有关幂等算子的Drazin逆等价性
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摘要
研究Hilbert空间上幂等算子的Drazin逆等价性性质。引入Hilbert空间上有界线性算子的空间分解理论和算子矩阵分块的技巧,给出幂等算子在一定空间分解下的算子矩阵表示,进而研究幂等算子和、差和积的Drazin逆的几个等价条件,最后给出幂等算子积条件下Drazin逆的等价条件。
Properties of two idempotent operators on a Hilbert space are studied.Based on the theory of space decomposition of linear and bounded operators and the technique of operator matrix blocking, the representations and characterization of two idempotent operators on a given Hilbert space are given. And several equivalent conditions for the Drazin invertibility of the sum, difference of two idempotents on a Hilbert space are studied. Furthermore, equivalent condition for the Drazin invertibility of the product of idempotents is obtained.
Properties of two idempotent operators on a Hilbert space are studied.Based on the theory of space decomposition of linear and bounded operators and the technique of operator matrix blocking, the representations and characterization of two idempotent operators on a given Hilbert space are given. And several equivalent conditions for the Drazin invertibility of the sum, difference of two idempotents on a Hilbert space are studied. Furthermore, equivalent condition for the Drazin invertibility of the product of idempotents is obtained.
引文
[1] DRAZIN M P.Pseudo-inverses in associative rings and semi groups[J].Amer Math Monthly,1958,65(7):506-514.
[2] BAKSALARY J K,BAKSALARY O M.Idempotency of linear combinations of two idempotent matrices[J].Linear Algebra Appl,2000,321(1):3-7.
[3] DU H K,YAO X Y,DENG C Y.Invertibility of linear combinations of two idempotents[J].Proceeding of the American Mathematical Society,2006,134(5):1451-1457.
[4] KOLIHA J J,RAKOCEVIC V.The difference and sum of projections[J].Linear Algebra Appl,2004,388(5):279-288.
[5] BAKSALARY O M.Idempotency of linear combinations of three idempotent matrices,two of which are disjoint[J].Linear Algebra Appl,2004,388(1):67-78.
[6] BAKSALARY J K,BAKSALARY O M.Idempotency of linear combinations of two idempotent matrices[J].Linear Algebra Appl,2000,321(1):3-7.
[7] DENG C Y,DU H K.Common complements of two subspaces and an answer of Grob′s question[J].Acta Math Sinica,2006,49(5):1009-1112.
[8] LU J M,DU H K,WEI Y M.Drazin invertibility of operators AB and BA[J].J Math Rese Expo,2008,28(4):1017-1020.
[9] DENG C Y,WEI Y M.Characterizations and representations of the Drazin inverse involving idempotents[J].Linear Algebra Appl,2009,431(9):1526-1538.
[10] DJORDJEVIC D C,STANIMIROVIC P S.On the generalized Drazin inverse and generalized resolvent[J].Czechoslovak Math J,2001,51(3):617-634.
[11] DENG C Y.The Drazin inverses of products and differences of orthogonal projections[J].J Math Anal Appl,2007,335(1):64-71.
[12] KOLIHA J J,RAKOCEVIC V.Differentiability of the g-Drazin inverse[J].Studia Math,2005,168(3):193-201.
[13] RAKOEVIC V,WEI Y.The perturbation theory for the Drazin inverse and its applications II[J].J Austral Math Soc,2001,70(2):189-197.
[14] DU H K,DENG C Y.The representation and characterization of the Drazin inverses on a Hilbert space[J].Linear Algebra Appl,2005,407(407):117-124.
[15] WEI Y M,QIAO S.The Drazin inverse of updating of a square matrix with application to perturbation formula[J].Appl Math Comput,2000,108(2):77-83.
[16] ZHENG Z,ZHANG Y,JING S.Nonlinear impulsive differential and integral inequalities with nonlocal jump conditions[J].Journal of Inequalities & Applications,2018,2018(1):170-185.
[17] HENDERSON J,LUCA R.Existence and multiplicity of positive solutions for a system of difference equations with coupled boundary conditions[J].J Appl Anal Comput,2017,7(1):134-146.
[2] BAKSALARY J K,BAKSALARY O M.Idempotency of linear combinations of two idempotent matrices[J].Linear Algebra Appl,2000,321(1):3-7.
[3] DU H K,YAO X Y,DENG C Y.Invertibility of linear combinations of two idempotents[J].Proceeding of the American Mathematical Society,2006,134(5):1451-1457.
[4] KOLIHA J J,RAKOCEVIC V.The difference and sum of projections[J].Linear Algebra Appl,2004,388(5):279-288.
[5] BAKSALARY O M.Idempotency of linear combinations of three idempotent matrices,two of which are disjoint[J].Linear Algebra Appl,2004,388(1):67-78.
[6] BAKSALARY J K,BAKSALARY O M.Idempotency of linear combinations of two idempotent matrices[J].Linear Algebra Appl,2000,321(1):3-7.
[7] DENG C Y,DU H K.Common complements of two subspaces and an answer of Grob′s question[J].Acta Math Sinica,2006,49(5):1009-1112.
[8] LU J M,DU H K,WEI Y M.Drazin invertibility of operators AB and BA[J].J Math Rese Expo,2008,28(4):1017-1020.
[9] DENG C Y,WEI Y M.Characterizations and representations of the Drazin inverse involving idempotents[J].Linear Algebra Appl,2009,431(9):1526-1538.
[10] DJORDJEVIC D C,STANIMIROVIC P S.On the generalized Drazin inverse and generalized resolvent[J].Czechoslovak Math J,2001,51(3):617-634.
[11] DENG C Y.The Drazin inverses of products and differences of orthogonal projections[J].J Math Anal Appl,2007,335(1):64-71.
[12] KOLIHA J J,RAKOCEVIC V.Differentiability of the g-Drazin inverse[J].Studia Math,2005,168(3):193-201.
[13] RAKOEVIC V,WEI Y.The perturbation theory for the Drazin inverse and its applications II[J].J Austral Math Soc,2001,70(2):189-197.
[14] DU H K,DENG C Y.The representation and characterization of the Drazin inverses on a Hilbert space[J].Linear Algebra Appl,2005,407(407):117-124.
[15] WEI Y M,QIAO S.The Drazin inverse of updating of a square matrix with application to perturbation formula[J].Appl Math Comput,2000,108(2):77-83.
[16] ZHENG Z,ZHANG Y,JING S.Nonlinear impulsive differential and integral inequalities with nonlocal jump conditions[J].Journal of Inequalities & Applications,2018,2018(1):170-185.
[17] HENDERSON J,LUCA R.Existence and multiplicity of positive solutions for a system of difference equations with coupled boundary conditions[J].J Appl Anal Comput,2017,7(1):134-146.