In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of o...In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of operators, where the operator ?was defined by . In this note, a partial answer for the question is given.展开更多
Let A ∈ B（X） and B ∈ B（Y）, Me be an operator on Banach space X ＋ Y given by Mc =（A C 0 B）A generalized Drazin spectrum defined by σgD（T） = {λ∈C ： T-λI is not generalized Drazin invertible} is considere...Let A ∈ B（X） and B ∈ B（Y）, Me be an operator on Banach space X ＋ Y given by Mc =（A C 0 B）A generalized Drazin spectrum defined by σgD（T） = {λ∈C ： T-λI is not generalized Drazin invertible} is considered in this paper. It is shown that展开更多
A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of s...A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.展开更多
In this paper,the author discusses some singular integral operators,singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels,and obtain some useful prop...In this paper,the author discusses some singular integral operators,singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels,and obtain some useful properties for them.These results improve both the classical theory of singular integral equation and the classical theory of singular quadrature.展开更多
In this paper, we discuss some singulal integral operators, singular quadrature operators and discrethation matrices associated with singular integral equations of the first kind, and obtain some useful Properties for...In this paper, we discuss some singulal integral operators, singular quadrature operators and discrethation matrices associated with singular integral equations of the first kind, and obtain some useful Properties for them. Using these operators we give a unified framework for various collocation methods of numerical solutions of singular integral equations of the fine kind, which appears very simple.展开更多
Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathe...Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathematically.展开更多
For classical transformation （q1,q2） → （Aq1 ＋ Bq2, Cq1 ＋ Dq2）, where AD - CB ≠ 1, we find its quantum mechanical image by using LDU decomposition of the matrix （A B C D ）. The explicit operators L, D, and U ...For classical transformation （q1,q2） → （Aq1 ＋ Bq2, Cq1 ＋ Dq2）, where AD - CB ≠ 1, we find its quantum mechanical image by using LDU decomposition of the matrix （A B C D ）. The explicit operators L, D, and U axe derived and their physical meaning is revealed, this also provides a new way for disentangling some exponential operators.展开更多
Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In ...Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n× n?structured matrices M?, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.展开更多
Let Mc = （ A0CB ） be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that σr（A） ∪ σr（ B）= σr （Mc） ∪ W ,where W is the union of certain of the holes in σr（Mc...Let Mc = （ A0CB ） be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that σr（A） ∪ σr（ B）= σr （Mc） ∪ W ,where W is the union of certain of the holes in σr（Mc） which happen to be subsets of σr（A） ∩ σr（B）, and σr（A）, σr（B）, σr（Mc） can be equal to the Browder or essential spectra of A, B, Mc, respectively. We also show that the above result isn＇t true for the Kato spectrum, left （right） essential spectrum and left （right） spectrum.展开更多
In this paper, we study the perturbation of spectra for 2 ×2 operator matrices such as Mx ={A0 XB） AC and Mz = （Az CB） on the Hilbert space H K and the sets……
Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For gi...Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.展开更多
Let MX=（A C X B ）be a 2×2 operator matrix acting on the Hilbert space H＋K. For given A∈B（H）,B∈B（K）and C∈B（K,H）the set UX∈B（H,K）^σe（MX）is determined, whereσe（T）denotes the essential spectrum.
We shall point out the further properties of the opera-tion→.And some uses of the operation→in the Fuzzy Equa-tions are discussed.Moreover a sufficient and necessarycondition is obtained.
文摘In [Perturbation of Spectrums of 2 × 2 Operator Matrices, Proceedings of the American Mathematical Society, Vol. 121, 1994], the authors asked whether there was an operator ?such that ?for a given pair?(A,B)?of operators, where the operator ?was defined by . In this note, a partial answer for the question is given.
基金Supported by the National Natural Science Foundation of China(11301077,1117131,11171066 and11226113)Foundation of the Education Department of Fujian Province(JA12074)the Natural ScienceFoundation of Fujian Province(2012J05003)
文摘Let A ∈ B（X） and B ∈ B（Y）, Me be an operator on Banach space X ＋ Y given by Mc =（A C 0 B）A generalized Drazin spectrum defined by σgD（T） = {λ∈C ： T-λI is not generalized Drazin invertible} is considered in this paper. It is shown that
基金supported by the National Natural Science Foundation of China (Grant Nos. 11361034 and 11371185)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20111501110001)the Natural Science Foundation of Inner Mongolia, China (Grant Nos. 2012MS0105 and 2013ZD01 )
文摘A necessary and sufficient condition is obtained for the generalized eigenfunction systems of 2 ×2 operator matrices to be a block Schauder basis of some Hilbert space, which offers a mathematical foundation of solving symplectic elasticity problems by using the method of separation of variables. Moreover, the theoretical result is applied to two plane elasticity problems via the separable Hamiltonian systems.
基金Supported by the Netional Natural Science Foundation of China and the Foundation of the State Education Commission of China.
文摘In this paper,the author discusses some singular integral operators,singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels,and obtain some useful properties for them.These results improve both the classical theory of singular integral equation and the classical theory of singular quadrature.
