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.     

Spectral conditions for admissibility of evolution equations in Hilbert space

We study properties of solutions of the evolution equation , where B is a closable operator on the space AP(R,H) of almost periodic functions with values in a Hilbert space H such that B commutes with translations. The operator B generates a family of closed operators on H such that (whenever eiλtx∈D(B)). For a closed subset Λ⊂R, we prove that the following properties (i) and (ii) are equivalent: (i) for every function f∈AP(R,H) such that σ(f)⊆Λ, there exists a unique mild solution u∈AP(R,H) of Eq. (∗) such that σ(u)⊆Λ; (ii) is invertible for all λ∈Λ and .     

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.     

Some properties of sharp Hessenberg pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider an ordered pair of F-linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable on V; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V₀+V₁+?+V_i+1 for 0?i?d, where V_-1:=0 and V_d+1:=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and . We call such a pair a Hessenberg pair on V. It is known that if the Hessenberg pair A,A^∗ on V is irreducible then d=δ and for 0?i?d the dimensions of V_i and coincide. We say a Hessenberg pair A,A^∗ on V is sharp whenever it is irreducible and .In this paper, we give the definitions of a Hessenberg system and a sharp Hessenberg system. We discuss the connection between a Hessenberg pair and a Hessenberg system. We also define a finite sequence of scalars called the parameter array for a sharp Hessenberg system, which consists of the eigenvalue sequence, the dual eigenvalue sequence and the split sequence. We calculate the split sequence of a sharp Hessenberg system. We show that a sharp Hessenberg pair is a tridiagonal pair if and only if there exists a nonzero nondegenerate bilinear form on V that satisfies 〈Au,v〉=〈u,Av〉 and 〈A^∗u,v〉=〈u,A^∗v〉 for all u,v∈V.     

On continuous choice of retractions onto nonconvex subsets

For a Banach space B and for a class A of its bounded closed retracts, endowed with the Hausdorff metric, we prove that retractions on elements A∈A can be chosen to depend continuously on A, whenever nonconvexity of each A∈A is less than . The key geometric argument is that the set of all uniform retractions onto an α-paraconvex set (in the spirit of E. Michael) is -paraconvex subset in the space of continuous mappings of B into itself. For a Hilbert space H the estimate can be improved to and the constant can be replaced by the root of the equation α+α²+α³=1.     

A classification of sharp tridiagonal pairs

Let F denote a field and let V denote a vector space over F with finite positive dimension. We consider a pair of linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W, A^∗W⊆W, W≠0, W≠V. We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide. The pair A,A^∗ is called sharp whenever . It is known that if F is algebraically closed then A,A^∗ is sharp. In this paper we classify up to isomorphism the sharp tridiagonal pairs. As a corollary, we classify up to isomorphism the tridiagonal pairs over an algebraically closed field. We obtain these classifications by proving the μ-conjecture.     

Isomorphisms between unital C-algebras

It is shown that every almost linear bijection of a unital C^∗-algebra A onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all unitaries u∈A, all y∈A, and n=0,1,2,…, and that almost linear continuous bijection of a unital C^∗-algebra A of real rank zero onto a unital C^∗-algebra B is a C^∗-algebra isomorphism when h(n²uy)=h(n²u)h(y) for all , all y∈A, and n=0,1,2,…. Assume that X and Y are left normed modules over a unital C^∗-algebra A. It is shown that every surjective isometry , satisfying T(0)=0 and T(ux)=uT(x) for all x∈X and all unitaries u∈A, is an A-linear isomorphism. This is applied to investigate C^∗-algebra isomorphisms between unital C^∗-algebras.     

Lifting fixed points of completely positive mappings

Let 1?n?∞, and let be a row contraction on some Hilbert space H. Let F(T) be the space of all X∈B(H) such that . We show that, if non-zero, this space is completely isometric to the commutant of the Cuntz part of the minimal isometric dilation of .     

On a property of plane curves

Let γ:[0,1]→²[0,1] be a continuous curve such that γ(0)=(0,0), γ(1)=(1,1), and γ(t)∈²(0,1) for all t∈(0,1). We prove that, for each n∈N, there exists a sequence of points Ai, 0?i?n+1, on γ such that A₀=(0,0), An+1=(1,1), and the sequences and , 0?i?n, are positive and the same up to order, where π₁, π₂ are projections on the axes.     

Weighted Davenport’s constant and the weighted EGZ Theorem

Let G^∗,G be finite abelian groups with nontrivial homomorphism group . Let Ψ be a non-empty subset of . Let D_Ψ(G) denote the minimal integer, such that any sequence over G^∗ of length D_Ψ(G) must contain a nontrivial subsequence s₁,…,s_r, such that for some ψ_i∈Ψ. Let E_Ψ(G) denote the minimal integer such that any sequence over G^∗ of length E_Ψ(G) must contain a nontrivial subsequence of length |G|,s₁,…,s_|G|, such that for some ψ_i∈Ψ. In this paper, we show that E_Ψ(G)=|G|+D_Ψ(G)−1.     

