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1.
We show that for a linear space of operators M ? B(H1, H2) the following assertions are equivalent. (i) M is reflexive in the sense of Loginov-Shulman. (ii) There exists an order-preserving map Ψ = (ψ1, ψ2) on a bilattice Bil(M) of subspaces determined by M with P ≤ ψ1(P,Q) and Q ≤ ψ2(P,Q) for any pair (P,Q) ∈ Bil(M), and such that an operator TB(H1, H2) lies in M if and only if ψ2(P,Q)Tψ1(P,Q) = 0 for all (P,Q) ∈ Bil(M). This extends the Erdos-Power type characterization of weakly closed bimodules over a nest algebra to reflexive spaces.  相似文献   

2.
Let ? be a trace on the unital C*-algebra A and M ? be the ideal of the definition of the trace ?. We obtain a C*analogue of the quantum Hall effect: if P,QA are idempotents and P ? QM ? , then ?((P ? Q)2n+1) = ?(P ? Q) ∈ R for all nN. Let the isometries UA and A = A*∈ A be such that I+A is invertible and U-AM ? with ?(U-A) ∈ R. Then I-A, I?UM ? and ?(I?U) ∈ R. Let nN, dimH = 2n + 1, the symmetry operators U, VB(H), and W = U ? V. Then the operator W is not a symmetry, and if V = V*, then the operator W is nonunitary.  相似文献   

3.
For an n×n complex matrix A with ind(A) = r; let AD and Aπ = IAAD be respectively the Drazin inverse and the eigenprojection corresponding to the eigenvalue 0 of A: For an n×n complex singular matrix B with ind(B) = s, it is said to be a stable perturbation of A, if I–(BπAπ)2 is nonsingular, equivalently, if the matrix B satisfies the condition \(\mathcal{R}(B^s)\cap\mathcal{N}(A^r)=\left\{0\right\}\) and \(\mathcal{N}(B^s)\cap\mathcal{R}(A^r)=\left\{0\right\}\), introduced by Castro-González, Robles, and Vélez-Cerrada. In this paper, we call B an acute perturbation of A with respect to the Drazin inverse if the spectral radius ρ(BπAπ) < 1: We present a perturbation analysis and give suffcient and necessary conditions for a perturbation of a square matrix being acute with respect to the matrix Drazin inverse. Also, we generalize our perturbation analysis to oblique projectors. In our analysis, the spectral radius, instead of the usual spectral norm, is used. Our results include the previous results on the Drazin inverse and the group inverse as special cases and are consistent with the previous work on the spectral projections and the Moore-Penrose inverse.  相似文献   

4.
We prove that for any \({A,B\in\mathbb{R}^{n\times n}}\) such that each matrix S satisfying min(A, B) ≤ S ≤ max(A, B) is nonsingular, all four matrices A ?1 B, AB ?1, B ?1 A and BA ?1 are P-matrices. A practical method for generating P-matrices is drawn from this result.  相似文献   

5.
Let τ be a faithful normal semifinite trace on a von Neumann algebra M, let p, 0 < p < ∞, be a number, and let Lp(M, τ) be the space of operators whose pth power is integrable (with respect to τ). Let P and Q be τ-measurable idempotents, and let AP ? Q. In this case, 1) if A ≥ 0, then A is a projection and QA = AQ = 0; 2) if P is quasinormal, then P is a projection; 3) if QM and ALp(M, τ), then A2Lp(M, τ). Let n be a positive integer, n > 2, and A = AnM. In this case, 1) if A ≠ 0, then the values of the nonincreasing rearrangement μt(A) belong to the set {0} ∪ [‖An?2?1, ‖A‖] for all t > 0; 2) either μt(A) ≥ 1 for all t > 0 or there is a t0 > 0 such that μt(A) = 0 for all t > t0. For every τ-measurable idempotent Q, there is aunique rank projection PM with QP = P, PQ = Q, and PM = QM. There is a unique decomposition Q = P + Z, where Z2 = 0, ZP = 0, and PZ = Z. Here, if QLp(M, τ), then P is integrable, and τ(Q) = τ(P) for p = 1. If AL1(M, τ) and if A = A3 and A ? A2M, then τ(A) ∈ R.  相似文献   

6.
For an acyclic quiver Q and a finite-dimensional algebra A, we give a unified form of the indecomposable injective objects in the monomorphism category Mon(Q,A) and prove that Mon(Q,A) has enough injective objects.  相似文献   

7.
The cusp was recently shown to admit the structure of a quantum homogeneous space, that is, its coordinate ring B can be embedded as a right coideal subalgebra into a Hopf algebra A such that A is faithfully flat as a B-module. In the present article such a Hopf algebra A is constructed for the coordinate ring B of the nodal cubic, thus further motivating the question which affine varieties are quantum homogeneous spaces.  相似文献   