文摘In this paper, we discuss some singulal integral operators, singular quadrature operators and discrethation matrices associated with singular integral equations of the first kind, and obtain some useful Properties for them. Using these operators we give a unified framework for various collocation methods of numerical solutions of singular integral equations of the fine kind, which appears very simple.
文摘Matrix structuring is a very beautiful way to place Bernoulli numbers, by which a new view to the numbers is opened. Natural Numbers are mathematics seeds and Natural Number System （NNS） breeds the whole world mathematically.
基金Supported by the President Foundation of Chinese Academy of Science and Specialized Research Fund for the Doctorial Progress of Higher Education under Grant No.20070358009
文摘For classical transformation （q1,q2） → （Aq1 ＋ Bq2, Cq1 ＋ Dq2）, where AD - CB ≠ 1, we find its quantum mechanical image by using LDU decomposition of the matrix （A B C D ）. The explicit operators L, D, and U axe derived and their physical meaning is revealed, this also provides a new way for disentangling some exponential operators.
文摘Newton’s iteration is a fundamental tool for numerical solutions of systems of equations. The well-known iteration ?rapidly refines a crude initial approximation X0?to the inverse of a general nonsingular matrix. In this paper, we will extend and apply this method to n× n?structured matrices M?, in which matrix multiplication has a lower computational cost. These matrices can be represented by their short generators which allow faster computations based on the displacement operators tool. However, the length of the generators is tend to grow and the iterations do not preserve matrix structure. So, the main goal is to control the growth of the length of the short displacement generators so that we can operate with matrices of low rank and carry out the computations much faster. In order to achieve our goal, we will compress the computed approximations to the inverse to yield a superfast algorithm. We will describe two different compression techniques based on the SVD and substitution and we will analyze these approaches. Our main algorithm can be applied to more general classes of structured matrices.
基金the National Natural Science Foundation of China (Grants NO.10471025,NO.10771034)the Natural Science Foundation of Fujian Province of China(Grant NO.S0650009)+1 种基金the Education Department Foundation of Fujian Province of China (Grants NO.JA05211,NO.JB06026)the Foundation of Technology and Development of Fuzhou University(Grant NO.2007-XY-11)
文摘Let Mc = （ A0CB ） be a 2 × 2 upper triangular operator matrix acting on the Banach space X × Y. We prove that σr（A） ∪ σr（ B）= σr （Mc） ∪ W ,where W is the union of certain of the holes in σr（Mc） which happen to be subsets of σr（A） ∩ σr（B）, and σr（A）, σr（B）, σr（Mc） can be equal to the Browder or essential spectra of A, B, Mc, respectively. We also show that the above result isn＇t true for the Kato spectrum, left （right） essential spectrum and left （right） spectrum.
基金Supported by the National Natural Science Foundation of China(No.10962004)Tianyuan Fund for Mathematics(No.11126307)+2 种基金the National Natural Science Foundation of Inner Mongolia(No.2011MS0104, 2012MS0105)the Research Program of Science at Universities of Inner Mongolia Autonomous Region(No.NJZZ11011)Program of Higher-level Talents of Inner Mongolia University(No.Z20100116)
文摘In this paper, we study the perturbation of spectra for 2 ×2 operator matrices such as Mx ={A0 XB） AC and Mz = （Az CB） on the Hilbert space H K and the sets……
基金the National Natural Science Foundation of China (No.10562002)the Specialized Research Foundation for the Doctoral Program of Higher Education (No.20070126002)the Scientific Research Foun-dation for the Returned Overseas Chinese Scholars
文摘Let H1, H2 and H3 be infinite dimensional separable complex Hilbert spaces. We denote by M（D,V,F） a 3×3 upper triangular operator matrix acting on Hi ＋H2＋ H3 of theform M（D,E,F）=（A D F 0 B F 0 0 C）.For given A ∈ B（H1）, B ∈ B（H2） and C ∈ B（H3）, the sets ∪D,E,F^σp（M（D,E,F））,∪D,E,F ^σr（M（D,E,F））,∪D,E,F ^σc（M（D,E,F）） and ∪D,E,F σ（M（D,E,F）） are characterized, where D ∈ B（H2,H1）, E ∈B（H3, H1）, F ∈ B（H3,H2） and σ（·）, σp（·）, σr（·）, σc（·） denote the spectrum, the point spectrum, the residual spectrum and the continuous spectrum, respectively.
基金Foundation item： the National Natural Science Foundation of China （No. 10726043）.
文摘Let MX=（A C X B ）be a 2×2 operator matrix acting on the Hilbert space H＋K. For given A∈B（H）,B∈B（K）and C∈B（K,H）the set UX∈B（H,K）^σe（MX）is determined, whereσe（T）denotes the essential spectrum.
文摘We shall point out the further properties of the opera-tion→.And some uses of the operation→in the Fuzzy Equa-tions are discussed.Moreover a sufficient and necessarycondition is obtained.
