[59]A. Gheondea, S. Gudder. Sequential product of quantum effects [J]. Proc. Amer. Math. Soc.,2004,132:503-512.
[60]S. Gudder. An order for quantum observables[J]. Math. Slovaca,2006,56:573-589.
[61]S. Pulmannova, E. Vincekova. Remarks on the order for quantum observables[J]. Math. Slovaca,2007,57:589-600.
[62]W. Liu, J. D. Wu. A representation theorem of bounded quantum observables[J]. J. Math. Phys.,2008,49:073521-073521-5.
[63]S. Gudder. A structure for quantum measurements [J]. Rep. Math. Phys.,2005,55: 249-267.
[64]S. Gudde, G. Nagy. Sequential quantum measurements [J]. J. Math. Phys.,2001,42: 5212-5222.
[65]A. Arias, A. Gheordea, S. Gudder. Fixed points of quantum operation [J]. J. Math. Phys., 2002,43(12):5872-5881.
[66]T. Moreland, S. Gudder. Infima of Hilbert space effects [J]. Linear Algebra Appl.,1999, 286:1-17.
[67]S. Gudder. Lattice properties of quantum effects[J]. J. Math. Phys.,1996,37:2637-2642.
[68]杜鸿科,邓春源,李启慧.量子效应的下确界问题[J].中国科学A辑,2006,36(3)：320-332.
[69]杜鸿科,窦艳妮,逻辑序下自伴算子下确界的谱表示[J].数学学报(中文版).2009,52(6)：1141-1146.
[70]J. D. Stampfli. The norm of a derivation[J]. Pacific J. Math.,1970,33:737-747.
[71]R. Bhatia, X. Zhan. Norm inequalities for operators with positive real part[J]. J. Oper. Theory,2003,50:67-76.
[72]F. Kittaneh. On some operator inequalities [J]. Linear Algebra Appl.,1994,208:19-28.
[73]F. Kittaneh. Commutator inequalities associated with the polar decomposition [J]. Proc. Amer. Math. Soc.,2002,130:1279-1283.
[74]E. A. Martins, F. C. Silva. On the eigenvalues of Jordan products[J]. Linear Algebra Appl.,2003,359 249-262.
[75]T. Ogasawara. A theorem on operator algebras[J]. J. Sci. Hiroshima Univ. Ser. A.1955, 18:307-309.
[76]R. Bhatia, F. Kittaneh. On the singular values of a product of operators[J]. SIAM J. Matrix Anal. Appl.,1990,11:272-277.
[77]Y. Q. Wang, H. K. Du. Norms of commutators of self-adjoint operators[J]. J. Math. Anal. Appl.,2008,342:747-751.
[78]X. Zhan. Matrix Inequalities[M], Berlin:Springer-Verlag,2002.
[79]H. K. Du, Y. N. Dou, Y. Q. Wang. Maximum norms of commutators of positive contrac-tive operators[J]. Submitted.
[80]Y. Q. Wang, H. K. Du.A characterization of maximum norms of commutators of positive contractive operators[J]. J. Math. Anal. Appl.,2008,348:990-995.
[81]R. Bhatia, F. Kittaneh. Cartesian decomposition and Schatten norms[J]. Linear Algebra Appl.,2000,318:109-116.
[82]H. K. Du, Y. Q. Wang, S. F. Ji. Minimum spectra of Jordan products of positive operators[J]. submitted.
[83]M. D. Choi. Magic in Non-Commutative Computations, Lecture on the International Conference on Banach spaces and Operator spaces(ICBSOS), Nankai University,2007.
[84]F. Kittaneh. Norm inequalities for commutators of positive operators and applications[J]. Math. Z.,2008,258:845-849.
[85]F. Kittaneh. Inequalities for commutators of positive operators[J]. J. Funct. Anal.,2007, 250:132-143.
[86]X. Zhan. On some matrix inequalities [J]. Linear Algebra Appl.,2004,376:299-303.
[87]R. T. Behrens, L. L. Scharf. Signal processing applications of oblique projection opera-tors[J]. IEEE Trans Signal Processing,1994,42(6):1413-1424.
[88]P. Vandaele, M. Moonen. Two deterministic blind channel estimation algorithms based on oblique projections[J]. Signal Processing,2000,80(3):481-495.
