算子矩阵谱的摄动与Drazin算子的有关探讨
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摘要
本硕士学位论文着重对算子矩阵谱的摄动问题和幂等算子线性组合Drazin可逆性及Drazin逆表示问题作比较详细的探讨,取得一部分结果.
首先,探讨了2×2阶上三角算子矩阵的连续谱σc。、点谱σp和剩余谱σr的相关问题,完全刻画出∩C∈B(Y,X)σc(MC)、∩C∈B(K,H)σP(MC)和∩C∈B(K,H)σr(MC)的表示(分别见定理1.2.4、定理1.2.7、定理1.2.10)；肯定地回答了算子连续谱σc的可达问题(推论1.2.5).
其次,在可分的Hilbert空间上,给出了三类新算子的概念：Kato一Weyl算子,Kato-左Weyl算子和Kato-右Weyl算子(见定义2.1.2-定义2.1.4)；进而探讨了可分Hilbert空间上的2×2阶上三角算子矩阵是Kato-(左、右)Weyl算子的一些充分条件或者必要条件(见定理2.2.5、定理2.2.7、推论2.2.11.、定理2.2.12、定理2.2.15、定理2.2.16)；还获得了算子Kato-(左、右)Weyl谱的多种摄动的稳定性(见定理2.3.1、定理2.3.2、推论2.3.3).
最后,借助算子矩阵分块的思想方法,进一步地探讨了在某些条件下Hilbert空间上幂等算子线性组合的Drazin可逆性并且给出了Drazin逆的表示(见定理3.2.3、定理3.2.5、定理3.2.7、定理3.3.1).此外,还讨论了Banach空间上Drazin可逆算子在0点的特征投影(见定理4.2.7、推论4.2.8、定理4.2.10).
本文有两个特色：其一,构造了一些反例,能够有效地用来辨析算子矩阵谱理论探讨中的某些问题；其二,定义了若干类新算子,相应地对算子三种新的谱成分σ*∈{σwk,σlwk,σrwk}进行了系统的研讨.
In this paper, we mainly study the problem of the perturbations of spectrum of operator matrix and the problem of the Drazin invertibility and Drazin inverses of linear combinations of two idempotents, leading to some new results about these thesises.
Firstly, we investigate some related spectrum issues for the continuous spectrumσc, point spectrumσp and residual spectrumσr of the 2×2 upper operator matrix. Then the sets∩C∈B(Y,X)σc(Mc),∩C∈B(K,H)σpp(Mc) and∩C∈B(K,H)σr(Mc) are, re-spectively, characterized completely(see theorem 1.2.4、1.2.7 and 1.2.10). We also give an affirmative answer for the attainment problem of the continuous spectrumσc(see proposition 1.2.5).
Secondly, three new classes of operators—Kato-Weyl operator, Kato-left Weyl operator and Kato-right Weyl operator—are defined in separable Hilbert spaces(see definitions 2.1.2-2.1.4). Then it follows some sufficient or necessary conditions for the 2×2 upper operator matrix to be Kato-Weyl operator, Kato-left Weyl operator, Kato-right Weyl operator, respectively. And we obtain some stabilities of perturbations of the newly defined operators(see theorem 2.3.1、2.3.2 and 2.3.3).
Finally, by virtue of the thought and methods of block of operator matrix, we have some more generalized results of the Drazin invertibility and Drazin inverses of linear combinations of two idempotents under some conditions in separable Hilbert space(see theorem 3.2.3,3.2.5 and 3.2.7). Also the characterizations of eigenprojections at zero of Drazin invertible operators are extended from Hilbert space into Banach space.
In addition, this article has two features:one is that we construct some examples, which are of great help for the study of the spectrum of operator matrices; the other one is that we define three new classes of operators and the corresponding spectrumσ*∈{σω
首先,探讨了2×2阶上三角算子矩阵的连续谱σc。、点谱σp和剩余谱σr的相关问题,完全刻画出∩C∈B(Y,X)σc(MC)、∩C∈B(K,H)σP(MC)和∩C∈B(K,H)σr(MC)的表示(分别见定理1.2.4、定理1.2.7、定理1.2.10)；肯定地回答了算子连续谱σc的可达问题(推论1.2.5).
