Isometric factorization of weakly compact operators and the approximation property

Using an isometric version of the Davis, Figiel, Johnson, and Pe?czyński factorization of weakly compact operators, we prove that a Banach spaceX has the approximation property if and only if, for every Banach spaceY, the finite rank operators of norm ≤1 are dense in the unit ball ofW(Y,X), the space of weakly compact operators fromY toX, in the strong operator topology. We also show that, for every finite dimensional subspaceF ofW(Y,X), there are a reflexive spaceZ, a norm one operatorJ:Y→Z, and an isometry Φ :F →W(Y,X) which preserves finite rank and compact operators so thatT=Φ(T) oJ for allT∈F. This enables us to prove thatX has the approximation property if and only if the finite rank operators form an ideal inW(Y,X) for all Banach spacesY.     

Local solvability on {\mathbb{H}_1}: non-homogeneous operators

Local solvability and non-solvability are classified for left-invariant differential operators on the Heisenberg group ${\mathbb{H}_1}$ of the form L?=?P _n (X, Y)?+?Q(X, Y) where the P _n are certain homogeneous polynomials of order n?≥ 2 and Q is of lower order with ${X=\partial_x,\,Y=\partial_y+x\partial_w}$ on ${\mathbb{R}^3}$ . We extend previous studies of operators of the form P _n (X, Y) via representations involving ordinary differential operators with a parameter.     

The regular part of a semigroup of transformations with restricted range

Let T(X) be the full transformation semigroup on the set X and let T(X,Y) be the semigroup consisting of all total transformations from X into a fixed subset Y of X. It is known that $$F(X, Y)=\{\alpha\in T(X, Y): X\alpha\subseteq Y\alpha\},$$ is the largest regular subsemigroup of T(X,Y) and determines Green??s relations on T(X,Y). In this paper, we show that F(X,Y)?T(Z) if and only if X=Y and |Y|=|Z|; or |Y|=1=|Z|, and prove that every regular semigroup S can be embedded in F(S ¹,S). Then we describe Green??s relations and ideals of F(X,Y) and apply these results to get all of its maximal regular subsemigroups when Y is a nonempty finite subset of X.     

Hereditary order convexity inL(X,Y)

LetX be a topological vector space,Y an ordered topological vector space andL(X,Y) the space of all linear and continuous mappings fromX intoY. The hereditary order-convex cover [K] ^h of a subsetK ofL(X,Y) is defined by [K] ^h ={A∈L(X,Y):Ax∈[Kx] for allx∈X}, where[Kx] is the order-convex ofKx. In this paper we study the hereditary order-convex cover of a subset ofL(X,Y). We show how this cover can be constructed in specific cases and investigate its structural and topological properties. Our results extend to the spaceL(X,Y) some of the known properties of the convex hull of subsets ofX ^*.     

Estimation in nonparametric location-scale regression models with censored data

Consider the random vector (X, Y), where X is completely observed and Y is subject to random right censoring. It is well known that the completely nonparametric kernel estimator of the conditional distribution ${F(\cdot|x)}$ of Y given X = x suffers from inconsistency problems in the right tail (Beran 1981, Technical Report, University of California, Berkeley), and hence any location function m(x) that involves the right tail of ${F(\cdot|x)}$ (like the conditional mean) cannot be estimated consistently in a completely nonparametric way. In this paper, we propose an alternative estimator of m(x), that, under certain conditions, does not share the above inconsistency problems. The estimator is constructed under the model Y = m(X) + σ(X)ε, where ${\sigma(\cdot)}$ is an unknown scale function and ε (with location zero and scale one) is independent of X. We obtain the asymptotic properties of the proposed estimator of m(x), we compare it with the completely nonparametric estimator via simulations and apply it to a study of quasars in astronomy.     