[89]C. Y. Peng, X. D. Zhang. On recursive Oblique Projectors [J]. IEEE Signal Processing Letters,2005,12(6):433-436.
[90]M. H. Chen, Z. Y. Wang. Subspace tracking in colored noise based on oblique projection. International Conference on Acoustics, Speech and Signal Processing (ICASSP),Ⅲ:556-559,2006.
[91]B. Remy, B. Guillaume. Oblique projections for direction-of-arrival estimation with prior knowledge. IEEE Trans Signal Process,2008,56:3174-1387.
[92]A. Cerny. Characterization of the oblique projector U(VU)+V with application to con-strained least squares[J]. Linear Algebra Appl.,2009,431(9):1564-1570.
[93]Y. N. Chen, H. K. Du, H. Y. Zhang. Path connectivity of idempotents on a Hilbert space[J]. Math. Soc.,2008,136:3483-3492.
[94]H. K. Du, X. Y. Yao, C. Y. Deng. Invertibility of linear combinations of two idempo-tents[J]. Proc. Amer. Math. Soc.,2006,134:1451-1457.
[95]H. K. Du, Y. N. Dou, H. Y. Zhang. Representations and characterizations of oblique projections[J].完成.
[96]J. K. Baksalary, O. M. Baksalary. Idempotency of linear combinations of two idempo-tency matrices[J], Linear Algebra Appl.,2000,321:3-7.
[97]V. Ptak. Extremal operators and oblique projections [J]. Casopis Pest Mat.,1985,110: 343-350.
[98]邓春源,杜鸿科.两子空间的公共补与Gro β问题[J].数学学报(中文版),2006,49：1099-1112.
[99]J. J. Koliha, V. Rakocevic. The nullity and rank of linear combinations of idempotent matrices[J]. Linear Algebra Appl.,2006,418(1):11-14.
[100]J. B. Conway. A Coursw in Functional Analysis[M]. New York:Springer-Verlag,1990.
[101]H. K. Du. Operator matrix forms of positive operator matrices[J]. Chinese Quart. J. Math.,1992,7:9-11.
[102]R.G.Douglas. Banach Algebra Techniques in Operator Theory[M]. New-York:Springer-Verlag,1998.
[103]P. R. Halmos. A Hilbert Space Problem Book[M]. New York:Springer-Verlag,1982.
[104]R. G. Douglas. On majorization, factorization and range inclusion of operators on Hilbert space[J]. Proc. Amer. Math. Soc.,1966,17:413-415.
[105]R. A. Horn, C. R. Johnson. Matrix Analysis[M]. Cambridge, England:University of Cambridge.1985.
[106]P. A. Fillmore, J. G. Stampfli, J. P. Williams. On the essential numerical range, the essential spectrum and a pooblem of Halmos[J]. Acta. Sci. Math.,1972,33:179-192.
[107]G. K. Pedersen. Analysis Now[M]. BeiJing:Springer-Verlag,1990.
[108]杜鸿科.关于控制算子的若干注记[J].数学研究与评论,1987,7(2)：263-268.
[109]H. K. Du, G. X. Ji. Norm attainability of elementary operators and derivations [J]. Northeast. Math. J.,1994,10(3):396-400.
[110]张恭庆,林源渠.泛函分析讲义[M].北京：北京大学出版社,2003.
[111]M. A. Nielsen, I. L. Chuang.(赵千川译).量子计算和量子信息[M].北京：清华大学出版社.2006.
[2]M. Barraa, M. Boumazgour. A note on the spectrum of an upper triangular operator matrices[J]. Proc. Amer. Math. Soc.,2003,131:3083-3088.
[3]X. H. Cao. Browder essential approximate point spectra and hypercyclicity for operator matrices[J]. Linear Algebra Appl.,2007,426(2-3):317-324.
[4]X. H. Cao, M. Z. Guo, B. Meng. Semi-Fredholm spectrum and Weyl's theorem for operator matrices[J]. Acta Math. Sinica,2006,22(1):169-178.
[5]D. S. Djordjevic. Perturbations of spectra of operator matrices[J]. J. Operator Theory, 2002,48:467-486.