其次,在可分的Hilbert空间上,给出了三类新算子的概念：Kato一Weyl算子,Kato-左Weyl算子和Kato-右Weyl算子(见定义2.1.2-定义2.1.4)；进而探讨了可分Hilbert空间上的2×2阶上三角算子矩阵是Kato-(左、右)Weyl算子的一些充分条件或者必要条件(见定理2.2.5、定理2.2.7、推论2.2.11.、定理2.2.12、定理2.2.15、定理2.2.16)；还获得了算子Kato-(左、右)Weyl谱的多种摄动的稳定性(见定理2.3.1、定理2.3.2、推论2.3.3).
最后,借助算子矩阵分块的思想方法,进一步地探讨了在某些条件下Hilbert空间上幂等算子线性组合的Drazin可逆性并且给出了Drazin逆的表示(见定理3.2.3、定理3.2.5、定理3.2.7、定理3.3.1).此外,还讨论了Banach空间上Drazin可逆算子在0点的特征投影(见定理4.2.7、推论4.2.8、定理4.2.10).
本文有两个特色：其一,构造了一些反例,能够有效地用来辨析算子矩阵谱理论探讨中的某些问题；其二,定义了若干类新算子,相应地对算子三种新的谱成分σ*∈{σwk,σlwk,σrwk}进行了系统的研讨.
In this paper, we mainly study the problem of the perturbations of spectrum of operator matrix and the problem of the Drazin invertibility and Drazin inverses of linear combinations of two idempotents, leading to some new results about these thesises.
Firstly, we investigate some related spectrum issues for the continuous spectrumσc, point spectrumσp and residual spectrumσr of the 2×2 upper operator matrix. Then the sets∩C∈B(Y,X)σc(Mc),∩C∈B(K,H)σpp(Mc) and∩C∈B(K,H)σr(Mc) are, re-spectively, characterized completely(see theorem 1.2.4、1.2.7 and 1.2.10). We also give an affirmative answer for the attainment problem of the continuous spectrumσc(see proposition 1.2.5).
Secondly, three new classes of operators—Kato-Weyl operator, Kato-left Weyl operator and Kato-right Weyl operator—are defined in separable Hilbert spaces(see definitions 2.1.2-2.1.4). Then it follows some sufficient or necessary conditions for the 2×2 upper operator matrix to be Kato-Weyl operator, Kato-left Weyl operator, Kato-right Weyl operator, respectively. And we obtain some stabilities of perturbations of the newly defined operators(see theorem 2.3.1、2.3.2 and 2.3.3).
Finally, by virtue of the thought and methods of block of operator matrix, we have some more generalized results of the Drazin invertibility and Drazin inverses of linear combinations of two idempotents under some conditions in separable Hilbert space(see theorem 3.2.3,3.2.5 and 3.2.7). Also the characterizations of eigenprojections at zero of Drazin invertible operators are extended from Hilbert space into Banach space.
In addition, this article has two features:one is that we construct some examples, which are of great help for the study of the spectrum of operator matrices; the other one is that we define three new classes of operators and the corresponding spectrumσ*∈{σω
引文
[1]钟怀杰.Banach空间结构和算子理想[M],北京：科学出版社,2005.
[2]Pietro Aiena. Predholm and Local Spectral Theory with Application to Multipliers[M], Dordrecht:Kluwer Acad. Publishers,2004.
[3]J. K. Baksalary, O. M. Baksalary, H. Ozdemir. A note on linear combinations of com-muting tripotent matrices[J], Linear Algebra Appl,2004,388:45-51.
[4]J. K. Baksalary, O. M. Baksalary. Idempotency of linear combinations of two idempotent matrice[J], Linear Algebra Appl,2000,321:3-7.
