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

The relationship between {1, 3, 4}-inverses of AB and the product of {1, 3, 4}-inverses of A and B have been studied in this paper. The necessary and sufficient conditions for B{1,3,4}A{1,3,4}⊆(AB){1,3,4}, B{1,3,4}A{1,3,4}⊇(AB){1,3,4} and B{1,3,4}A{1,3,4}=(AB){1,3,4} are given.     

The intersection of left (right) spectra of 2 × 2 upper triangular operator matrices

When A∈B(H) and B∈B(K) are given, we denote by M_C the operator matrix acting on the infinite-dimensional separable Hilbert space H⊕K of the form In this paper, for given A and B, the sets and ?_C∈Inv(K,H)σ_l(M_C) are determined, where σ_l(T),B_l(K,H) and Inv(K,H) denote, respectively, the left spectrum of an operator T, the set of all the left invertible operators and the set of all the invertible operators from K into H.     

An elementary operator with log-hyponormal, p-hyponormal entries

A Hilbert space operator A∈B(H) is p-hyponormal, A∈(p-H), if |A^∗|^2p?|A|^2p; an invertible operator A∈B(H) is log-hyponormal, A∈(?-H), if log(TT^∗)?log(T^∗T). Let d_AB=δ_AB or ?_AB, where δ_AB∈B(B(H)) is the generalised derivation δ_AB(X)=AX-XB and ?_AB∈B(B(H)) is the elementary operator ?_AB(X)=AXB-X. It is proved that if A,B^∗∈(?-H)∪(p-H), then, for all complex λ, , the ascent of (d_AB-λ)?1, and d_AB satisfies the range-kernel orthogonality inequality ‖X‖?‖X-(d_AB-λ)Y‖ for all X∈(d_AB-λ)^-1(0) and Y∈B(H). Furthermore, isolated points of σ(d_AB) are simple poles of the resolvent of d_AB. A version of the elementary operator E(X)=A₁XA₂-B₁XB₂ and perturbations of d_AB by quasi-nilpotent operators are considered, and Weyl’s theorem is proved for d_AB.     

Elementary operators on self-adjoint operators

Let H be a Hilbert space and let A and B be standard ∗-operator algebras on H. Denote by As and Bs the set of all self-adjoint operators in A and B, respectively. Assume that and are surjective maps such that M(AM^∗(B)A)=M(A)BM(A) and M^∗(BM(A)B)=M^∗(B)AM^∗(B) for every pair A∈As, B∈Bs. Then there exist an invertible bounded linear or conjugate-linear operator and a constant c∈{−1,1} such that M(A)=cTAT^∗, A∈As, and M^∗(B)=cT^∗BT, B∈Bs.     

A localization inequality for set functions

We prove the following theorem, which is an analog for discrete set functions of a geometric result of Lovász and Simonovits. Given two real-valued set functions f₁,f₂ defined on the subsets of a finite set S, satisfying for i∈{1,2}, there exists a positive multiplicative set function μ over S and two subsets A,B⊆S such that for i∈{1,2}μ(A)f_i(A)+μ(B)f_i(B)+μ(A∪B)f_i(A∪B)+μ(A∩B)f_i(A∩B)?0. The Ahlswede-Daykin four function theorem can be deduced easily from this.     

Generalized composition operators from F(p, q, s) spaces to Bloch-type spaces

Let H(B) denote the space of all holomorphic functions on the unit ball B of Cⁿ. Let φ be a holomorphic self-map of B and g ∈ H(B) such that g(0) = 0. In this paper, we investigate the boundedness and compactness of the generalized composition operator

     

Groupoid normalizers of tensor products

We consider an inclusion B⊆M of finite von Neumann algebras satisfying B^′∩M⊆B. A partial isometry v∈M is called a groupoid normalizer if vBv^∗,v^∗Bv⊆B. Given two such inclusions Bi⊆Mi, i=1,2, we find approximations to the groupoid normalizers of in , from which we deduce that the von Neumann algebra generated by the groupoid normalizers of the tensor product is equal to the tensor product of the von Neumann algebras generated by the groupoid normalizers. Examples are given to show that this can fail without the hypothesis , i=1,2. We also prove a parallel result where the groupoid normalizers are replaced by the intertwiners, those partial isometries v∈M satisfying vBv^∗⊆B and v^∗v,vv^∗∈B.     

A measure-theoretic proof of Turing incomparability

We prove that if S is an ω-model of weak weak König’s lemma and , is incomputable, then there exists , such that A and B are Turing incomparable. This extends a recent result of Ku?era and Slaman who proved that if S₀ is a Scott set (i.e. an ω-model of weak König’s lemma) and A∈S₀, A⊆ω, is incomputable, then there exists B∈S₀, B⊆ω, such that A and B are Turing incomparable.     