[6]S. V. Djordjevic, Y. M. Han. A note on Weyl's theorem for operator matrices[J]. Proc. Amer. Math. Soc.,2003,131:2543-2547.
[7]B. A. Barnes. Riesz points of upper triangular operator matrices[J]. Proc. Amer. Math. Soc.,2005,133:1343-1347.
[8]X. H. Cao, B. Meng. Essential approximate point spectra and Weyl's theorem for oper-ator matrices[J]. J. Math. Anal. Appl.,2005,304:759-771.
[9]W. Y. Lee. Weyl's theorem for operator matrices[J]. Int. Equ. Oper. Theory,1998,32: 319-331.
[10]W. Y. Lee. Weyl's spectra for operator matrices[J]. Proc. Amer. Math. Soc.,2000,129: 131-138.
[11]Y. Li, X. H. Sun, H. K. Du. The intersection of left (right) weyl spectrum of 2 x 2 upper triangular operator matrices [J]. Acta Math. Sinica,2005,48(4):653-660.
[12]候国林,阿拉坦仓.2×2阶上三角算子矩阵的谱扰动[J].系统科学与数学.2006,26(3)：257-263.
[13]海国君,阿拉坦仓.2×2阶上三角型算子矩阵的可能谱.中国科技论文在线.http://www.paper.edu.cn/paper.php?serialn umber=200805-625.2008年5月22.
[14]J. C. Hou, H. K. Du. Norm inequalities of positive operator matrices[J]. Int. Equ. Oper. Theory.,1995,22(3):281-294.
[15]H. Y. Zhang, H. K. Du. Browder spectra of upper-triangular operator matrices [J]. J. Math. Anal. Appl.,2006,323(1):700-707.
[16]C. Benhida, E. H. Zerouali, F. Zguittl. Spectra of upper triangular operator matrices [J]. Proc. Amer. Math. Soc.,2005,133(10):3013-3020.
[17]张海燕,张希花,杜鸿科.2×2阶上三角算子矩阵的Drazin谱[J].数学物理学报.2009,29A(2)：272-282.
[18]邓春源,杜鸿科.Hilbert空间上线性算子的Drazin可逆性[J].数学学报,2007,50(6)：1262-1270.
[19]J. K. Han, H. Y. Lee. Invertible completions of 2×2 upper triangular operator matrices[J]. Proc. Amer. Math. Soc.,2000,128:119-123.
[20]K. Takahashi. Invertible completions of operator matrices[J]. Int. Equ. Oper. Theory, 1995,21:355-361.
[21]M. Barraa, M. Boumazgour. On the perturbations of spectra of upper triangular operator matrices[J]. J. Math. Anal. Appl.,2008,347(1):315-322.
[22]I. S. Hwang, W. Y. Lee. The boundedness below of 2 x 2 upper triangular operator matrices[J]. Int. Equ. Oper. Theory,2001,39:267-276.
[23]Y. N. Dou, G. C. Du, C. F. Shao, H. K. Du. Closedness of ranges of upper-triangular operators[J]. J. Math. Anal. Appl.,2009,356:13-20.
[24]S. Gudder, G. Nagy. Sequential quantum measurements [J]. J. Math. Phys.,2001,42: 5212-5222.
[25]H. K. Du, Y. N. Dou, H. Y. Zhang, L. M. Shen. A generalization of Gudder-Nagy Theorem with numerical ranges of operators[J]. submitted.
[26]M. D. Choi, D. W. Kribs, K. Zyczkowskid. Higher-rank numerical ranges and compres-sion problems[J]. Linear Algebra Appl.,2006,418(2-3):828-839.
[27]C. K. Lia, Y. T. Poonb, N. S. Sze. Higher rank numerical ranges and low rank pertur-bations of quantum channels[J]. J. Math. Anal. Appl.,2008,348(2):843-855.
[28]T. S. herbruggen, G. Dirr, U. Helmke, S J. Glaser. The significance of the C-numerical range and the local C-numerical range in quantum control and quantum information [J]., Linear and Multilinear Algebra,2008,56(1-2):3-26.
