On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.     

Representation,interpolation, and reiteration theorems for generalized approximation spaces

We show that the representation theorem for classical approximation spaces can be generalized to spaces A(X,l ^q (ℬ))={f∈X:{E _n (f)}∈l ^q (ℬ)} in which the weighted l ^q -space l ^q (ℬ) can be (more or less) arbitrary. We use this theorem to show that generalized approximation spaces can be viewed as real interpolation spaces (defined with K-functionals or main-part K-functionals) between couples of quasi-normed spaces which satisfy certain Jackson and Bernstein-type inequalities. Especially, interpolation between an approximation space and the underlying quasi-normed space leads again to an approximation space. Together with a general reiteration theorem, which we also prove in the present paper, we obtain formulas for interpolation of two generalized approximation spaces. Received: December 6, 2001; in final form: April 2, 2002?Published online: March 14, 2003     

NON-EMPTY CROSS-2-INTERSECTING FAMILIES OF SUBSETS

Let Cdenote the set of all k-subests of an n-set.Assume Alohtain in Ca,and A lohtain in (A,B) is called a cross-2-intersecting family if |A B≥2 for and A∈A,B∈B.In this paper,the best upper bounds of the cardinalities for non-empty cross-2-intersecting familles of a-and b-subsets are obtained for some a and b,A new proof for a Frankl-Tokushige theorem[6] is also given.     

Computing the nucleolus of some combinatorially-structured games

In this paper we introduce the ℬ-prenucleolus for a transferable utility game (N,v), where ℬ⊆2 ^N . The ℬ-prenucleolus is a straightforward generalization of the ordinary prenucleolus, where only the coalitions in ℬ determine the outcome. We impose a combinatorial structure on the collection ℬ which enables us to compute the ℬ-prenucleolus in ?(n ³|ℬ|) time. The algorithm can be used for computing the nucleolus of several classes of games, among which is the class of minimum cost spanning tree games. Received: September 4, 1995 / Accepted: May 5, 1997?Published online June 8, 2000     

Non-linear operators on sets of measures

Summary If M[ℬ, U(C, C)] is the collection of U(C, C)-valued (non-linear) set functions defined on the Borel subsetsℬ of the compact Hausdorff space S, one may define operators on M[ℬ, U(C, C)] which are ? of the Hammerstein type ?. We initiate a study of a concept analogous to the second dual of a space of continuous functions by inquiring as to what representation theorems one may obtain for these operators. A ? Lebesgue type ? decomposition theorem for elements of M[ℬ, U(C, C)] is obtained. A ? density ? theorem is also obtained for the space M[ℬ, U(C, C)]. Entrata in Redazione il 6 marzo 1974.     

Stinespring representability and Kadison’s Schwarz inequality in non-unital Banach star algebras and applications

A completely positive operator valued linear map ϕ on a (not necessarily unital) Banach *-algebra with continuous involution admits minimal Stinespring dilation iff for some scalark > 0, ϕ(x)*ϕ(x) ≤ kϕ(x*x) for allx iff ϕ is hermitian and satisfies Kadison’s Schwarz inequality ϕ(h) ² ≤ kϕ(h ²) for all hermitianh iff ϕ extends as a completely positive map on the unitizationA _e of A. A similar result holds for positive linear maps. These provide operator state analogues of the corresponding well-known results for representable positive functionals. Further, they are used to discuss (a) automatic Stinespring representability in Banach *-algebras, (b) operator valued analogue of Bochner-Weil-Raikov integral representation theorem, (c) operator valued analogue of the classical Bochner theorem in locally compact abelian groupG, and (d) extendability of completely positive maps from *-subalgebras. Evans’ result on Stinespring respresentability in the presence of bounded approximate identity (BAI) is deduced. A number of examples of Banach *-algebras without BAI are discussed to illustrate above results.     

Approximating the Pathway Axis and the Persistence Diagrams for a Collection of Balls in 3-Space

Given a collection ℬ of balls in a three-dimensional space, we wish to explore the cavities, voids, and tunnels in the complement space of ∪ℬ. We introduce the pathway axis of ℬ as a useful subset of the medial axis of the complement of ∪ℬ and prove that it satisfies several desirable geometric properties. We present an algorithm that constructs the pathway graph of ∪ℬ, a piecewise-linear approximation of the pathway axis. At the heart of our approach is an approximation scheme that constructs a collection K{\mathcal{K}} of same-size balls that approximate ℬ so that the Hausdorff distance between ∪ℬ and èK\bigcup{\mathcal{K}} is bounded by a prescribed parameter. We prove a bound on the ratio between the number of balls in K{\mathcal{K}} and the number of balls in ℬ. We employ this bound and the approximation scheme to show how to approximate the persistence diagrams for ∪ℬ, which can be used to extract major topological features such as the large voids and tunnels in the complement of ∪ℬ. We show that our approach is superior in terms of complexity to the standard point-sample approaches for the two problems that we address in this paper: approximating the pathway axis of ℬ and approximating the persistence diagrams for ∪ℬ. In a companion paper we introduce MolAxis, a tool for the identification of channels in macromolecules that demonstrates how the pathway graph and the persistence diagrams are used to identify plausible pathways in the complement of molecules.     