The structure of a tridiagonal pair

Let K denote a field and let V denote a vector space over K with finite positive dimension.We consider a pair of K-linear transformations A:V→V and A^∗:V→V that satisfy the following conditions: (i) each of A,A^∗ is diagonalizable; (ii) there exists an ordering of the eigenspaces of A such that A^∗V_i⊆V_i-1+V_i+V_i+1 for 0?i?d, where V_-1=0 and V_d+1=0; (iii) there exists an ordering of the eigenspaces of A^∗ such that for 0?i?δ, where and ; (iv) there is no subspace W of V such that AW⊆W,A^∗W⊆W,W≠0,W≠V.We call such a pair a tridiagonal pair on V. It is known that d=δ and for 0?i?d the dimensions of coincide.In this paper we show that the following (i)-(iv) hold provided that K is algebraically closed: (i) Each of has dimension 1.(ii) There exists a nondegenerate symmetric bilinear form 〈,〉 on V such that 〈Au,v〉=〈u,Av〉 and 〈A^∗u,v〉=〈u,A^∗v〉 for all u,v∈V.(iii) There exists a unique anti-automorphism of End(V) that fixes each of A,A^∗.(iv) The pair A,A^∗ is determined up to isomorphism by the data , where θ_i (resp.) is the eigenvalue of A (resp.A^∗) on V_i (resp.), and is the split sequence of A,A^∗ corresponding to and .     

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let M be a von Neumann algebra equipped with a normal faithful semifinite trace τ, and let E be an r.i. space on (0,∞). Let E(M) be the associated symmetric space of measurable operators. Then to any bounded linear map T from E(M) into a Hilbert space H corresponds a positive norm one functional f∈E(2)^∗(M) such that

     

On an integral-type operator from logarithmic Bloch-type spaces to mixed-norm spaces on the unit ball

Let H(B) denote the space of all holomorphic functions on the open unit ball B of Cⁿ and g∈H(B). We characterize the boundedness and compactness of the following integral-type operator

     

Generalized homothetic biorders

In this paper, we study the binary relations R on a nonempty N^∗-set A which are h-independent and h-positive (cf. the introduction below). They are called homothetic positive orders. Denote by B the set of intervals of R having the form [r,+∞[ with 0<r≤+∞ or ]q,∞[ with q∈Q_≥0. It is a Q_>0-set endowed with a binary relation > extending the usual one on R_>0 (identified with a subset of B via the map r?[r,+∞[). We first prove that there exists a unique map Φ_R:A×A→B such that (for all and all ) we have Φ(mx,ny)=mn⁻¹⋅Φ(x,y) and . Then we give a characterization of the homothetic positive orders R on A such that there exist two morphisms of N^∗-sets satisfying . They are called generalized homothetic biorders. Moreover, if we impose some natural conditions on the sets u₁(A) and u₂(A), the representation (u₁,u₂) is “uniquely” determined by R. For a generalized homothetic biorder R on A, the binary relation R₁ on A defined by is a generalized homothetic weak order; i.e. there exists a morphism of N^∗-sets u:A→B such that (for all ) we have . As we did in [B. Lemaire, M. Le Menestrel, Homothetic interval orders, Discrete Math. 306 (2006) 1669-1683] for homothetic interval orders, we also write “the” representation (u₁,u₂) of R in terms of u and a twisting factor.     

On the classification of minimal mass blowup solutions of the focusing mass-critical Hartree equation

Consider the focusing mass-critical nonlinear Hartree equation iut+Δu=−(⁻²|⋅|∗²|u|)u for spherically symmetric initial data with ground state mass M(Q) in dimension d?5. We show that any global solution u which does not scatter must be the solitary wave eitQ up to phase rotation and scaling.     

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.     

The Erd?s-Ko-Rado properties of various graphs containing singletons

Let G=(V,E) be a graph. For r≥1, let be the family of independent vertex r-sets of G. For v∈V(G), let denote the star. G is said to be r-EKR if there exists v∈V(G) such that for any non-star family A of pair-wise intersecting sets in . If the inequality is strict, then G is strictlyr-EKR.Let Γ be the family of graphs that are disjoint unions of complete graphs, paths, cycles, including at least one singleton. Holroyd, Spencer and Talbot proved that, if G∈Γ and 2r is no larger than the number of connected components of G, then G is r-EKR. However, Holroyd and Talbot conjectured that, if G is any graph and 2r is no larger than μ(G), the size of a smallest maximal independent vertex set of G, then G is r-EKR, and strictly so if 2r<μ(G). We show that in fact, if G∈Γ and 2r is no larger than the independence number of G, then G is r-EKR; we do this by proving the result for all graphs that are in a suitable larger set Γ^′?Γ. We also confirm the conjecture for graphs in an even larger set Γ^″?Γ^′.     

A Morita type equivalence for dual operator algebras

We generalize the main theorem of Rieffel for Morita equivalence of W^∗-algebras to the case of unital dual operator algebras: two unital dual operator algebras A,B have completely isometric normal representations α,β such that α(A)=[M^∗β(B)M]^−w∗ and β(B)=[Mα(A)M^∗]^−w∗ for a ternary ring of operators M (i.e. a linear space M such that MM^∗M⊂M) if and only if there exists an equivalence functor which “extends” to a ∗-functor implementing an equivalence between the categories and . By we denote the category of normal representations of A and by the category with the same objects as and Δ(A)-module maps as morphisms (Δ(A)=A∩A^∗). We prove that this functor is equivalent to a functor “generated” by a B,A bimodule, and that it is normal and completely isometric.     

Sum of graphs of continuous functions and boundedness of additive operators

Assume that and are uniformly continuous functions, where D₁,D₂⊂X are nonempty open and arc-connected subsets of a real normed space X. We prove that then either f and g are affine functions, that is f(x)=x^∗(x)+a and g(x)=x^∗(x)+b with some x^∗∈X^∗ and a,b∈R or the algebraic sum of graphs of functions f and g has a nonempty interior in a product space X×R treated as a normed space with a norm .     

On maximal injective subalgebras of tensor products of von Neumann algebras

Let Mi be a von Neumann algebra, and Bi be a maximal injective von Neumann subalgebra of Mi, i=1,2. If M₁ has separable predual and the center of B₁ is atomic, e.g., B₁ is a factor, then is a maximal injective von Neumann subalgebra of . This partly answers a question of Popa.