[5]J. K. Baksalary, O.M. Baksalary, G.P.H. Styan. Idempotency of linear combinations of an idempotent matrix and a tripotent matrix[J], Linear Algebra Appl,2002,54:21-34.
[6]O. M. Baksalary, J. Benftez. Idempotency of linear combinations of three idempotent matrices, two of which are commuting[J], Linear Algebra Appl,2007,424:320-337.
[7]J. Benitez, N. Thome. Idempotency of linear combinations of an idempotent matrix and a t-potent matrix that commute[J], Linear Algebra Appl,2005,403:414-418.
[8]X. H. Cao. Browder spectra for upper triangular operator matrices[J], J. Math. Anal. Appl,2008,342:477-484.
[9]Caradus S.R. Pfaffenberger W.E. Yood B. Calkin Algebras and Algebras of Operators on Banach Spaces[M], New York:Marcel Dekker,1974.
[10]X. L. Chen, S. F. Zhang, H. J. Zhong. On the filling in holes problem of operator matrices [J], Linear Algebra Appl,2009,430:558-563.
[11]C. Y. Deng. The Drazin inverses of products and differences of orthogonal projections[J], J. Math. Anal. Appl,2007,335:64-71.
[12]C. Y. Deng. The Drazin inverses of sum and difference of idempotents, LinearAlgebra Appl[J],2009,430:1282-1291.
[13]D. S. Djordjevic. Perturbations of spectra of operator matrices[J], J. Operator Theory, 2002,48:467-486.
[14]D.S.Djordjevic, P.S.Stanimirovic. On the generalized Drazin inverse and generalized re-solvent[J], Czechoslovak Math. J,2001,51(126):617-634.
[15]R.G.Douglas. On majorization facterization and range inclusion of operators in Hilbert space[J], Proc.Amer.Math.Soc,1966,17:413-416.
[16]H.R.Dowson. Spectral theory of linear operators[M], New York:Academic Press,1978.
[17]M.P. Drazin. Pseudoinverse in associative rings and semigroups [J], Amer. Math. Monthly,1958,65:506-514.
[18]H. K. Du, J. Pan. Perturbation of spectrums of 2 x 2 operator matrices [J], Proc. Amer. Math. Soc,1994,121:761-766.
[19]H. K. Du, X. Yao, C. Deng. Invertibility of linear combinations of two idempotents[J], Proc. Amer. Math. Soc,2006,134:1451-1457.
[20]J. K. Han, H. Y. Lee, W. Y. Lee. Invertible completions of 2 x 2 upper triangular operator matrices[J], Proc. Amer. Math. Soc,1999,128:119-123.
[21]J.J.Koliha, V. Rakccevic. The nullity and rank of linear combinations of idempotent matrices [J], Linear Algebra Appl,2006,418:11-14.
[22]J.J. Koliha, V. Rakccevic. Stability theorems for linear combinations of idempotents[J], Integral Equations Operator Theory, to appear.
[23]J.J. Koliha, V. Rakccevic, I. Straskraba. The difference and sum of projectors [J], Linear Algebra Appl,2004,388:279-288.
[24]G. L. Hou, Alatancang. Perturbation of spectrums of upper triangular operator matri-ces[J], J. Sys. Sci. Math. Scis (in Chinese),2006,26:257-263.
[25]I. S. Hwang, W. Y. Lee. The boundedness below of 2 x 2 upper triangular operator matrices [J], Integr. Equ. Oper. Theory,2001,39:267-276.
[26]Y. Q. Ji. Quasitriangular+small compact=strongly irreducible[J],Trans. Amer. Math. Soc,1999,351:4657-4673.
[27]W. Y. Lee. Weyl's theorem for operator matrices[J], Integr. equ. oper. theory,1998,32: 319-331.
[28]W. Y. Lee. Weyl spectra of operator matrices[J], Proc. Amer. Math. Soc,2000,129: 131-138.