The existence spectrum for overlarge sets of pure directed triple systems

A directed triple system of order v,denoted by DTS(v),is a pair (X,B) where X is a v-set and B is a collection of transitive triples on X such that every ordered pair of X belongs to exactly one triple of B.A DTS(v) (X,A) is called pure and denoted by PDTS(v) if (a,b,c) ∈ A implies (c,b,a) ∈/ A.An overlarge set of PDTS(v),denoted by OLPDTS(v),is a collection {(Y \{yi},Aij) : yi ∈ Y,j ∈ Z3},where Y is a (v+1)-set,each (Y \{yi},Aij) is a PDTS(v) and these Ais form a partition of all transitive triples on Y .In this paper,we shall discuss the existence problem of OLPDTS(v) and give the following conclusion: there exists an OLPDTS(v) if and only if v ≡ 0,1 (mod 3) and v 3.     

Degree Sum Conditions for Cyclability in Bipartite Graphs

We denote by G[X, Y] a bipartite graph G with partite sets X and Y. Let d _G (v) be the degree of a vertex v in a graph G. For G[X, Y] and ${S \subseteq V(G),}$ we define ${\sigma_{1,1}(S):=\min\{d_G(x)+d_G(y) : (x,y) \in (X \cap S,Y) \cup (X, Y \cap S), xy \not\in E(G)\}}$ . Amar et al. (Opusc. Math. 29:345–364, 2009) obtained σ _1,1(S) condition for cyclability of balanced bipartite graphs. In this paper, we generalize the result as it includes the case of unbalanced bipartite graphs: if G[X, Y] is a 2-connected bipartite graph with |X| ≥ |Y| and ${S \subseteq V(G)}$ such that σ _1,1(S) ≥ |X| + 1, then either there exists a cycle containing S or ${|S \cap X| > |Y|}$ and there exists a cycle containing Y. This degree sum condition is sharp.     

A non-archimedean Dugundji extension theorem

We prove a non-archimedean Dugundji extension theorem for the spaces C*(X, C* (X, K) of continuous bounded functions on an ultranormal space X with values in a non-archimedean non-trivially valued complete field K. Assuming that K is discretely valued and Y is a closed subspace of X we show that there exists an isometric linear extender T: C* (Y, K) → K* (X, K) if X is collectionwise normal or Y is Lindelöf or K is separable. We provide also a self contained proof of the known fact that any metrizable compact subspace Y of an ultraregular space X is a retract of X.     

Existence theorems and necessary conditions for a general formulation of the minimum effort problem

The basic problem considered may be described briefly as follows. LetX,Y, andZ be normed linear spaces,T:D(T)→Y,S:D(S)→Z linear operators withD(T) \( \subseteq\) X andD(S) \( \subseteq\) X,Ω \( \subseteq\) X a convex set containing the zero elementθ, andJ a real-valued convex function defined onX×Y such that

1. J(x,y)?-0 for (x,y)teX×Y,
2. J(θ,θ)=0,
3. J(x,y)→+∞, as (∥x∥²+∥y∥²)^1/2→+∞.

Givenζ∈Y andη∈S[core _T Ω∩;D(S)], find an elementx=x ₀ which minimizesJ(x,ζ?Tx) on the set {x∈[Ω∩;D(S)∩;D(T)]:Sx=η}. The abovementioned problem, together with certain special cases, is analyzed using the classical techniques of functional analysis. Existence problems are considered for a certain class of closed linear operators. In particular, existence of an optimal solution is determined by evaluating a generalized Minkowski functional at the point (ζ,η) inY×Z. A necessary condition is presented for special cases, and corresponding characterizations of optimal solutions are made in terms of the adjoint operators. These results are applicable to linear minimum effort problems, constrained variational problems, optimal control of distributive systems, and certain ill-posed variational problems.     

