Hilbert空间下的再生核理论研究
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摘要
这是一篇关于Hilbert空间下的再生核理论的研究综述。本文主要分为三部分：第一部分主要介绍了再生核的产生,以及介绍一些相关需要的定义,定理；第二部分主要介绍了当我们所研究的函数是解析时,单位圆盘上的再生核表示形式以及相关定理,结论；第三部分主要介绍了当我们将上述形式推广到多维时,相关定理及其部分应用。我们最后利用一些已有的几何方面理论对一些再生核空间进行分类,这给出了一种新的证明方法。
This is a report about Reproducing kernel on Hilbert Spaces.It consists of three parts.The first part focuses on the regeneration of reproducing kernel, and introduces some definitions needed. The second part is mainly about some representation theorems and results of reproducing kernel when the Hilbert space consists of analytic functions on the unit disk. In part three, we extend the results of part 2 to multi-dimensional case.At last we will give some classification of some types of reproducing kernel Hilbert space by using some geometrical methods in operator theory. This gives a new proof of the classification.
This is a report about Reproducing kernel on Hilbert Spaces.It consists of three parts.The first part focuses on the regeneration of reproducing kernel, and introduces some definitions needed. The second part is mainly about some representation theorems and results of reproducing kernel when the Hilbert space consists of analytic functions on the unit disk. In part three, we extend the results of part 2 to multi-dimensional case.At last we will give some classification of some types of reproducing kernel Hilbert space by using some geometrical methods in operator theory. This gives a new proof of the classification.
引文
[1]Jim Agler and John E, McCarthy, Pick interpolation and Hilbert function spaces[M],Graduate Studies in Mathematics, American Mathematical Society, Providence.RI,2002,xx+308 pp.
[2]Aronszajn.N, Theory of Producing Kernels[J],Transactions of the American Mathematical So-ciety,1950,339-363.
[3]Conway.John.B, A Course in Functional Analysis (Sencond Edition)[M],Graduate Texts in Mathematics 96, Springer-Verlag,1990.
[4]Cowen.M.J and Douglas.R.G, Complex Geometry and Operator Theory, Acta Math.141(1978)no.3-4,187-261.
[5]Cowen.M.J and Douglas.R.G, Operators Possessing an Open Set of Eigenvalues, Memorical Conf.for Fejer-Riesz, Colloq.Math.Soc.J.Bolyai,1980,323-341.
[6]Chen Xiaoman and Guo Kunyu, Analytic Hilbert Modules[M],Chapman & Hall/CRC Res. Notes Math.vol.433,2003.
[7]Douglas.Ronald.G, Banach Algera Techniques in Operator Theory(Sencond Edition)[M],Grad-uate Texts in Mathematics 179,Springer-Verlag, July 1998.
[8]Douglas.Ronald.G and Vern I Paulsen, Hilbert modules over Function Algebras[M],Pitman Research Notes in Mathematics Series 217.Longman Scientific & Technical, Harlow,copublished in the United States with John Wiley & Sons, Inc, New York,1989.
[9]Ronald.G.Douglas and G.Misra, Equivalence of Quotient Hilbert Modules-Ⅱ[J],Trans. Amer. Math. Soc.360,2008,no.4,2229-2264.
[10]Kenneth Hoffman, Banach Spaces of Analytic Functions[M],Dover Publications,Inc,New York,1962.
[11]Guo Kuyun, Reproducing Function Spaces on the Complex Plane[J],2002,12,1-8.(print)
[12]Guo Kunyu, Junyun Hu and Xianmin Xu, Toeplitz Algebras,Subnormal Tupeles and Rigidity on ReproducingC[z1,…,zd]-Modules[J],Journal of Functional Analysis, June 2003,214-247.
[13]Paulsen Vern I, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces [J], 2006,2,20,1-34.(print)
[14]Rudin Walter, Function Theory in Polydiscs,Benjiamin[M],New York,1979.
[15]Rudin Walter, Function Theory in the Unit Ball of C[M],Springer-Verlag,New York,1980.
[16]Zhu.K.H, Operator Theory on Function Spaces[M],Marcel Dekker, New York,1990.
[17]郭坤宇,泛函分析的概念和方法[M],复旦大学数学科学学院,2006,8.
[18]童裕孙,泛函分析教程(第二版)[M],复旦大学出版社,2003,6.
[19]张锦豪,邱维元,复变函数论[M],高等教育出版社,2006,12.
[2]Aronszajn.N, Theory of Producing Kernels[J],Transactions of the American Mathematical So-ciety,1950,339-363.
[3]Conway.John.B, A Course in Functional Analysis (Sencond Edition)[M],Graduate Texts in Mathematics 96, Springer-Verlag,1990.
[4]Cowen.M.J and Douglas.R.G, Complex Geometry and Operator Theory, Acta Math.141(1978)no.3-4,187-261.
[5]Cowen.M.J and Douglas.R.G, Operators Possessing an Open Set of Eigenvalues, Memorical Conf.for Fejer-Riesz, Colloq.Math.Soc.J.Bolyai,1980,323-341.
[6]Chen Xiaoman and Guo Kunyu, Analytic Hilbert Modules[M],Chapman & Hall/CRC Res. Notes Math.vol.433,2003.
[7]Douglas.Ronald.G, Banach Algera Techniques in Operator Theory(Sencond Edition)[M],Grad-uate Texts in Mathematics 179,Springer-Verlag, July 1998.
[8]Douglas.Ronald.G and Vern I Paulsen, Hilbert modules over Function Algebras[M],Pitman Research Notes in Mathematics Series 217.Longman Scientific & Technical, Harlow,copublished in the United States with John Wiley & Sons, Inc, New York,1989.
[9]Ronald.G.Douglas and G.Misra, Equivalence of Quotient Hilbert Modules-Ⅱ[J],Trans. Amer. Math. Soc.360,2008,no.4,2229-2264.
[10]Kenneth Hoffman, Banach Spaces of Analytic Functions[M],Dover Publications,Inc,New York,1962.
[11]Guo Kuyun, Reproducing Function Spaces on the Complex Plane[J],2002,12,1-8.(print)
[12]Guo Kunyu, Junyun Hu and Xianmin Xu, Toeplitz Algebras,Subnormal Tupeles and Rigidity on ReproducingC[z1,…,zd]-Modules[J],Journal of Functional Analysis, June 2003,214-247.
[13]Paulsen Vern I, An Introduction to the Theory of Reproducing Kernel Hilbert Spaces [J], 2006,2,20,1-34.(print)
[14]Rudin Walter, Function Theory in Polydiscs,Benjiamin[M],New York,1979.
[15]Rudin Walter, Function Theory in the Unit Ball of C[M],Springer-Verlag,New York,1980.
[16]Zhu.K.H, Operator Theory on Function Spaces[M],Marcel Dekker, New York,1990.
[17]郭坤宇,泛函分析的概念和方法[M],复旦大学数学科学学院,2006,8.
[18]童裕孙,泛函分析教程(第二版)[M],复旦大学出版社,2003,6.
[19]张锦豪,邱维元,复变函数论[M],高等教育出版社,2006,12.