[29]A. Y. Kazakov. An elementary constructive approach to the higher-rank numerical ranges of unitary matrices and quantum error-correcting codes[J]. Journal of Physics A:Mathematical and Theoretical,2008,41:255306(1-9).
[30]M. D. Choi, J. A. Holbrook, D.W. Kribs, K. zyczkowski. Higher-rank numerical ranges of unitary and normal matrices[J]. Operators and Matrices,2007,1(3):409-426
[31]R. A. Horn, C. R. Johnson, Topics in Matrix Analysis[M]. Cambridge, England:Univer-sity of Cambridge.1991.
[32]K. E. Gustafson, D. K. M.RaO. Numerical Range, The Field of Values of Linear Operators and Matriees[M], New York:Springer-Verlag,1997.
[33]P. J. Psarrakos, M. J. Tsatsomeros. Numerical range:(in) a matrix nutshell, part 1[M]. Mathematical Notes from Washington State University,2002,45(2).
[34]A. W. Wafula, J. M. Khalagai. Invariance of numerical range under isometric transfor-mation[J]. Afr. J. Math. Comput. Sci. Res.,2010,3(2):036-037.
[35]Y. N. Dou, H. K. Du. An alternative proof of a theorem about the corner of numerical ranges[J]. Submitted.
[36]Y. N. Dou, H. K. Du. An answer of an open problem concerning numerical ranges [J]. Submitted.
[37]H. J. Woerdeman. The higher rank numerical range is convex[J]. Linear and Multilinear Algebra,2008,56(1-2):65-67.
[38]D. Farenick. Matricial extensions of the numerical range:A brief survey [J]. Linear and Multilinear Algebra[J].1993,34(3-4),197-211.
[39]M. D. Choi, C. K. Li. Numerical ranges of the powers of an operator [J]. J. Math. Anal. Appl.,2010,365(2):458-466.
[40]P. Y. Wu. Numerical range of Aluthge transform of operator[J]. Linear Algebra Appl., 2002,357(1-3):295-298.
[41]O. Axelsson, B. Polman, On the numerical radius of matrices and its application for iterative solution methods[J]. Linear and Multilinear Algebra.1994,37:225-238.
[42]B. Kuzma, G. Lesnjak, C. K. Li, T. Pete, L.Rodman. Conditions for linear dependence of two operators[J]. Operator Theory:Advances and Applications,25 pages, accepted for publications.
[43]C. K. Fong. On the essential maximal numerical range[J]. Acta Sci. Math.,1979,41: 307-315.
[44]K. C. Das, B. D. Craven. Linearity and weak convergence on the boundary of numerical range[J]. J. Austral. Math. Soc. (Series A),1983,35:221-226.
[45]C. R. Crawford. A stable generalized eigenvalue problem[J]. SIAM J. Numer. Anal. 1976,13(6):854-860.
[46]G. W. Stewart. Perturbation bounds for the definete generalized eigenvalue problem[J]. Linear Algebra Appl.,1979,23:69-85.
[47]G. L. Long. The general quantum interference principle and the duality computer [J]. Commun. Theor. Phys.(Beijing, China),2006,45(5):825-844.
[48]S. Gudder. Duality quantum computers and quantum operations[J]. Int. J. Theor. Phys., 2008,47(1):268-279.
[49]S. Gudder. Mathematical theory of duality quantum computers [J]. Quantum Information Processing,2007,6(1):37-48.
[50]Y. Q. Wang, H. K. Du, Y. N. Dou. Note on generalized quantum gates and quantum operations[J]. Inter. J. Theor. Phys.,2008,47:2268-2278.
[51]H. K. Du, Y. Q. Wang, J. L. Xu. Applications of the generalized Luders theorem [J]. J. Math. Phys.,2008,49(1):013507.
[52]H. K. Du, Y. N. Dou. A spectral characterization for generalized quantum gates [J]. J. Math. Phys.2009,50:032101(1-6).
[53]G. L. Long, Y. F. Zhou, J. Q. Jin, Y. Sun, H. W. Lee. Density matrix in quantum mechanics and distinctness of ensembles having the same compressed density matrix[J]. Foundations of Physics,2006,36(8):1217-1243.
[54]G. L. Long, Y. Liu. Duality and recycling computing in quantum computers[J]. Commun. Theor. Phys.(Beijing, China),2008,50(6):1303-1306.
