Operators on the space of vector-valued totally measurable functions

Let L(X,Y) stand for the space of all bounded linear operators between real Banach spaces X and Y, and let Σ be a σ-algebra of sets. A bounded linear operator T from the Banach space B(Σ,X) of X-valued Σ-totally measurable functions to Y is said to be σ-smooth if ‖T(fn)Y_‖→0 whenever a sequence of scalar functions (‖fn(⋅)X_‖) is order convergent to 0 in B(Σ). It is shown that a bounded linear operator is σ-smooth if and only if its representing measure is variationally semi-regular, i.e., as An↓∅ (here stands for the semivariation of m on A∈Σ). As an application, we show that the space Lσs(B(Σ,X),Y) of all σ-smooth operators from B(Σ,X) to Y provided with the strong operator topology is sequentially complete. We derive a Banach-Steinhaus type theorem for σ-smooth operators from B(Σ,X) to Y. Moreover, we characterize countable additivity of measures in terms of continuity of the corresponding operators .     

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.     

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.     

Properties of two-dimensional sets with small sumset

We give tight lower bounds on the cardinality of the sumset of two finite, nonempty subsets A,B⊆R² in terms of the minimum number h₁(A,B) of parallel lines covering each of A and B. We show that, if h₁(A,B)?s and |A|?|B|?2s²−3s+2, then

     

Simultaneous quasidiagonalization of complex matrices

Let

be the complex algebra generated by a pair of n × n Hermitian matrices A, B. A recent result of Watters states that A, B are simultaneously unitarily quasidiagonalizable [i.e., A and B are simultaneously unitarily similar to direct sums C₁⊕…⊕C_t,D₁⊕…⊕D_t for some t, where C_i, D_i are k_i × k_i and k_i?2(1?i?t)] if and only if [p(A, B), A]² and [p(A, B), B]² belong to the center of

for all polynomials p(x, y) in the noncommuting variables x, y. In this paper, we obtain a finite set of conditions which works. In particular we show that if A, B are positive semidefinite, then A, B are simultaneously quasidiagonalizable if (and only if) [A, B]², [A², B]² and [A, B²]² commute with A, B.     

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.     

Turán-Type Results for Complete <Emphasis Type="Italic">h</Emphasis>-Partite Graphs in Comparability and Incomparability Graphs

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, y ∈ P are joined by an edge if x< _i y or y< _i x holds for some 1 ≤ i ≤ r. 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 ≤ j ≤ r, 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 < j ≤ h, 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.     

Spanning multi-paths in hypercubes

A set A of vertices of a hypercube is called balanced if . We prove that for every natural number n there exists a natural number π₁(n) such that for every hypercube Q with dim(Q)?π₁(n) there exists a family of pairwise vertex-disjoint paths P_i between A_i and B_i for i=1,2,…,n with if and only if {A_i,B_i∣i=1,2,…,n} is a balanced set.     

Filtered ends of infinite covers and groups

Let f:A→B be a covering map. We say that A has e filtered ends with respect to f (or B) if, for some filtration {K_n} of B by compact subsets, A−f⁻¹(K_n) “eventually” has e components. The main theorem states that if Y is a (suitable) free H-space, if K<H has infinite index, and if Y has a positive finite number of filtered ends with respect to H?Y, then Y has one filtered end with respect to K?Y. This implies that if G is a finitely generated group and K<H<G are subgroups each having infinite index in the next, then implies that , where is the number of filtered ends of a pair of groups in the sense of Kropholler and Roller.     

An Erd?s-Ko-Rado theorem for partial permutations

Let [n] denote the set of positive integers {1,2,…,n}. An r-partial permutation of [n] is a pair (A,f) where A⊆[n], |A|=r and f:A→[n] is an injective map. A set A of r-partial permutations is intersecting if for any (A,f), (B,g)∈A, there exists x∈A∩B such that f(x)=g(x). We prove that for any intersecting family A of r-partial permutations, we have .It seems rather hard to characterize the case of equality. For 8?r?n-3, we show that equality holds if and only if there exist x₀ and ε₀ such that A consists of all (A,f) for which x₀∈A and f(x₀)=ε₀.     

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.     

Robustness of A-optimal designs

Suppose that Y=(Y_i) is a normal random vector with mean Xb and covariance σ²I_n, where b is a p-dimensional vector (b_j),X=(X_ij) is an n×p matrix. A-optimal designs X are chosen from the traditional set D of A-optimal designs for ρ=0 such that X is still A-optimal in D when the components Y_i are dependent, i.e., for i≠i′, the covariance of Y_i,Y_i^′ is ρ with ρ≠0. Such designs depend on the sign of ρ. The general results are applied to X=(X_ij), where X_ij∈{-1,1}; this corresponds to a factorial design with -1,1 representing low level or high level respectively, or corresponds to a weighing design with -1,1 representing an object j with weight b_j being weighed on the left and right of a chemical balance respectively.     

