Table of Content

  

20 April 2012
Volume 47 Issue 4

Articles

Jordan derivations of Jordan algebras

JI Pei-sheng, SUN Xiao-lu, JIANG Hua

J4. 2012, 47(4):  1-4. 

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Let A be a unital Jordan algebra. A linear map d: A→A is called a Jordan derivation on A if it satisfies that d(ab)=d(a)。b+a。d(b) for all a,b∈A. Expressions of the Jordan derivations of Jordan algebras of all self-adjoint operators and Spin factors are given. It is proved that all local Jordan derivations and 2-local Jordan derivations on Spin factors are Jordan derivations.

Additive Jordan derivable maps of certain rings

CAO Zong-xia, ZHANG Jian-hua

J4. 2012, 47(4):  5-10. 

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Under some mild conditions on a unital ring R, we show that every additive map δ from R into itself satisfies δ(S。T)=δ(S)。T+S。δ(T) for any S,T∈R with ST=P if and only if δ is a Jordan derivation, where S。T=ST+TS is the Jordan product and P is a nontrivial idempotent of ring R.

Judgement for the property (ω′) and the generalized property (ω′)

WANG Gai-ling, CAO Xiao-hong*

J4. 2012, 47(4):  11-16. 

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The property （ω′） and the generalized property（ω′） are variants of Weyl′s theorem. The sufficient and necessary conditions are established for which the property （ω′） and the generalized property（ω′）for T and perturbations by power finite rank operators commuting with T by means of a new spectrum defined on a bounded linear operator  T acting on a Hilbert space by the essential spectrum, singlevalued extension property and the operator of consistence in Fredholm and index.

Generalized Kato type and generalized property (ω′)

LIU Ai-fang, LI Na-na, CAO Xiao-hong*

J4. 2012, 47(4):  17-23. 

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The definition of the generalized Kato type is given.  New spectra are defined in view of the generalized Kato property and the property of consistence in Fredholm and index.  The sufficient and necessary conditions  are established for a bounded linear operator defined on a Hilbert space for which the generalized property (ω′) holds. Also, the stability of a generalized property (ω′) is discussed. In addition, as a consequence of the main result, the generalized property (ω′) and its perturbation for H(p) operators are studied.

Judgement for the stability of Weyl′s theorem

SHI Wei-juan, CAO Xiao-hong*

J4. 2012, 47(4):  24-27. 

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An operator T is said to satisfy Browder′s theorem if σ(T)＼σw(T)π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum respectively, and π00(T)=｛λ∈iso σ(T),0<dimN(T-λI)<∞｝. If σ(T)＼σw(T)=π00(T), T satisfies Weyl′s theorem.  Using the characteristics of semi-Fredholm domain, the stability of Browder′s theorem and Weyl′s theorem under compact perturbations are investigated, and  those operators which have this stability are characterized.

The disjointness and invariant problems of frames on Hilbert K-modules

DONG Fang-fang

J4. 2012, 47(4):  33-36. 

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 The frames of modules on compact operator algebras which are called Kmodules are considered. The theories of function analysis and operator algebras are used and the equivalent conditions of disjointness of frames on Hilbert Kmodule are given and proved. Finally, Two important conclusions about invariant problems on Hilbert C*-subalgebras are obtained by the index theories of C*-subalgebras When Hilbert C*-subalgebras are considered as a Hilbert K-module on itself.

Generalized *-Lie derivable mappings

ZHANG Fang-juan

J4. 2012, 47(4):  37-41. 

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Closedness of ranges of upper triangular operator matrix

DU Gui-chun1,2, SHAO Chun-fang1

J4. 2012, 47(4):  42-46. 

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Solutions to the operator equation AXB*-BX*A*=C

XU Jun-lian1，2

J4. 2012, 47(4):  47-52. 

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The reverse order law for ｛1,3,4｝-inverse of the product of two operators

DUAN Ying-tao

J4. 2012, 47(4):  53-56. 

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Wavelet preservers on the Hilbert space L²(R)

YIN Jun-cheng1,2, CAO Huai-xin1

J4. 2012, 47(4):  57-61. 

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The operational properties of mother wavelets in the Hilbert space L²(R) and operators that preserve mother wavelets are discussed. Denoted by W(L²(R)) the set of all mother wavelets in L²(R), it is proved that GW(L²(R)):=W(L²(R))∪｛0｝  becomes an abelian normed algebra with the usual addition, scalar multiplication, convolution and the norm ‖·‖1. A bounded linear operator A on L2(R) is said to be a wavelet preserver if it maps W(L²(R)) into W(L²(R)). It is proved that the set WP(L²(R)) of all wavelet preservers is a unital multiplicative semigroup. Furthermore, an operator in B(L²(R)) is said to be a generalized wavelet preserver (GWP) if it maps GW(L²(R)) into GW(L²(R)). It is shown that the set GWP(L²(R)) of all GWPs is a unital normed subalgebra of the Banach algebra B(L²(R)). Finally, a sufficient condition for an operator to be a wavelet preserver is obtained.

Weak boundedness of Littlewood-Paley operators on  weighted Herz spaces

ZHAO Kai, SUN Xiao-hua, DU Hong-bin

J4. 2012, 47(4):  62-65. 

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By the decompositons of weighted Herz spaces, using the properties of weighted functions and the estimates of some inequalities, the boundedness of Littlewood-Paley g functions from weighted Herz spaces to weighted weak Herz spaces is obtained. This result enriches the theory of LittlewoodPaley functions.

An integral-type operator from the Dirichlet  space to the Bloch-type space

MA Xin-guang

J4. 2012, 47(4):  66-69. 