[29]Y.Li, X.H.Sun, H.K.Du. Intersections of the Left and Right Essential Spectra of 2*2 Upper Triangular Operator Matrices[J], Bull.London Math.Soc,2004,36:811-819.
[30]Y. Li, H.K. Du. The intersection of essential approximate point Spectra of Operator Matrices[J], J. Math. Anal. Appl,2006,323:1171-1183.
[31]H. Ozdemir, A.Y. Ozban. On idempotency of linear combinations of idempotent matri-ces[J], Appl. Math. Comput,2004,159:439-448.
[32]M. Sarduvan, H. Ozdemir. On linear combinations of two tripotent, idempotent and involutive matrices [J], Appl. Math. Comput,2008,200:401-406.
[33]H.Y. Zhang, H. K. Du. Browder spectra of upper-triangular Operator Matrices[J], J. Math. Anal. Appl,2006,323:700-707.
[34]S. F. Zhang, H. J. Zhong, Q. F. Jiang. Drazin spectrum of operator matrices on the Banach space[J], Linear Algebra Appl,2008,429:2067-2075.
[35]陈晓玲.算子矩阵谱理论的有关探讨[D],福建师范大学硕士学位论文,2006.
[36]张世芳.算子矩阵谱的相关探讨[D],福建师范大学硕士学位论文,2008.
[37]江泽坚,孙善利.泛函分析(第二版)[M],北京：高等教育出版社,2005.
[38]李愿.2*2上三角算子矩阵的左(右)Weyl谱的交[J],数学学报,2005,48：653-660.
[39]张希花,陈燕妮等.Drazin可逆算子在0点的特征投影刻画[J],陕西师范大学学报(自然科学版),2006,12：21-24.
[2]Pietro Aiena. Predholm and Local Spectral Theory with Application to Multipliers[M], Dordrecht:Kluwer Acad. Publishers,2004.
[3]J. K. Baksalary, O. M. Baksalary, H. Ozdemir. A note on linear combinations of com-muting tripotent matrices[J], Linear Algebra Appl,2004,388:45-51.
[4]J. K. Baksalary, O. M. Baksalary. Idempotency of linear combinations of two idempotent matrice[J], Linear Algebra Appl,2000,321:3-7.
[5]J. K. Baksalary, O.M. Baksalary, G.P.H. Styan. Idempotency of linear combinations of an idempotent matrix and a tripotent matrix[J], Linear Algebra Appl,2002,54:21-34.
[6]O. M. Baksalary, J. Benftez. Idempotency of linear combinations of three idempotent matrices, two of which are commuting[J], Linear Algebra Appl,2007,424:320-337.
[7]J. Benitez, N. Thome. Idempotency of linear combinations of an idempotent matrix and a t-potent matrix that commute[J], Linear Algebra Appl,2005,403:414-418.
[8]X. H. Cao. Browder spectra for upper triangular operator matrices[J], J. Math. Anal. Appl,2008,342:477-484.
[9]Caradus S.R. Pfaffenberger W.E. Yood B. Calkin Algebras and Algebras of Operators on Banach Spaces[M], New York:Marcel Dekker,1974.
[10]X. L. Chen, S. F. Zhang, H. J. Zhong. On the filling in holes problem of operator matrices [J], Linear Algebra Appl,2009,430:558-563.
[11]C. Y. Deng. The Drazin inverses of products and differences of orthogonal projections[J], J. Math. Anal. Appl,2007,335:64-71.
[12]C. Y. Deng. The Drazin inverses of sum and difference of idempotents, LinearAlgebra Appl[J],2009,430:1282-1291.
[13]D. S. Djordjevic. Perturbations of spectra of operator matrices[J], J. Operator Theory, 2002,48:467-486.
[14]D.S.Djordjevic, P.S.Stanimirovic. On the generalized Drazin inverse and generalized re-solvent[J], Czechoslovak Math. J,2001,51(126):617-634.