[55]P. Busch, J. Singh. Luders theorem for unsharp quantum measurements[J]. Phys. Letters A,1998,249:10-12.
[56]Z. Li, H. X. Zhu, H. Zhang, H. K. Du. Two combinations of unitary operators and frame representations [J]. Northeastern Math. J.,2005,21:212-220.
[57]G. L. Long. Mathematical theory of the duality computer in the density matrix formal-ism, Quantum Information Processing[J].2007,6:49-54.
[58]M. Nielsen, J. Chuang. Quantum Computation and Quantum Information[M]. Cam-bridge, England:University of Cambridge.2000.
[59]A. Gheondea, S. Gudder. Sequential product of quantum effects [J]. Proc. Amer. Math. Soc.,2004,132:503-512.
[60]S. Gudder. An order for quantum observables[J]. Math. Slovaca,2006,56:573-589.
[61]S. Pulmannova, E. Vincekova. Remarks on the order for quantum observables[J]. Math. Slovaca,2007,57:589-600.
[62]W. Liu, J. D. Wu. A representation theorem of bounded quantum observables[J]. J. Math. Phys.,2008,49:073521-073521-5.
[63]S. Gudder. A structure for quantum measurements [J]. Rep. Math. Phys.,2005,55: 249-267.
[64]S. Gudde, G. Nagy. Sequential quantum measurements [J]. J. Math. Phys.,2001,42: 5212-5222.
[65]A. Arias, A. Gheordea, S. Gudder. Fixed points of quantum operation [J]. J. Math. Phys., 2002,43(12):5872-5881.
[66]T. Moreland, S. Gudder. Infima of Hilbert space effects [J]. Linear Algebra Appl.,1999, 286:1-17.
[67]S. Gudder. Lattice properties of quantum effects[J]. J. Math. Phys.,1996,37:2637-2642.
[68]杜鸿科,邓春源,李启慧.量子效应的下确界问题[J].中国科学A辑,2006,36(3)：320-332.
[69]杜鸿科,窦艳妮,逻辑序下自伴算子下确界的谱表示[J].数学学报(中文版).2009,52(6)：1141-1146.
[70]J. D. Stampfli. The norm of a derivation[J]. Pacific J. Math.,1970,33:737-747.
[71]R. Bhatia, X. Zhan. Norm inequalities for operators with positive real part[J]. J. Oper. Theory,2003,50:67-76.
[72]F. Kittaneh. On some operator inequalities [J]. Linear Algebra Appl.,1994,208:19-28.
[73]F. Kittaneh. Commutator inequalities associated with the polar decomposition [J]. Proc. Amer. Math. Soc.,2002,130:1279-1283.
[74]E. A. Martins, F. C. Silva. On the eigenvalues of Jordan products[J]. Linear Algebra Appl.,2003,359 249-262.
[75]T. Ogasawara. A theorem on operator algebras[J]. J. Sci. Hiroshima Univ. Ser. A.1955, 18:307-309.
[76]R. Bhatia, F. Kittaneh. On the singular values of a product of operators[J]. SIAM J. Matrix Anal. Appl.,1990,11:272-277.
[77]Y. Q. Wang, H. K. Du. Norms of commutators of self-adjoint operators[J]. J. Math. Anal. Appl.,2008,342:747-751.
[78]X. Zhan. Matrix Inequalities[M], Berlin:Springer-Verlag,2002.
[79]H. K. Du, Y. N. Dou, Y. Q. Wang. Maximum norms of commutators of positive contrac-tive operators[J]. Submitted.
[80]Y. Q. Wang, H. K. Du.A characterization of maximum norms of commutators of positive contractive operators[J]. J. Math. Anal. Appl.,2008,348:990-995.
[81]R. Bhatia, F. Kittaneh. Cartesian decomposition and Schatten norms[J]. Linear Algebra Appl.,2000,318:109-116.
[82]H. K. Du, Y. Q. Wang, S. F. Ji. Minimum spectra of Jordan products of positive operators[J]. submitted.
[83]M. D. Choi. Magic in Non-Commutative Computations, Lecture on the International Conference on Banach spaces and Operator spaces(ICBSOS), Nankai University,2007.
