B（H）上保持逼近酉相似及算子乘积投影的映射
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摘要
算子代数上的保持问题是近年来算子代数理论中比较活跃的研究课题之一,在算子代数分类的研究中有至关重要的理论价值和应用价值.本文的研究内容涉及算子代数上保持逼近酉相似的线性映射和算子代数上保持乘积非零投影的非线性满射两个方面.本文在研究方法上着重使用了算子分块技巧,根据所研究的内容,对给定的算子进行适当的分块.通过对它们的研究使得算子之间的内在关系变得更加清晰,由此揭示所涉及到的保持映射更多信息.全文分三章：
第一章绪论介绍了本文选题的意义和背景以及文中用到的符号,定义和后两章需要的一些概念及结论.首先我们介绍了本文选题的意义和背景,接着引入逼近酉相似,极小逼近酉相似,不变子空间,酉轨道,一秩投影等概念,最后给出了一些熟知的命题和定理.
第二章讨论了B(H)上双边保持逼近酉相似的线性满射φ的特征.令H为无限维复Hilbert空间,B(H)表示H上所有有界线性算子全体.通过引入极小逼近酉相似不变子空间的概念以及应用算子酉轨道的闭包特征,证明了φ具有如下形式：φ(X)=cU*XU或者φ(X)=cU*XtU((?)X∈B(H)),其中c为非零常数,U∈B(H)为酉算子,Xt为X关于H中任意给定的正规正交基的转置矩阵.
第三章首先研究了算子乘积投影的相关性质,其次刻画了B(H)上保持乘积非零投影的非线性满射,最后给出了B(H)上保持乘积非零投影的非线性满射的结构.令H为复Hilbert空间且dim H≥2,B(H)表示H上所有有界线性算子全体.设φ是B(H)上双边保持乘积非零投影的非线性满射,通过应用算子分块技巧证明了φ具有如下结构：φ(A)=λU*AU((?)A∈B(H)),其中λ为非零常数满足λ2=1,U∈B(H)为酉算子或反酉算子.从而说明了算子乘积非零投影是B(H)上的等距不变量.
Preserver problems on operator algebras represent one of the most active re-search topics in operator algebra and also have important value in both theory and application for the classification of operator algebra. The research of this thesis fo-cuses on linear maps preserving asymptotic unitary similarity on operator algebras and non-linear surjective maps preserving the non-zero projections of products of operators on operator algebras. Using the technique of operator blocks, more char-acters of discussed maps can be found. This article is divided into three chapters.
In chapter 1, the signigication and background of this thesis selecting subject are introduced. In addition, we offer some necesary notations, definitions, and some conceptions and conclusions in the later chapters. Firstly, we give the signigication and background of this thesis selecting subject. Subsequently, we introduce the definitions of asymptotic unitary similarity, minimal asymptotic u-similarity, invari-ant subspaces, the unitary orbit, one-rank projections etc. Finally, we give some well-known propositions and theorems.
In chapter 2, the characters of the linear surjective mapφon B(H) preserv-ing asymptotic unitary similarity in both directions are studied. Let H be an infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded linear opeators on H. By giving the definition of minimal asymptotic uni-tary similarity-invariant subspaces and applying the closure of the unitary orbit of a linear operator, it is proved that the mapφhas the following representations:φ(X)= cU*XU orφ(X)=cU*XtU((?)X∈B(H)), where c is a non-zero constant, U∈B(H) is a unitary operator, and Xt denotes the transpose of X relative to a fixed but arbitrary orthonormal basis of H.
In chapter 3, we analyse the related characters of the projections of products of two operators at first. Then, maps preserving the non-zero projections of products of two operators are characterized and the structures of the mentioned maps are obtained at last. Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H≤2. Letφbe a surjective map on B(H) preserving non-zero projections of products of two operators in both directions. By using the technique of operator blocks, it is proved that the mapφhas the following structure:φ(A)=λU*AU for all A in B(H) for some non-zero constantλwithλ2=1, where U is a unitary or an anti-unitary operator on H. These show that non-zero projections of products of two operators are isometric invariants of B(H).
第一章绪论介绍了本文选题的意义和背景以及文中用到的符号,定义和后两章需要的一些概念及结论.首先我们介绍了本文选题的意义和背景,接着引入逼近酉相似,极小逼近酉相似,不变子空间,酉轨道,一秩投影等概念,最后给出了一些熟知的命题和定理.