Distortion theorem for Bloch mappings on the unit ball <Emphasis Type="Italic">ℬ</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

In this paper, we obtain a version of subordination lemma for hyperbolic disk relative to hyperbolic geometry on the unit disk D. This subordination lemma yields the distortion theorem for Bloch mappings f ∈ H（B^n） satisfying ｜｜f｜｜0 = 1 and det f＇（0） = α ∈ （0, 1], where｜｜f｜｜0 = sup{（1 - ｜z｜^2 ）n＋1/2n det（f＇（z））[1/n ： z ∈ B^n}. Here we establish the distortion theorem from a unified perspective and generalize some known results. This distortion theorem enables us to obtain a lower bound for the radius of the largest univalent ball in the image of f centered at f（0）. When a = 1, the lower bound reduces to that of Bloch constant found by Liu. When n = 1, our distortion theorem coincides with that of Bonk, Minda and Yanagihara.     

A proof of the montague-plemmons-schein theorem on maximal subgroups of the semigroup of binary relations

The theorem in question is that the group of automorphisms of a partially ordered set (X,π), π denoting the order relation on the set X, is isomorphic to the maximal subgroup of ℬ_x containing π, where ℬ_x is the semigroup of all binary relations on X. This theorem is due to Montague and Plemmons [1] for the case X finite or countably infinite, and was extended by Schein to the general case, using a theorem due to Zaretsky [4]. A proof of the general case, based on [1] and results due to Plemmons and West [3], is also given in the preceding note by Plemmons and Schein [2]. The purpose of this note is to give an entirely self-contained proof of this intersesting theorem.     

Large antipodal families

A family {A _i | i ∈ I} of sets in ℝ ^d is antipodal if for any distinct i, j ∈ I and any p ∈ A _i , q ∈ A _j , there is a linear functional ϕ:ℝ ^d → ℝ such that ϕ(p) ≠ ϕ(q) and ϕ(p) ≤ ϕ(r) ≤ ϕ(q) for all r ∈ ∪ _i∈I A _i . We study the existence of antipodal families of large finite or infinite sets in ℝ³. The research was supported by the Hungarian-South African Intergovernmental Scientific and Technological Cooperation Programme, NKTH Grant no. ZA-21/2006 and South African National Research Foundation Grant no. UID 61853, as well as Hungarian National Foundation for Scientific Research Grants no. NK 67867, no. T47102, and no. K72537.     

Additive Maps Preserving Nilpotent Operators or Spectral Radius

Let X be a （real or complex） Banach space with dimension greater than 2 and let B0（X） be the subspace of B（X） spanned by all nilpotent operators on X. We get a complete classification of surjective additive maps Ф on B0（X） which preserve nilpotent operators in both directions. In particular, if X is infinite-dimensional, we prove that Ф has the form either Ф（T） = cATA^-1 or Ф（T） = cAT＇A^-1, where A is an invertible bounded linear or conjugate linear operator, c is a scalar, T＇ denotes the adjoint of T. As an application of these results, we show that every additive surjective map on B（X） preserving spectral radius has a similar form to the above with ｜c｜ = 1.     

Representations of inverse semigroups by one-to-one partial transformations of a set

We show that each representation ϕ, say, of an inverse semigroup S, by means of transformations of a set X, determines a representation ϕ^* by means of partial one-to-one transformations of X, in such a fashion that sϕ ↦ sϕ^*, for s ∈ S, is an isomorphism of Sϕ upon Sϕ^*. An immediate corollary is the classical faithful representation of an inverse semigroup as a semigroup of partial one-to-one transformations.     

Radon-Nikodým theorem for biweights and regular biweights on partial *-algebras

The first purpose of this paper is to investigate Radon-Nikodym theorem for biweights on partial *-algebra. Secondly, we study regularity of biweights on partial *-algebraA and show that a biweightϕ onA is decomposed intoϕ=ϕ _r+ϕ _s, whereϕ _r is a regular biweight onA andϕ _s is a singular biweight onA.     