Trotter-Kato product formula and fractional powers of self-adjoint generators

Let A and B be non-negative self-adjoint operators in a Hilbert space such that their densely defined form sum obeys dom(H^α)⊆dom(A^α)∩dom(B^α) for some α∈(1/2,1). It is proved that if, in addition, A and B satisfy dom(A^1/2)⊆dom(B^1/2), then the symmetric and non-symmetric Trotter-Kato product formula converges in the operator norm:

||(e−tB/2ne−tA/ne−tB/2n)n−e−tH||=O(n−(2α−1))||(e−tA/ne−tB/n)n−e−tH||=O(n−(2α−1))     

Super-stationary set, subword problem and the complexity

Let Ω⊂{0,1}^N be a nonempty closed set with N={0,1,2,…}. For N={N₀<N₁<N₂<?}⊂N and ω∈{0,1}^N, define ω[N]∈{0,1}^N by and

     

A bilinear version of Orlicz-Pettis theorem

Given three Banach spaces X, Y and Z and a bounded bilinear map , a sequence x=n(xn)⊆X is called B-absolutely summable if is finite for any y∈Y. Connections of this space with are presented. A sequence x=n(xn)⊆X is called B-unconditionally summable if is finite for any y∈Y and z^∗∈Z^∗ and for any M⊆N there exists xM∈X for which _∑n∈M〈B(xn,y),z^∗〉=〈B(xM,y),z^∗〉 for all y∈Y and z^∗∈Z^∗. A bilinear version of Orlicz-Pettis theorem is given in this setting and some applications are presented.     

Vertex Turán problems in the hypercube

Let Qn be the n-dimensional hypercube: the graph with vertex set n{0,1} and edges between vertices that differ in exactly one coordinate. For 1?d?n and F⊆d{0,1} we say that S⊆n{0,1} is F-free if every embedding satisfies i(F)?S. We consider the question of how large S⊆n{0,1} can be if it is F-free. In particular we generalise the main prior result in this area, for F=²{0,1}, due to E.A. Kostochka and prove a local stability result for the structure of near-extremal sets.We also show that the density required to guarantee an embedded copy of at least one of a family of forbidden configurations may be significantly lower than that required to ensure an embedded copy of any individual member of the family.Finally we show that any subset of the n-dimensional hypercube of positive density will contain exponentially many points from some embedded d-dimensional subcube if n is sufficiently large.     

The least squares g-inverses for sum of matrices

In this article, we exhibit under suitable conditions a neat relationship between the least squares g-inverse for a sum of two matrices and the least squares g-inverses of the individual terms. We give a necessary and sufficient condition for the set equations (A?+?B){1,?3}?=?A{1,?3}?+?B{1,?3} and (A?+?B){1,?4}?=?A{1,?4}?+?B{1,?4}.     

On the exponential sum—product problem

Let g be an element of order T over a finite field Fp of p elements, where p is a prime. We show that for a very wide class of sets A, B ∈ {1, . . . , T} at least one of the sets

{gab:a∈A,b∈B}and{ga+gb:a∈A,b∈B}     

New families of graphs without short cycles and large size

We denote by ex(n;{C₃,C₄,…,C_s}) or f_s(n) the maximum number of edges in a graph of order n and girth at least s+1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g−1 on fewer vertices. For an infinite sequence of values of n and s∈{4,6,10} the obtained graphs are denser than the known constructions of graphs of the same girth s+1. We also give another different construction of dense graphs for an infinite sequence of values of n and s∈{7,11}. These two methods improve the known lower bounds on f_s(n) for s∈{4,6,7,10,11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that for s∈{5,7,11}, and for s∈{6,10}.     

The sum of unitary similarity orbits containing only special operators

Let B(H) be the algebra of bounded linear operator acting on a Hilbert space H (over the complex or real field). Characterization is given to A₁,…,A_k∈B(H) such that for any unitary operators is always in a special class S of operators such as normal operators, self-adjoint operators, unitary operators. As corollaries, characterizations are given to A∈B(H) such that complex, real or nonnegative linear combinations of operators in its unitary orbit U(A)={U^∗AU:Uunitary} always lie in S.     

Finite depth and Jacobson-Bourbaki correspondence

We introduce a notion of depth three tower C⊆B⊆A with depth two ring extension A|B being the case B=C. If and B|C is a Frobenius extension with A|B|C depth three, then A|C is depth two. If A, B and C correspond to a tower G>H>K via group algebras over a base ring F, the depth three condition is the condition that K has normal closure K^G contained in H. For a depth three tower of rings, a pre-Galois theory for the ring and coring (A⊗_BA)^C involving Morita context bimodules and left coideal subrings is applied to specialize a Jacobson-Bourbaki correspondence theorem for augmented rings to depth two extensions with depth three intermediate division rings.     

Density of sets of natural numbers and the Lévy group

Let N denote the set of positive integers. The asymptotic density of the set A⊆N is d(A)=limn→∞|A∩[1,n]|/n, if this limit exists. Let AD denote the set of all sets of positive integers that have asymptotic density, and let SN denote the set of all permutations of the positive integers N. The group L^? consists of all permutations f∈SN such that A∈AD if and only if f(A)∈AD, and the group L^* consists of all permutations f∈L^? such that d(f(A))=d(A) for all A∈AD. Let be a one-to-one function such that d(f(N))=1 and, if A∈AD, then f(A)∈AD. It is proved that f must also preserve density, that is, d(f(A))=d(A) for all A∈AD. Thus, the groups L^? and L^* coincide.