ADDITIVE MAPS ON SOME OPERATOR ALGEBRAS BEHAVING LIKE(α, β)-DERIVATIONS OR GENERALIZED(α, β)-DERIVATIONS AT ZERO-PRODUCT ELEMENTS

Let A be a subalgebra of B(X) containing the identity operator I and an idempotent P. Suppose that α, β : A → A are ring epimorphisms and there exists some nest N on X such that α(P)(X) and β(P)(X) are non-trivial elements of N. Let A contain all rank one operators in AlgN and δ : A → B(X) be an additive mapping. It is shown that, if δ is(α, β)-derivable at zero point, then there exists an additive(α, β)-derivation τ : A → B(X)such that δ(A) = τ(A) + α(A)δ(I) for all A ∈ A. It is also shown that if δ is generalized(α, β)-derivable at zero point, then δ is an additive generalized(α, β)-derivation. Moreover,by use of this result, the additive maps(generalized)(α, β)-derivable at zero point on several nest algebras, are also characterized.     

Characterization of derivations on B(X) by local actions

Let A be a unital algebra and M be a unital A-bimodule. A linear map δ : A →M is said to be Jordan derivable at a nontrivial idempotent P ∈ A if δ(A) ? B + A ? δ(B) =δ(A ? B) for any A, B ∈ A with A ? B = P, here A ? B = AB + BA is the usual Jordan product. In this article, we show that if A = Alg N is a Hilbert space nest algebra and M = B(H), or A = M = B(X), then, a linear map δ : A → M is Jordan derivable at a nontrivial projection P ∈ N or an arbitrary but fixed nontrivial idempotent P ∈ B(X) if and only if it is a derivation. New equivalent characterization of derivations on these operator algebras was obtained.     

PERTURBATION OF A_0 ~+h

Let Z be a topological space and mapping A2 :Z→B(H) with closed range R(A2 ) be continuous . Some necessary and sufficient conditions of the continuity of M-P inverses A z+ are given in [1], [2]. It is one of them that AZ+ is continuous if ana only if AZ+ is locallybounded. In this paper, we discuss the following problem: if limA n = A0 in B(H) and ||An+||is unbounded (i.e. the above necessary and sufficient condition fails), what h in H will make the equations: limAm+ h = A0+ h or w-limAn+ h= A 0+ h be true. For this purpose three theorems and an error estimation are given in this paper.     

Geometry of 2×2 Hermitian matrices over any division ring

Let D be a division ring with an involution-,H2(D) be the set of 2 × 2 Hermitian matrices over D. Let ad(A,B) = rank(A-B) be the arithmetic distance between A,B ∈ H2(D) . In this paper,the fundamental theorem of the geometry of 2 × 2 Hermitian matrices over D(char(D) = 2) is proved:if  :H2(D) → H2(D) is the adjacency preserving bijective map,then  is of the form (X) = tP XσP +(0) ,where P ∈ GL2(D) ,σ is a quasi-automorphism of D. The quasi-automorphism of D is studied,and further results are obtained.     

C*-Isomorphisms Associated with Two Projections on a Hilbert C*-Module*

Motivated by two norm equations used to characterize the Friedrichs angle, this paper studies C^*-isomorphisms associated with two projections by introducing the matched triple and the semi-harmonious pair of projections. A triple (P, Q, H) is said to be matched if H is a Hilbert C^*-module, P and Q are projections on H such that their infimum P ∧ Q exists as an element of L(H), where L(H) denotes the set of all adjointable operators on H. The C^*-subalgebras of L(H) generated by elements in {P - P ∧ Q, Q - P ∧ Q, I} and {P, Q, P ∧ Q, I} are denoted by i(P, Q, H) and o(P, Q, H), respectively. It is proved that each faithful representation (π, X) of o(P, Q, H) can induce a faithful representation (π, X e) of i(P, Q, H) such that e π(P - P ∧ Q) = π(P) - π(P) ∧ π(Q),eπ(Q - P ∧ Q) = π(Q) - π(P) ∧ π(Q).When (P, Q) is semi-harmonious, that is, R(P + Q) and R(2I - P - Q) are both orthogonally complemented in H, it is shown that i(P, Q, H) and i(I - Q, I - P, H) are unitarily equivalent via a unitary operator in L(H). A counterexample is constructed, which shows that the same may be not true when (P, Q) fails to be semi-harmonious. Likewise, a counterexample is constructed such that (P, Q) is semi-harmonious, whereas (P, I - Q) is not semi-harmonious. Some additional examples indicating new phenomena of adjointable operators acting on Hilbert C^*-modules are also provided.     