Error growth analysis via stability regions for discretizations of initial value problems

This paper deals with numerical methods for the solution of linear initial value problems. Two main theorems are presented on the stability of these methods. Both theorems give conditions guaranteeing a mild error growth, for one-step methods characterized by a rational function ϕ(z). The conditions are related to the stability regionS={z:z∈ℂ with |ϕ(z)|≤1}, and can be viewed as variants to the resolvent condition occurring in the reputed Kreiss matrix theorem. Stability estimates are presented in terms of the number of time stepsn and the dimensions of the space. The first theorem gives a stability estimate which implies that errors in the numerical process cannot grow faster than linearly withs orn. It improves previous results in the literature where various restrictions were imposed onS and ϕ(z), including ϕ′(z)≠0 forz∈σS andS be bounded. The new theorem is not subject to any of these restrictions. The second theorem gives a sharper stability result under additional assumptions regarding the differential equation. This result implies that errors cannot grow faster thann ^β, with fixed β<1. The theory is illustrated in the numerical solution of an initial-boundary value problem for a partial differential equation, where the error growth is measured in the maximum norm.     

Minimum and Surjectivity Moduli Preserving in Banach Space

Let m(T) and q(T) be respectively the minimum and the surjectivity moduli of T∈ℬ(X), where ℬ(X) denotes the algebra of all bounded linear operators on a complex Banach space X. If there exists a semi-invertible but non-invertible operator in ℬ(X) then, given a surjective unital linear map φ: ℬ(X)⟶ℬ(X), we prove that m(T)=m(φ(T)) for all T∈ℬ(X), if and only if, q(T)=q(φ(T)) for all T∈ℬ(X), if and only if, there exists a bijective isometry U∈ℬ(X) such that φ(T)=UTU ⁻¹ for all T∈ℬ(X).     

The Existence and Uniqueness of the Solution for Neutral Stochastic Functional Differential Equations with Infinite Delay in Abstract Space

The main aim of this paper is to prove the existence and uniqueness of the solution for neutral stochastic functional differential equations with infinite delay, which the initial data belong to the phase space ℬ((−∞,0];ℝ ^d ). The vital work of this paper is to extend the initial function space of the paper (Wei and Wang, J. Math. Anal. Appl. 331:516–531, 2007) and give some examples to show that the phase space ℬ((−∞,0];ℝ ^d ) exists. In addition, this paper builds a Banach space ℳ²((−∞,T],ℝ ^d ) with a new norm in order to discuss the existence and uniqueness of the solution for such equations with infinite delay.     

On the dimensions of anisotropic quadratic forms in I4

Let WF denote the Witt ring of a field F of characteristic ≠2 and let I ⁿ F denote the n-th power of the ideal IF of even-dimensional forms in WF. The Arason-Pfister Hauptsatz states that if 0≠ϕ∈I ⁿ F is anisotropic then dim ϕ≥ 2 ⁿ . Pfister also showed that if ϕ∈I ³ F is anisotropic and dim ϕ>8 then dim ϕ≥12. We extend this result to I ⁴ F and show that if ϕ∈I ⁴ F is anisotropic and dim ϕ>16 then dim ϕ≥24 and we provide some results on anisotropic 24-dimensional forms in I ⁴ F. Oblatum 5-IV-1996 & 11-III-1997     

Any two irreducible Markov chains are finitarily orbit equivalent

Two invertible dynamical systems (X, gA, μ, T) and (Y, ℬ, ν, S), where X, Y are metrizable spaces and T, S are homeomorphisms on X and Y, are said to be finitarily orbit equivalent if there exists an invertible measure preserving mapping ϕ from a subset X ₀ of X of full measure to a subset Y ₀ of Y of full measure such that ϕ|x ₀ is continuous in the relative topology on X ₀, ϕ ⁻¹|Y ₀ is continuous in the relative topology on Y ₀ and ϕ(Orb _T (x)) = Orb _Sϕ (x) for μ-a.e. x ∈ X. In this article a finitary orbit equivalence mapping is shown to exist between any two irreducible Markov chains.     

Über Maximale Monotonie von Operatoren des Typs L*ϕ·L

In a Hilbert space H, we consider operators of type A=L^*ϕ·L, where L is a closed, linear operator and ϕ is a maximal cyclically monotone, coercive operator. The operators ϕ, L, L^* and their inverses are not necessarily everywhere defined. Our principle result is a nonlinear extension of an earlier theorem of v. Neumann for A=L^*L.Theorem: Suppose that either (L^*)⁻¹ is bounded or that both L⁻¹ is bounded and, D(ϕ) υ N (L^*). The L^*ϕ·L, is maximal cyclically monotone. Maximality of sums is also considered, and the theory is applied to concrete differential operators of the form , with monotone functions f₁ and various boundary conditions.      

The Exact Hausdorff Measure Function of the Level Sets of Multi-parameter Symmetric Stable Process

In this paper, we investigate the Hausdorff measure for level sets of N-parameter Rd-valued stable processes, and develop a means of seeking the exact Hausdorff measure function for level sets of N-parameter Rd-valued stable processes. We show that the exact Hausdorff measure function of level sets of N-parameter Rd-valued symmetric stable processes of index α is Ф（r） = r^N-d/α （log log l/r）d/α when Nα 〉 d. In addition, we obtain a sharp lower bound for the Hausdorff measure of level sets of general （N, d, α） strictly stable processes.