8.
In this paper LJ-spaces are introduced and studied. They are a common generalization of Lindelöf spaces and J-spaces researched by E. Michael. A space X is called an LJ-space if, whenever {A, B} is a closed cover of X with AB compact, then A or B is Lindelöf. Semi-strong LJ-spaces and strong LJ-spaces are also defined and investigated. It is demonstrated that the three spaces are different and have interesting properties and behaviors.  相似文献   

9.
A classical tensor product \({A \otimes B}\) of complete lattices A and B, consisting of all down-sets in \({A \times B}\) that are join-closed in either coordinate, is isomorphic to the complete lattice Gal(A,B) of Galois maps from A to B, turning arbitrary joins into meets. We introduce more general kinds of tensor products for closure spaces and for posets. They have the expected universal property for bimorphisms (separately continuous maps or maps preserving restricted joins in the two components) into complete lattices. The appropriate ingredient for quantale constructions is here distributivity at the bottom, a generalization of pseudocomplementedness. We show that the truncated tensor product of a complete lattice B with itself becomes a quantale with the closure of the relation product as multiplication iff B is pseudocomplemented, and that the tensor product has a unit element iff B is atomistic. The pseudocomplemented complete lattices form a semicategory in which the hom-set between two objects is their tensor product. The largest subcategory of that semicategory has as objects the atomic boolean complete lattices, which is equivalent to the category of sets and relations. More general results are obtained for closure spaces and posets.  相似文献   

10.
11.
We study the well-posedness of the third-order degenerate differential equation \(\left( {{P_3}} \right):\alpha {\left( {Mu} \right)^{\prime \prime \prime }}\left( t \right) + {\left( {Mu} \right)^{\prime \prime }}\left( t \right) = \beta Au\left( t \right) + f\left( t \right)\), (t ∈ [0, 2p]) with periodic boundary conditions \(Mu\left( 0 \right) = Mu\left( {2\pi } \right),\;Mu'\left( 0 \right) = Mu'\left( {2\pi } \right),\;Mu''\left( 0 \right) = Mu''\left( {2\pi } \right)\), in periodic Lebesgue–Bochner spaces Lp(T,X), periodic Besov spaces Bp,qs(T,X) and periodic Triebel–Lizorkin spaces Fp,qs(T,X), where A, B and M are closed linear operators on a Banach space X satisfying D(A) \( \cap \)D(B) ? D(M) and α, β, γ ∈ R. Using known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of (P3) in the above three function spaces.  相似文献   

12.
István Tomon 《Order》2016,33(3):537-556
We consider an h-partite version of Dilworth’s theorem with multiple partial orders. Let P be a finite set, and let <1,...,< r be partial orders on P. Let G(P, <1,...,< r ) be the graph whose vertices are the elements of P, and x, yP are joined by an edge if x< i y or y< i x holds for some 1 ≤ ir. We show that if the edge density of G(P, <1, ... , < r ) is strictly larger than 1 ? 1/(2h ? 2) r , then P contains h disjoint sets A 1, ... , A h such that A 1 < j ... < j A h holds for some 1 ≤ jr, and |A 1| = ... = |A h | = Ω(|P|). Also, we show that if the complement of G(P, <) has edge density strictly larger than 1 ? 1/(3h ? 3), then P contains h disjoint sets A 1, ... , A h such that the elements of A i are incomparable with the elements of A j for 1 ≤ i < jh, and |A 1| = ... = |A h | = |P|1?o(1). Finally, we prove that if the edge density of the complement of G(P, <1, <2) is α, then there are disjoint sets A, B ? P such that any element of A is incomparable with any element of B in both <1 and <2, and |A| = |B| > n 1?γ(α), where γ(α) → 0 as α → 1. We provide a few applications of these results in combinatorial geometry, as well.  相似文献   

13.
The article is devoted to the theory of elliptic functions of level n. An elliptic function of level n determines a Hirzebruch genus called an elliptic genus of level n. Elliptic functions of level n are also of interest because they are solutions of the Hirzebruch functional equations. The elliptic function of level 2 is the Jacobi elliptic sine function, which determines the famous Ochanine–Witten genus. It is the exponential of the universal formal group of the form F(u, v) = (u2 ? v2)/(uB(v) ? vB(u)), B(0) = 1. The elliptic function of level 3 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uA(v)2 ? vA(u)2), A(0) = 1, A″(0) = 0. In the present study we show that the elliptic function of level 4 is the exponential of the universal formal group of the form F(u, v) = (u2A(v) ? v2A(u))/(uB(v) ? vB(u)), where A(0) = B(0) = 1 and for B′(0) = A″(0) = 0, A′(0) = A1, and B″(0) = 2B2 the following relation holds: (2B(u) + 3A1u)2 = 4A(u)3 ? (3A12 ? 8B2)u2A(u)2. To prove this result, we express the elliptic function of level 4 in terms of the Weierstrass elliptic functions.  相似文献   