[84]F. Kittaneh. Norm inequalities for commutators of positive operators and applications[J]. Math. Z.,2008,258:845-849.
[85]F. Kittaneh. Inequalities for commutators of positive operators[J]. J. Funct. Anal.,2007, 250:132-143.
[86]X. Zhan. On some matrix inequalities [J]. Linear Algebra Appl.,2004,376:299-303.
[87]R. T. Behrens, L. L. Scharf. Signal processing applications of oblique projection opera-tors[J]. IEEE Trans Signal Processing,1994,42(6):1413-1424.
[88]P. Vandaele, M. Moonen. Two deterministic blind channel estimation algorithms based on oblique projections[J]. Signal Processing,2000,80(3):481-495.
[89]C. Y. Peng, X. D. Zhang. On recursive Oblique Projectors [J]. IEEE Signal Processing Letters,2005,12(6):433-436.
[90]M. H. Chen, Z. Y. Wang. Subspace tracking in colored noise based on oblique projection. International Conference on Acoustics, Speech and Signal Processing (ICASSP),Ⅲ:556-559,2006.
[91]B. Remy, B. Guillaume. Oblique projections for direction-of-arrival estimation with prior knowledge. IEEE Trans Signal Process,2008,56:3174-1387.
[92]A. Cerny. Characterization of the oblique projector U(VU)+V with application to con-strained least squares[J]. Linear Algebra Appl.,2009,431(9):1564-1570.
[93]Y. N. Chen, H. K. Du, H. Y. Zhang. Path connectivity of idempotents on a Hilbert space[J]. Math. Soc.,2008,136:3483-3492.
[94]H. K. Du, X. Y. Yao, C. Y. Deng. Invertibility of linear combinations of two idempo-tents[J]. Proc. Amer. Math. Soc.,2006,134:1451-1457.
[95]H. K. Du, Y. N. Dou, H. Y. Zhang. Representations and characterizations of oblique projections[J].完成.
[96]J. K. Baksalary, O. M. Baksalary. Idempotency of linear combinations of two idempo-tency matrices[J], Linear Algebra Appl.,2000,321:3-7.
[97]V. Ptak. Extremal operators and oblique projections [J]. Casopis Pest Mat.,1985,110: 343-350.
[98]邓春源,杜鸿科.两子空间的公共补与Gro β问题[J].数学学报(中文版),2006,49：1099-1112.
[99]J. J. Koliha, V. Rakocevic. The nullity and rank of linear combinations of idempotent matrices[J]. Linear Algebra Appl.,2006,418(1):11-14.
[100]J. B. Conway. A Coursw in Functional Analysis[M]. New York:Springer-Verlag,1990.
[101]H. K. Du. Operator matrix forms of positive operator matrices[J]. Chinese Quart. J. Math.,1992,7:9-11.
[102]R.G.Douglas. Banach Algebra Techniques in Operator Theory[M]. New-York:Springer-Verlag,1998.
[103]P. R. Halmos. A Hilbert Space Problem Book[M]. New York:Springer-Verlag,1982.
[104]R. G. Douglas. On majorization, factorization and range inclusion of operators on Hilbert space[J]. Proc. Amer. Math. Soc.,1966,17:413-415.
[105]R. A. Horn, C. R. Johnson. Matrix Analysis[M]. Cambridge, England:University of Cambridge.1985.
[106]P. A. Fillmore, J. G. Stampfli, J. P. Williams. On the essential numerical range, the essential spectrum and a pooblem of Halmos[J]. Acta. Sci. Math.,1972,33:179-192.
[107]G. K. Pedersen. Analysis Now[M]. BeiJing:Springer-Verlag,1990.
[108]杜鸿科.关于控制算子的若干注记[J].数学研究与评论,1987,7(2)：263-268.
[109]H. K. Du, G. X. Ji. Norm attainability of elementary operators and derivations [J]. Northeast. Math. J.,1994,10(3):396-400.
[110]张恭庆,林源渠.泛函分析讲义[M].北京：北京大学出版社,2003.
[111]M. A. Nielsen, I. L. Chuang.(赵千川译).量子计算和量子信息[M].北京：清华大学出版社.2006.