[15]R.G.Douglas. On majorization facterization and range inclusion of operators in Hilbert space[J], Proc.Amer.Math.Soc,1966,17:413-416.
[16]H.R.Dowson. Spectral theory of linear operators[M], New York:Academic Press,1978.
[17]M.P. Drazin. Pseudoinverse in associative rings and semigroups [J], Amer. Math. Monthly,1958,65:506-514.
[18]H. K. Du, J. Pan. Perturbation of spectrums of 2 x 2 operator matrices [J], Proc. Amer. Math. Soc,1994,121:761-766.
[19]H. K. Du, X. Yao, C. Deng. Invertibility of linear combinations of two idempotents[J], Proc. Amer. Math. Soc,2006,134:1451-1457.
[20]J. K. Han, H. Y. Lee, W. Y. Lee. Invertible completions of 2 x 2 upper triangular operator matrices[J], Proc. Amer. Math. Soc,1999,128:119-123.
[21]J.J.Koliha, V. Rakccevic. The nullity and rank of linear combinations of idempotent matrices [J], Linear Algebra Appl,2006,418:11-14.
[22]J.J. Koliha, V. Rakccevic. Stability theorems for linear combinations of idempotents[J], Integral Equations Operator Theory, to appear.
[23]J.J. Koliha, V. Rakccevic, I. Straskraba. The difference and sum of projectors [J], Linear Algebra Appl,2004,388:279-288.
[24]G. L. Hou, Alatancang. Perturbation of spectrums of upper triangular operator matri-ces[J], J. Sys. Sci. Math. Scis (in Chinese),2006,26:257-263.
[25]I. S. Hwang, W. Y. Lee. The boundedness below of 2 x 2 upper triangular operator matrices [J], Integr. Equ. Oper. Theory,2001,39:267-276.
[26]Y. Q. Ji. Quasitriangular+small compact=strongly irreducible[J],Trans. Amer. Math. Soc,1999,351:4657-4673.
[27]W. Y. Lee. Weyl's theorem for operator matrices[J], Integr. equ. oper. theory,1998,32: 319-331.
[28]W. Y. Lee. Weyl spectra of operator matrices[J], Proc. Amer. Math. Soc,2000,129: 131-138.
[29]Y.Li, X.H.Sun, H.K.Du. Intersections of the Left and Right Essential Spectra of 2*2 Upper Triangular Operator Matrices[J], Bull.London Math.Soc,2004,36:811-819.
[30]Y. Li, H.K. Du. The intersection of essential approximate point Spectra of Operator Matrices[J], J. Math. Anal. Appl,2006,323:1171-1183.
[31]H. Ozdemir, A.Y. Ozban. On idempotency of linear combinations of idempotent matri-ces[J], Appl. Math. Comput,2004,159:439-448.
[32]M. Sarduvan, H. Ozdemir. On linear combinations of two tripotent, idempotent and involutive matrices [J], Appl. Math. Comput,2008,200:401-406.
[33]H.Y. Zhang, H. K. Du. Browder spectra of upper-triangular Operator Matrices[J], J. Math. Anal. Appl,2006,323:700-707.
[34]S. F. Zhang, H. J. Zhong, Q. F. Jiang. Drazin spectrum of operator matrices on the Banach space[J], Linear Algebra Appl,2008,429:2067-2075.
[35]陈晓玲.算子矩阵谱理论的有关探讨[D],福建师范大学硕士学位论文,2006.
[36]张世芳.算子矩阵谱的相关探讨[D],福建师范大学硕士学位论文,2008.
[37]江泽坚,孙善利.泛函分析(第二版)[M],北京：高等教育出版社,2005.
[38]李愿.2*2上三角算子矩阵的左(右)Weyl谱的交[J],数学学报,2005,48：653-660.
[39]张希花,陈燕妮等.Drazin可逆算子在0点的特征投影刻画[J],陕西师范大学学报(自然科学版),2006,12：21-24.