Non-negative Spectral Measures and Representations of C*-Algebras

Regular normalized W-valued spectral measures on a compact Hausdorff space X are in one-to-one correspondence with unital *-representations ${\rho : C(X, \mathbb{C}) \to W}$ , where W stands for a von Neumann algebra. In this paper we show that for every compact Hausdorff space X and every von Neumann algebras W ₁, W ₂ there is a one-to-one correspondence between unital *-representations ${\rho : C(X, W_1) \to W_2}$ and special B(W ₁, W ₂)-valued measures on X that we call non-negative spectral measures. Such measures are special cases of non-negative measures that we introduced in our previous paper (Cimpri? and Zalar, J Math Anal Appl 401:307–316, 2013) in connection with moment problems for operator polynomials.     

A law of the iterated logarithm for random walk in random scenery with deterministic normalizers

LetX, X _i ,i≥1, be a sequence of independent and identically distributed ? ^d -valued random vectors. LetS _o=0 and \(S_n = \sum\nolimits_{i = 1}^n {X_i } \) forn≤1. Furthermore letY, Y(α), α∈? ^d , be independent and identically distributed ?-valued random variables, which are independent of theX _i . Let \(Z_n = \sum\nolimits_{i = 0}^n {Y(S_i )} \) . We will call (Z _n ) arandom walk in random scenery. In this paper, we consider the law of the iterated logarithm for random walk in random scenery where deterministic normalizers are utilized. For example, we show that if (S _n ) is simple, symmetric random walk in the plane,E[Y]=0 andE[Y ²]=1, then $$\mathop {\overline {\lim } }\limits_{n \to \infty } \frac{{Z_n }}{{\sqrt {2n\log (n)\log (\log (n))} }} = \sqrt {\frac{2}{\pi }} a.s.$$      

Vector measures: where are their integrals?

Let ν be a vector measure with values in a Banach space Z. The integration map $I_\nu: L^1(\nu)\to Z$ , given by $f\mapsto \int f\,d\nu$ for f ∈ L ¹(ν), always has a formal extension to its bidual operator $I_\nu^{**}: L^1(\nu)^{**}\to Z^{**}$ . So, we may consider the “integral” of any element f ^** of L ¹(ν)^** as I _ν ^** (f ^**). Our aim is to identify when these integrals lie in more tractable subspaces Y of Z ^**. For Z a Banach function space X, we consider this question when Y is any one of the subspaces of X ^** given by the corresponding identifications of X, X′′ (the Köthe bidual of X) and X′^* (the topological dual of the Köthe dual of X). Also, we consider certain kernel operators T and study the extended operator I _ν ^** for the particular vector measure ν defined by ν(A) := T(χ _A ).     

Multiple critical points for non-differentiable parametrized functionals and applications to differential inclusions

In this paper we deal with a class of non-differentiable functionals defined on a real reflexive Banach space X and depending on a real parameter of the form ${\mathcal{E}_\lambda(u)=L(u)-(J_1\circ T)(u)-\lambda (J_2\circ S)(u)}$ , where ${L:X \rightarrow \mathbb R}$ is a sequentially weakly lower semicontinuous C ¹ functional, ${J_1:Y\rightarrow\mathbb R, J_2:Z\rightarrow \mathbb R}$ (Y, Z Banach spaces) are two locally Lipschitz functionals, T : X → Y, S : X → Z are linear and compact operators and λ > 0 is a real parameter. We prove that this kind of functionals posses at least three nonsmooth critical points for each λ > 0 and there exists λ* > 0 such that the functional ${\mathcal{E}_{\lambda^\ast}}$ possesses at least four nonsmooth critical points. As an application, we study a nonhomogeneous differential inclusion involving the p(x)-Laplace operator whose weak solutions are exactly the nonsmooth critical points of some “energy functional” which satisfies the conditions required in our main result.     

Erratic behavior of CAT(0) geodesics under G-equivariant quasi-isometries

We show that, given any connected, compact space ${Z \subset \mathbb{R}^n}$ , there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry ${f\colon X\rightarrow Y}$ , and a geodesic ray c in X such that the closure of f (c), intersected with ${\partial Y}$ , is homeomorphic to Z. This characterizes all homeomorphism types of ??geodesic boundary images?? that arise in this manner.     