第二章讨论了B(H)上双边保持逼近酉相似的线性满射φ的特征.令H为无限维复Hilbert空间,B(H)表示H上所有有界线性算子全体.通过引入极小逼近酉相似不变子空间的概念以及应用算子酉轨道的闭包特征,证明了φ具有如下形式：φ(X)=cU*XU或者φ(X)=cU*XtU((?)X∈B(H)),其中c为非零常数,U∈B(H)为酉算子,Xt为X关于H中任意给定的正规正交基的转置矩阵.
第三章首先研究了算子乘积投影的相关性质,其次刻画了B(H)上保持乘积非零投影的非线性满射,最后给出了B(H)上保持乘积非零投影的非线性满射的结构.令H为复Hilbert空间且dim H≥2,B(H)表示H上所有有界线性算子全体.设φ是B(H)上双边保持乘积非零投影的非线性满射,通过应用算子分块技巧证明了φ具有如下结构：φ(A)=λU*AU((?)A∈B(H)),其中λ为非零常数满足λ2=1,U∈B(H)为酉算子或反酉算子.从而说明了算子乘积非零投影是B(H)上的等距不变量.
Preserver problems on operator algebras represent one of the most active re-search topics in operator algebra and also have important value in both theory and application for the classification of operator algebra. The research of this thesis fo-cuses on linear maps preserving asymptotic unitary similarity on operator algebras and non-linear surjective maps preserving the non-zero projections of products of operators on operator algebras. Using the technique of operator blocks, more char-acters of discussed maps can be found. This article is divided into three chapters.
In chapter 1, the signigication and background of this thesis selecting subject are introduced. In addition, we offer some necesary notations, definitions, and some conceptions and conclusions in the later chapters. Firstly, we give the signigication and background of this thesis selecting subject. Subsequently, we introduce the definitions of asymptotic unitary similarity, minimal asymptotic u-similarity, invari-ant subspaces, the unitary orbit, one-rank projections etc. Finally, we give some well-known propositions and theorems.
In chapter 2, the characters of the linear surjective mapφon B(H) preserv-ing asymptotic unitary similarity in both directions are studied. Let H be an infinite-dimensional complex Hilbert space and let B(H) denote the algebra of all bounded linear opeators on H. By giving the definition of minimal asymptotic uni-tary similarity-invariant subspaces and applying the closure of the unitary orbit of a linear operator, it is proved that the mapφhas the following representations:φ(X)= cU*XU orφ(X)=cU*XtU((?)X∈B(H)), where c is a non-zero constant, U∈B(H) is a unitary operator, and Xt denotes the transpose of X relative to a fixed but arbitrary orthonormal basis of H.
In chapter 3, we analyse the related characters of the projections of products of two operators at first. Then, maps preserving the non-zero projections of products of two operators are characterized and the structures of the mentioned maps are obtained at last. Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H≤2. Letφbe a surjective map on B(H) preserving non-zero projections of products of two operators in both directions. By using the technique of operator blocks, it is proved that the mapφhas the following structure:φ(A)=λU*AU for all A in B(H) for some non-zero constantλwithλ2=1, where U is a unitary or an anti-unitary operator on H. These show that non-zero projections of products of two operators are isometric invariants of B(H).
引文
[1]R. V. Kadison, J. R. Ringrose. Fundamentals of the theory of operator algebras [M]. volume Ⅰ, Ⅱ, Academic Press, INC., London,1986.
[2]候晋川,崔建莲.算子代数上的线性保持映射引论[M].科学出版社,北京,2002.
[3]C. K. Li, N. K. Tsing. Linear preserver problems:A brief introduction and some special techniques [J]. Linear Algebra and its Applications,1992,162-164:217-235.
[4]G. X. Ji, H. K. Du. Similaity-invariant subspaces and similarity-preserving linear maps [J]. Acta Mathematica Sinica (English Series),2002,18:489-498.
[5]G. X. Ji. Similarity-preserving linear maps on B(H) [J]. Linear Algebra and its Applications,2003,360:249-257.
[6]G. X. Ji. Asymptotic similarity-preserving linear maps on B(H) [J]. Linear Algebra and its Applications,2003,368:371-378.
[7]T. Petek. Linear mappings preserving similarity on B(H) [J]. Studia Mathematica, 2004,161 (2):177-186.
[8]T. Petek. A note on unitary similarity preserving linear mappings on B(H) [J]. Linear Algebra and its Applications,2005,394:217-224.
[9]D. A. Herrero. Approximation of Hilbert space operators [M]. Longman Scientific and Technical, New York,1989.
[10]P. Semrl. Non-linear commutativity preserving maps [J]. Acta Scientiarum Math-ematicarum (Szeged),2005,71:781-819.
[11]L. Huang, J. C. Hou. Maps preserving spectral functions of operator products [J]. Chinese Annals of Mathematics, Series A,2007,28 (6):8-14.