14.
We establish that the reducibility exponent (Differentsial’nye Uravneniya, 2007, vol. 43, no. 2, pp. 191–202) of each linear system
$$\dot x = A(t)x, x \in \mathbb{R}^n , t \geqslant 0$$
, with piecewise continuous bounded coefficient matrix A does not belong to the set of values of σ for which the perturbed system (1A+Q) with an arbitrary piecewise continuous perturbation Q satisfying the condition \(\overline {\lim } _{t \to + \infty } t^{ - 1} \ln \left\| {Q(t)} \right\| \leqslant - \sigma \) is reducible to the original system (1 A ) by some Lyapunov transformation.
  相似文献   

15.
Let μ be a nonnegative Borel measure on R d satisfying that μ(Q) ? l(Q)n for every cube Q ? R n , where l(Q) is the side length of the cube Q and 0 < n ? d.We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function B in the context of non-homogeneous spaces related to the measure μ. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result for certain fractional maximal operator proved in W.Wang, C. Tan, Z. Lou (2012).  相似文献   

16.
The purpose of this note is to show that there exist two Tychonoff spaces X, Y, a subset A of X and a subset B of Y such that A is weakly almost Lindelöf in X and B is weakly almost Lindelöf in Y, but A × B is not weakly almost Lindelöf in X × Y.  相似文献   

17.
We study the Möbius invariant spacesQ p andQ p, 0 of analytic functions. These scales of spaces include BMOA=Q1, VMOA=Q1, 0 and the Dirichlet space=Q0. Using the Bergman metric, we establish decomposition theorems for these spaces. We obtain also a fractional derivative characterization for bothQ p andQ p, 0 .  相似文献   

18.
Let A and B be non-empty subsets of a metric space. As a non-self mapping \({T:A\longrightarrow B}\) does not necessarily have a fixed point, it is of considerable interest to find an element x in A that is as close to Tx in B as possible. In other words, if the fixed point equation Tx = x has no exact solution, then it is contemplated to find an approximate solution x in A such that the error d(x, Tx) is minimum, where d is the distance function. Indeed, best proximity point theorems investigate the existence of such optimal approximate solutions, called best proximity points, to the fixed point equation Tx = x when there is no exact solution. As the distance between any element x in A and its image Tx in B is at least the distance between the sets A and B, a best proximity pair theorem achieves global minimum of d(x, Tx) by stipulating an approximate solution x of the fixed point equation Tx = x to satisfy the condition that d(x, Tx) = d(A, B). The purpose of this article is to establish best proximity point theorems for contractive non-self mappings, yielding global optimal approximate solutions of certain fixed point equations. Besides establishing the existence of best proximity points, iterative algorithms are also furnished to determine such optimal approximate solutions.  相似文献   

19.
We consider (in general noncoercive) mixed problems in a bounded domain D in ? n for a second-order elliptic partial differential operator A(x, ?). It is assumed that the operator is written in divergent form in D, the boundary operator B(x, ?) is the restriction of a linear combination of the function and its derivatives to ?D and the boundary of D is a Lipschitz surface. We separate a closed set Y ? ?D and control the growth of solutions near Y. We prove that the pair (A,B) induces a Fredholm operator L in suitable weighted spaces of Sobolev type, where the weight is a power of the distance to the singular set Y. Finally, we prove the completeness of the root functions associated with L.The article consists of two parts. The first part published in the present paper, is devoted to exposing the theory of the special weighted Sobolev–Slobodetskii? spaces in Lipschitz domains. We obtain theorems on the properties of these spaces; namely, theorems on the interpolation of these spaces, embedding theorems, and theorems about traces. We also study the properties of the weighted spaces defined by some (in general) noncoercive forms.  相似文献   

20.
Let H, A and B be subgroups of a group G. We call the pair (A, B) a θ-pair for H in G if: (i) \({\langle H, A\rangle=G}\) and B = (AH) G ; (ii) if A 1/B is a proper subgroup of A/B and \({{A_1/B \vartriangleleft G/B}}\), then \({G\neq \langle H, A_1\rangle}\). In this paper, we study the θ-pairs for 2-maximal subgroups of a group, which imply a group to be solvable or supersolvable.  相似文献   

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