Additive Maps Preserving the Star Partial Order on B(H)

Let B(H) be the C*-algebra of all bounded linear operators on a complex Hilbert space H. It is proved that an additive surjective map φ on B(H) preserving the star partial order in both directions if and only if one of the following assertions holds.(1) There exist a nonzero complex number α and two unitary operators U and V on H such that φ(X) = αUXV or φ(X) = αUX*V for all X ∈ B(H).(2)There exist a nonzero α and two anti-unitary operators U and V on H such thatφ(X) = αUXV or φ(X) = αUX*V for all X ∈ B(H).     

Finite groups whose minimal subgroups are weakly H-subgroups

Let G be a finite group.A subgroup H of G is called an H-subgroup in G if NG(H) ∩Hg≤H for all g∈G.A subgroup H of G is called a weakly H-subgroup in G if there exists a normal subgroup K of G such that G=HK and H∩K is an H-subgroup in G.In this paper,we investigate the structure of the finite group G under the assumption that every subgroup of G of prime order or of order 4 is a weakly H-subgroup in G.Our results improve and generalize several recent results in the literature.     

有限群的X-s-置换子群

Let X be a nonempty subset of a group G.A subgroup H of G is said to be X-s-permutable in G if there exists an element x ∈ X such that HPx = PxH for every Sylow subgroup P of G.In this paper,some new results are given under the assumption that some suited subgroups of G are X-s-permutable in G.     

K-POTENT PRESERVING LINEAR MAPS

Let B(X) be the Banach algebra of all bounded linear operators on a complex Banach space X. Let k ≥ 2 be an integer and φ a weakly continuous linear surjective map from B(X) into itself. It is shown that φ is k-potent preserving if and only if it is k-th-power preserving, and in turn, if and only if it is either an automorphism or an antiautomorphism on B(X) multiplied by a complex number λ satisfying λk-1= 1. Let A be a von Neumann algebra and B be a Banach algebra, it is also shown that a bounded surjective linear map from A onto B is k-potent preserving if and only if it is a Jordan homomorphism multiplied by an invertible element with (k - l)-th power I.     

主理想整环上n阶矩阵环中的Goldbach问题

Abstract: In this paper, we consider the Goldbach's problem for matrix rings, namely, we decompose an n ×n (n > 1) matrix over a principal ideal domain R into a sum of two matrices in Mn(R) with given determinants. We prove the following result: Let n > 1 be a natural number and A = (αij) be a matrix in Mn(R). Define d(A) := g.c.d{αij}. Suppose that p and q are two elements in R. Then (1) If n > 1 is even, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p-q; (2) If n > 1 is odd, then A can be written as a sum of two matrices X, Y in Mn(R) with det(X) = p and det(Y) = q if and only if d(A) |p + q. We apply the result to the matrices in Mn(Z) and Mn(Q[x]) and prove that if R = Z or Q[x], then any nonzero matrix A in Mn(R) can be written as a sum of two matrices in Mn(R) with prime determinants.     

An algebraic invariant for Jordan automorphisms on В(H): The set of idempotents

Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempo-tents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ {-1,1,2,3,1/2,1/3} and A, B ∈ B(H), A - λB ∈ I(H) (?) Φ(A) - λΦ(B) ∈ I(H), then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT-1 for all A ∈ B(H), or Φ(A) = TA*T-1 for all A ∈ B(H); if, in addition, A-iB ∈ I(H) (?) Φ(A) -ιΦ(B) ∈ I(H), here ι is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.     

Additive preservers of Drazin invertible operators with bounded index

Let B(X) be the algebra of all bounded linear operators on an infinite-dimensional complex or real Banach space X. Given an integer n ≥ 1, we show that an additive surjective map Φ on B(X)preserves Drazin invertible operators of index non-greater than n in both directions if and only if Φ is either of the form Φ(T) = αATA~(-1) or of the form Φ(T) = αBT~*B~(-1) where α is a non-zero scalar,A:X → X and B:X~*→ X are two bounded invertible linear or conjugate linear operators.     