Haagerup property for C-algebras

We define the Haagerup property for C^?-algebras A and extend this to a notion of relative Haagerup property for the inclusion B⊆A, where B is a C^?-subalgebra of A. Let Γ be a discrete group and Λ a normal subgroup of Γ, we show that the inclusion A_?α,rΛ⊆A_?α,rΓ has the relative Haagerup property if and only if the quotient group Γ/Λ has the Haagerup property. In particular, the inclusion has the relative Haagerup property if and only if Γ/Λ has the Haagerup property; has the Haagerup property if and only if Γ has the Haagerup property. We also characterize the Haagerup property for Γ in terms of its Fourier algebra A(Γ).     

The embeddability ordering of topological spaces

For K a set of topological spaces and X,Y∈K, the notation Xh_⊆Y means that X embeds homeomorphically into Y; and X∼Y means Xh_⊆Yh_⊆X. With , the equivalence relation ∼ on K induces a partial order h_? well-defined on K/∼ as follows: if Xh_⊆Y.For posets (P,P_?) and (Q,Q_?), the notation (P,P_?)?(Q,Q_?) means: there is an injection such that p₀P_?p₁ in P if and only if h(p₀)Q_?h(p₁) in Q. For κ an infinite cardinal, a poset (Q,Q_?) is a κ-universal poset if every poset (P,P_?) with |P|?κ satisfies (P,P_?)?(Q,Q_?).The authors prove two theorems which improve and extend results from the extensive relevant literature.

Theorem 2.2. There is a zero-dimensional Hausdorff space S with|S|=κsuch that(P(S)/∼,h_?)is a κ-universal poset.     

Both compact and sequentially compact sets in abelian topological group

We show that every abelian topological group contains many interesting sets which are both compact and sequentially compact. Then we can deduce some useful facts, e.g.,

(1)

if G is a Hausdorff abelian topological group and μ:N²→G is countably additive, then the range μ(N²)={μ(A):A⊆N} is compact metrizable;

(2)

if X is a Hausdorff locally convex space and {xj}⊂X, then F={_∑j∈Δxj:Δ⊂N, Δ is finite} is relatively compact in (X,weak) if and only if F is relatively compact in X, and if and only if F is relatively compact in (X,F(M)) where F(M) is the Dierolf topology which is the strongest 〈X,X^′〉-polar topology having the same subseries convergent series as the weak topology.

     

Continuity of the spectrum on a class of upper triangular operator matrices

Let B(H) denote the algebra of operators on an infinite dimensional complex Hilbert space H, and let A^○∈B(K) denote the Berberian extension of an operator A∈B(H). It is proved that the set theoretic function σ, the spectrum, is continuous on the set C(i)⊂B(Hi) of operators A for which σ(A)={0} implies A is nilpotent (possibly, the 0 operator) and at every non-zero λ∈σp(A^○) for some operators X and B such that λ∉σp(B) and σ(A^○)={λ}∪σ(B). If CS(m) denotes the set of upper triangular operator matrices , where Aii∈C(i) and Aii has SVEP for all 1?i?m, then σ is continuous on CS(m). It is observed that a considerably large number of the more commonly considered classes of Hilbert space operators constitute sets C(i) and have SVEP.     

Approximating reversal distance for strings with bounded number of duplicates

For a string A=a₁…a_n, a reversalρ(i,j), 1?i?j?n, transforms the string A into a string A^′=a₁…a_i-1a_ja_j-1…a_ia_j+1…a_n, that is, the reversal ρ(i,j) reverses the order of symbols in the substring a_i…a_j of A. In the case of signed strings, where each symbol is given a sign + or -, the reversal operation also flips the sign of each symbol in the reversed substring. Given two strings, A and B, signed or unsigned, sorting by reversals (SBR) is the problem of finding the minimum number of reversals that transform the string A into the string B.Traditionally, the problem was studied for permutations, that is, for strings in which every symbol appears exactly once. We consider a generalization of the problem, k-SBR, and allow each symbol to appear at most k times in each string, for some k?1. The main result of the paper is an O(k²)-approximation algorithm running in time O(n). For instances with , this is the best known approximation algorithm for k-SBR and, moreover, it is faster than the previous best approximation algorithm.     

Two Bilinear (3 × 3)-Matrix Multiplication Algorithms of Complexity 25

As is known, a bilinear algorithm for multiplying 3 × 3 matrices can be constructed by using ordered triples of 3 × 3 matrices A _ρ , B _ρ , C _ρ , \(\rho = \overline {1,r} ,\) where r is the complexity of the algorithm. Algorithms with various symmetries are being extensively studied. This paper presents two algorithms of complexity 25 possessing the following two properties (symmetries): (1) the matricesA₁,B₁, and C1 are identity, (2) if the algorithm involves a tripleA, B, C, then it also involves the triples B, C, A and C, A, B. For example, these properties are inherent in the well-known Strassen algorithm for multiplying 2 × 2 matrices. Many existing (3 × 3)-matrix multiplication algorithms have property (2). Methods for finding new algorithms are proposed. It is shown that the found algorithms are different and new.