On one sided ideals of a semiprime ring with generalized derivations

Let R be a ring with center Z(R). An additive mapping ${F : R \longrightarrow R}$ is said to be a generalized derivation on R if there exists a derivation ${d : R \longrightarrow R}$ such that F(xy) = F(x)y + xd(y), for all ${x, y \in R}$ (the map d is called the derivation associated with F). Let R be a semiprime ring and U be a nonzero left ideal of R. In the present note we prove that if R admits a generalized derivation F, d is the derivation associated with F such that d(U) ≠ (0) then R contains some nonzero central ideal, if one of the following conditions holds: (1) R is 2-torsion free and ${F(xy) \in Z(R)}$ , for all ${x, y \in U}$ , unless F(U)U = UF(U) = Ud(U) = (0); (2) ${F(xy) \mp yx \in Z(R)}$ , for all ${x,y \in U}$ ; (3) ${F(xy) \mp [x,y] \in Z(R)}$ , for all ${x,y \in U}$ ; (4) F ≠ 0 and F([x,y]) = 0, for all ${x, y \in U}$ , unless Ud(U) = (0); (5) F ≠ 0 and ${F([x, y]) \in Z(R)}$ , for all ${x, y \in U}$ , unless either d(Z(R))U = (0) or Ud(U) = (0)n.     

The structures of Hausdorff metric in non-Archimedean spaces

For non-Archimedean spaces X and Y, let $\mathcal{M}_\flat \left( X \right)$ , $\mathfrak{M}\left( {V \to W} \right)$ and $\mathfrak{D}_\flat \left( {X,Y} \right)$ be the ballean of X (the family of the balls in X), the space of mappings from X to Y, and the space of mappings from the ballean of X to Y, respectively. By studying explicitly the Hausdorff metric structures related to these spaces, we construct several families of new metric structures (e.g., $\hat \rho _u$ , $\hat \beta _{X,Y}^\lambda$ , $\hat \beta _{X,Y}^{ * \lambda }$ ) on the corresponding spaces, and study their convergence, structural relation, law of variation in the variable λ, including some normed algebra structure. To some extent, the class $\hat \beta _{X,Y}^\lambda$ is a counterpart of the usual Levy-Prohorov metric in the probability measure spaces, but it behaves very differently, and is interesting in itself. Moreover, when X is compact and Y = K is a complete non-Archimedean field, we construct and study a Dudly type metric of the space of K-valued measures on X.     

Factorization Theorems for Multiplication Operators on Banach Function Spaces

Let X Y and Z be Banach function spaces over a measure space \({(\Omega, \Sigma, \mu)}\) . Consider the spaces of multiplication operators \({X^{Y'}}\) from X into the Köthe dual Y′ of Y, and the spaces X ^Z and \({Z^{Y'}}\) defined in the same way. In this paper we introduce the notion of factorization norm as a norm on the product space \({X^Z \cdot Z^{Y'} \subseteq X^{Y'}}\) that is defined from some particular factorization scheme related to Z. In this framework, a strong factorization theorem for multiplication operators is an equality between product spaces with different factorization norms. Lozanovskii, Reisner and Maurey–Rosenthal theorems are considered in our arguments to provide examples and tools for assuring some requirements. We analyze the class \({d_{p,Z}^*}\) of factorization norms, proving some factorization theorems for them when p-convexity/p-concavity type properties of the spaces involved are assumed. Some applications in the setting of the product spaces are given.     

Asymptotic behavior of product of two heavy-tailed dependent random variables

Let X and Y be positive weakly negatively dependent(WND)random variables with finite expectations and continuous distribution functions F and G with heavy tails, respectively. The asymptotic behavior of the tail of distribution of XY is studied and some closure properties under some suitable conditions on ˉ F(x)= 1 F(x)and ˉ G(x)= 1 G(x)are provided. Moreover, subexponentiality of XY when X and Y are WND random variables is derived.