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The boundedness and compactness of operators is understood by calculating the norm and essential norm of operators. The norm and essential norm of an integral-type operator which maps Dirichlet space into Bloch -type space in the unit ball are discussed.

(A,B)-quantum measurements

WANG Wen-hua, CAO Huai-xin*, LI Wei

J4. 2012, 47(4):  70-76. 

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The concept of a (A,B)-quantum measurement is introduced and discussed. An equivalent relation between (A,B)-quantum measurement and g-frame with lower and upper bounds A and B is established. Some properties of direct sums and multiplications of (A,B)-quantum measurements are obtained. The measure frame operator and the dual quantum measurement of a (A,B)-quantum measurement are introduced and its canonical dual is constructed. Finally, a reconstruction formula of a quantum states is obtained in light of a (A,B)-quantum measurement.

Essential commutativity of composition operator and integral operator

ZHANG Liang, ZENG Hong-gang*

J4. 2012, 47(4):  77-79. 

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Essential commutativity of the operators is the important component part in operator fields. Generally, multiplication between operators is not essential commutative for the integral operators and composition operators between different spaces. Sufficient and necessary conditions of essential commutativity of integral operators and composition operators from F(p, q, s) space to weighted Bloch space are given.

Lipschitz estimates for commutators of oscillatory integral operators

DU Hong-shan, ZHANG Lei*

J4. 2012, 47(4):  80-83. 

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The boundedness of commutators formed by oscillatory integral operators and Lipschitz functions is established. A Lipschitz characterization of commutators of oscillatory integral operators with smooth C-Z Kernel is given. Also, the boundedness of commutators formed by oscillatory integral operators with standard C-Z Kernel and Lipschitz functions is obtained.

The approximation of definite integration by using Haar wavelet and operator matrix

GENG Wan-hai, CHEN Yi-ming, LIU Yu-feng, WANG Xiao-juan

J4. 2012, 47(4):  84-88. 

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The corresponding definite integral problem is transformed into calculate product of the constant matrix by combining Haar wavelet with operator matrix and the integrand is discrete felicitously. It is easy to achieve the product of constant matrix with MATLAB. The numerical examples show that the method is effective.

The representations of the Drazin inverse of differences of two elemen ts in Banach algebra

ZHANG Miao1, LIU Xiao-ji1,2*

J4. 2012, 47(4):  89-93. 

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By using the block matrix forms of the elements in Banach algebra, expressions of the Drazin inverse of difference of two elements are given. And then the related results  are extended.

On a generalization of (P)-covers of cyclic acts

ZHAO Mei-mei, QIAO Hu-sheng*

J4. 2012, 47(4):  94-96. 

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Let S be a monoid and I an ideal of S. Using the ideal I the condition (PI) is defined. The necessary and sufficient condition of cyclic act satisfying condition (PI) is given and the monoids over which every cyclic act has a (PI)-cover are investigated. If the ideal I equals to S, the condition (PI) and condition (P) are equivalent, so the main results in the reference are generalized.

A generalization of Macula′s disjunct matrices

YUAN Jun-xia1,2, ZHOU Hou-chun1*, XU Juan1

J4. 2012, 47(4):  97-100. 

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 A family of disjunct matrices based on the matchings on complete subgraphs of a multipartite complete graph is constructed, and it is proved that the method has good error-tolerant property and better row to column ratio.

A note on quasi-elastic spaces

CAO Zhen-bo, TAN An-hui, XU Yu-ming

J4. 2012, 47(4):  101-103. 

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The definitions of quasi-elastic spaces and unitary point extendable quasi pair-base are given. We prove that quasi-elastic spaces are elastic spaces, and spaces with unitary point extendable quasi pair-base are equivalent to proto-metrizable spaces.

Researches on some properties of L-fuzzy quantale

LIANG Shao-hui

J4. 2012, 47(4):  104-109. 

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The definition of L-fuzzy frame is generalized to L-fuzzy quantale, and some equivalent charaterizations of L-fuzzy quantale are given. The concept of L-filter and L-filtered quantale are introduced. It is proved that every L-filter can generate an L-filtered quantale, and there is an honest functor from L-FilQuant to L-FQuant. Finally, the concept of L-fuzzy ideal and L-fuzzy prime ideal are introduced, and their properties are discussed.

Filter topological spaces on R0-algebras

ZHOU Hong-jun

J4. 2012, 47(4):  110-115. 

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A topological space (M, TM) with the set of all MP-filters as its basis on an R0-algebra M is introduced. Characterizations of differentiate, closure and interior operators in (M, TM) are obtained. It is proved that (M, TM) is connected and covering-compact and satisfies the first separation axiom, and it satisfies the second separation axiom if and only if the set of all principle filters is countable. However, (M, TM) is neither T1 nor regular. It is also proved that (M, TM) is T0 if and only if the R0-algebra M reduces to a Boolean algebra. Finally, product spaces on product R0-algebras are investigated.

Sparse principal component analysis for symmetric matrix and  application in sufficient dimension reduction

SHAO Wei1, ZHU Li-ping2, LIU Fu-Guo2, WANG Qiu-Ping2

J4. 2012, 47(4):  116-120. 

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 Sparse principal component analysis(SPC) for symmetric matrix and application are discussed. Asymptotic properties are obtained. Monte Carlo based simulations are used to illustrate the efficacy of the new method.

The least-squares solutions and the optimal approximation of the inverse problem for row anti-symmetric matrices on linear manifolds

LIANG Mao-lin, DAI Li-fang, YANG Xiao-ya

J4. 2012, 47(4):  121-126. 

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The least-squares row anti-symmetric solutions of matrix equations AX=B, XC=D on linear manifolds are obtained by using the singular value decomposition. Also, for a given matrix , the unique optimal approximation solution in the least-squares solutions set is derived.