[12]C. K. Li, P. Semrl, N. S. Sze. Maps preserving the nilpotency of products of operators [J]. Linear Algebra and its Applications,2007,424:222-239.
[13]L. Fang, G. X. Ji, Y. F. Pang. Maps preserving the idempotency of products of operators [J]. Linear Algebra and its Applications,2007,426:40-52.
[14]G. X. Ji, F. L. Qu. Linear maps preserving projections of products of operators [J]. Acta Mathematica Sinica,2010,53 (2):315-322.
[15]C. Pearcy, D. Topping. Sums of small numbers of idempotents [J]. Michigan Math-ematical Journal,1967,14:453-465.
[16]L. Molnar. Maps on states preserving the relative entropy [J]. Journal of Mathe-matical Physics,2008,49:032-114.
[17]E. P. Wigner. Group theory and its application to the quantum theory of automic spectra [J]. Academic Press, INC., New York,1959:233-236.
[18]M. Bresar, P. Semrl. On locally linearly dependent operators and derivations [J]. Transactions of the American Mathematical Society,1999,35 (3):1257-1275.
[19]U. Uhlhorn. Representation of symmetry transformations in quantum mechanics [J]. Arkiy Fysik,1963,23:307-340.
[20]L. Baribeau, T Ransford. Non-linear spectrum preserving maps [J]. Bulletin of the London Mathematical Society,2000,32:8-14.
[21]M. Bresar, P. Semrl. Mappings which preserve idempotents, local automorphisms and local derivations [J]. Canadian Journal of Mathematics,1993,45 (3) 483-496.
[2]候晋川,崔建莲.算子代数上的线性保持映射引论[M].科学出版社,北京,2002.
[3]C. K. Li, N. K. Tsing. Linear preserver problems:A brief introduction and some special techniques [J]. Linear Algebra and its Applications,1992,162-164:217-235.
[4]G. X. Ji, H. K. Du. Similaity-invariant subspaces and similarity-preserving linear maps [J]. Acta Mathematica Sinica (English Series),2002,18:489-498.
[5]G. X. Ji. Similarity-preserving linear maps on B(H) [J]. Linear Algebra and its Applications,2003,360:249-257.
[6]G. X. Ji. Asymptotic similarity-preserving linear maps on B(H) [J]. Linear Algebra and its Applications,2003,368:371-378.
[7]T. Petek. Linear mappings preserving similarity on B(H) [J]. Studia Mathematica, 2004,161 (2):177-186.
[8]T. Petek. A note on unitary similarity preserving linear mappings on B(H) [J]. Linear Algebra and its Applications,2005,394:217-224.
[9]D. A. Herrero. Approximation of Hilbert space operators [M]. Longman Scientific and Technical, New York,1989.
[10]P. Semrl. Non-linear commutativity preserving maps [J]. Acta Scientiarum Math-ematicarum (Szeged),2005,71:781-819.
[11]L. Huang, J. C. Hou. Maps preserving spectral functions of operator products [J]. Chinese Annals of Mathematics, Series A,2007,28 (6):8-14.
[12]C. K. Li, P. Semrl, N. S. Sze. Maps preserving the nilpotency of products of operators [J]. Linear Algebra and its Applications,2007,424:222-239.
[13]L. Fang, G. X. Ji, Y. F. Pang. Maps preserving the idempotency of products of operators [J]. Linear Algebra and its Applications,2007,426:40-52.
[14]G. X. Ji, F. L. Qu. Linear maps preserving projections of products of operators [J]. Acta Mathematica Sinica,2010,53 (2):315-322.
[15]C. Pearcy, D. Topping. Sums of small numbers of idempotents [J]. Michigan Math-ematical Journal,1967,14:453-465.
[16]L. Molnar. Maps on states preserving the relative entropy [J]. Journal of Mathe-matical Physics,2008,49:032-114.
[17]E. P. Wigner. Group theory and its application to the quantum theory of automic spectra [J]. Academic Press, INC., New York,1959:233-236.
[18]M. Bresar, P. Semrl. On locally linearly dependent operators and derivations [J]. Transactions of the American Mathematical Society,1999,35 (3):1257-1275.
[19]U. Uhlhorn. Representation of symmetry transformations in quantum mechanics [J]. Arkiy Fysik,1963,23:307-340.
[20]L. Baribeau, T Ransford. Non-linear spectrum preserving maps [J]. Bulletin of the London Mathematical Society,2000,32:8-14.
[21]M. Bresar, P. Semrl. Mappings which preserve idempotents, local automorphisms and local derivations [J]. Canadian Journal of Mathematics,1993,45 (3) 483-496.