On solubility and supersolubility of some classes of finite groups

Let G be a finite group and H a subgroup of G. Then H is said to be S-permutable in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-permutable in G. Then we say that H is S-embedded in G if G has a normal subgroup T and an S-permutable subgroup C such that T ∩ H HsG and HT = C. Our main result is the following Theorem A. A group G is supersoluble if and only if for every non-cyclic Sylow subgroup P of the generalized Fitting subgrou...     

矩阵方程的最小二乘{P,Q,k+1}-自反解（英文）

Let P∈C^m×m and Q∈C^n×n be Hermitian and {k+1}-potent matrices,i.e.,P^k+1=P=P^*,Q^k+1=Q=Q^*,where(·)^* stands for the conjugate transpose of a matrix.A matrix X∈C^m×n is called {P,Q,k+1}-reflexive(anti-reflexive) if PXQ=X(PXQ=-X).In this paper,the least squares solution of the matrix equation AXB=C subject to {P,Q,k+1}-reflexive and anti-reflexive constraints are studied by converting into two simpler cases:k=1 and k...     

Some Conditions for K_0(B(X))=0

Let A be an unital Banach algebra with the unital element I. We denote the set of (n×n)-matrices over A by Mn(A). The set of idempotent elements of A is denoted by P1(A):={P∈A: P2=P}, and accordingly the set of idempotent elements of Mn(A) is denoted by Pn(A):=P1(Mn(A)). We define P∞(A) :=∪n∈N Pn(A). In the set of P∞(A), for P,Q ∈ Pn(A), we say that P is (algebraic) equivalent to Q, if there exist A,B∈ Mn(A) such that     

Zeros of Monomial Brauer Characters

Let G be a finite group and p be a fixed prime. A p-Brauer character of G is said to be monomial if it is induced from a linear p-Brauer character of some subgroup(not necessarily proper) of G. Denote by IBr_m(G) the set of irreducible monomial p-Brauer′characters of G. Let H = G′O~p′(G) be the smallest normal subgroup such that G/H is an abelian p′-group. Suppose that g ∈ G is a p-regular element and the order of gH in the factor group G/H does not divide |IBr_m(G)|. Then there exists ? ∈ IBr_m(G) such that ?(g) = 0.     

二项式展开的抵消系数以及$q$-摸拟

Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A(n,k;A) are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n k=0 A(n,k;A)(x - Ak)n-k = xn. (0.1) Similary, we define B(n,k;A) to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n k=0 B(n,k;A)(x - Ak)k = xn. (0.2) The main result of this paper is an explicit formula for B(n,k;A), which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations (0.1) and (0.2).     

The spectrum for overlarge sets of hybrid triple systems

A hybrid triple system of order v and index λ,denoted by HTS(v,λ),is a pair(X,B) where X is a v-set and B is a collection of cyclic triples and transitive triples on X,such that every ordered pair of X belongs to λ triples of B. An overlarge set of disjoint HTS(v,λ),denoted by OLHTS(v,λ),is a collection {(Y \{y},Ai)}i,such that Y is a(v+1)-set,each(Y \{y},Ai) is an HTS(v,λ) and all Ais form a partition of all cyclic triples and transitive triples on Y.In this paper,we shall discuss the existence problem of OLHTS(v,λ) and give the following conclusion: there exists an OLHTS(v,λ) if and only if λ=1,2,4,v ≡ 0,1(mod 3) and v≥4.     

New characterizations of finite supersoluble groups

Let A be a subgroup of a group G and X a nonempty subset of G.A is called an X- semipermutable subgroup of G if A has a supplement T in G such that for every subgroup T_1 of T there exists an clement x∈X such that AT_i~x=T_i~xA.On the basis of this concept we obtain some new characterizations of finite supersoluble groups.     

Ranks of the common solution to six quaternion matrix equations

A new expression is established for the common solution to six classical linear quaternion matrix equations A 1 X = C 1 , X B 1 = C 3 , A 2 X = C 2 , X B 2 = C 4 , A 3 X B 3 = C 5 , A 4 X B 4 = C 6 which was investigated recently by Wang, Chang and Ning (Q. Wang, H. Chang, Q. Ning, The common solution to six quaternion matrix equations with applications, Appl. Math. Comput. 195: 721-732 (2008)). Formulas are derived for the maximal and minimal ranks of the common solution to this system. Moreover, corresponding results on some special cases are presented. As an application, a necessary and sufficient condition is presented for the invariance of the rank of the general solution to this system. Some known results can be regarded as the special cases of the results in this paper.