Improved Bounds on the Union Complexity of Fat Objects

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ _s+2(n)log ² n). This improves the best known bound, and extends it to a more general class of objects. The research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.     

Compositions inside a rectangle and unimodality

Let c ^k,l (n) be the number of compositions (ordered partitions) of the integer n whose Ferrers diagram fits inside a k×l rectangle. The purpose of this note is to give a simple, algebraic proof of a conjecture of Vatter that the sequence c ^k,l (0),c ^k,l (1),…,c ^k,l (kl) is unimodal. The problem of giving a combinatorial proof of this fact is discussed, but is still open.     

Leibenzon's backward shift and composition operators

We apply Leibenzon's backward shift to show that the composition operator on the unit ball of always maps the weighted Hardy space into the Hardy class .

     

新型集合套范畴的弱Topos性

借助模糊集理论中的集合套理论，引入一种新型的集合范畴SEB3。它满足卡氏积封闭性，但不是Topos，而是介于卡氏积封闭性和Topos之间的一种结构，我们称之为弱Topos。     

Isolated points and essential components of composition operators on

We consider the topological space of all composition operators on the Banach algebra of bounded analytic functions on the unit disk. We obtain a function theoretic characterization of isolated points and show that each isolated composition operator is essentially isolated.

     

Convergent sequences of composition operators

Composition operators Cφ on the Hilbert Hardy space H² over the unit disk are considered. We investigate when convergence of sequences {φn} of symbols, (i.e., of analytic selfmaps of the unit disk) towards a given symbol φ, implies the convergence of the induced composition operators, Cφn→Cφ. If the composition operators Cφn are Hilbert-Schmidt operators, we prove that convergence in the Hilbert-Schmidt norm, ‖Cφn−CφHS_‖→0 takes place if and only if the following conditions are satisfied: ‖φn−φ_‖2→0, ∫1/(1−²|φ|)<∞, and ∫1/(1−²|φn|)→∫1/(1−²|φ|). The convergence of the sequence of powers of a composition operator is studied.     

Adjoints of a class of composition operators

Adjoints of certain operators of composition type are calculated. Specifically, on the classical Hardy space of the open unit disk operators of the form are considered, where is a finite Blaschke product. is obtained as a finite linear combination of operators of the form where and are rational functions, are associated Toeplitz operators and is defined by

     

Level sets and composition operators on the Dirichlet space

We consider composition operators in the Dirichlet space of the unit disc in the plane. Various criteria on boundedness, compactness and Hilbert-Schmidt class membership are established. Some of these criteria are shown to be optimal.     

Quadratic and Symmetric Bilinear Compositions of Quadratic Forms Over Commutative Rings

Over commutative rings in which 2 is a zero-divisor, to compose a quadratic form with symmetric bilinear forms or with quadratic forms is not quite the same. In this article, the relation between the two classes of compositions is clarified and the results applied to find the ranks of minimal compositions.     

BiBloch-type Maps: Existence and Beyond

It is shown that for f > 0 there are two constants c 1 , c 2 > 0 and a holomorphic map f from the unit disk ${\shadD} {\rm of} {\shadC}$ into ${\shadC}^2$ such that $ c_1(1-|z|)^{-\alpha }\le |\,f'(z)|\le c_2(1-|z|)^{-\alpha } $ for all $ z\in {\shadD}$ . Moreover, this existence is effectively used in the study of invariance of the Bloch-type spaces under composition, but also in the discussion of embedding the Bloch-type spaces via derivation into the Lebesgue, mixed-norm and Coifman-Meyer-Stein tent spaces.     

Derivative-free characterizations of bounded composition operators between Lipschitz spaces

We prove that maps into if and only if belongs to . In the case β < 1, we give another two equivalent conditions. Supported by MNZŽS Serbia, Project No. ON144010.     

Bounded composition operators with closed range on the Dirichlet space

For composition operators on spaces of analytic functions it is well known that norm estimates can be converted to Carleson measure estimates. The boundedness of the composition operator becomes equivalent to a Carleson measure inequality. The measure corresponding to a composition operator on the Dirichet space is , where is the cardinality of the preimage . The composition operator will have closed range if and only if the corresponding measure satisfies a ``reverse Carleson measure' theorem: for all . Assuming is bounded, a necessary condition for this inequality is a reverse of the Carleson condition: (C) for all Carleson squares . It has long been known that this is not sufficient for a completely general measure. Here we show that it is also not sufficient for the special measures . That is, we construct a function such that is bounded and satisfies (C) but the composition operator does not have closed range.

     

Products of integral-type operators and composition operators between Bloch-type spaces

A complete picture on the boundedness and compactness of the products of integral-type operators and composition operators between Bloch-type spaces of holomorphic functions on the unit disk is given in this paper.     

Extreme points of the closed convex hull of composition operators

We study the extreme points of the closed convex hull of the set of all composition operators on the space of bounded analytic functions and the disk algebra.     

直观实用的泊松图

本文介绍一种用来诊断数据是否来自泊松总体的方法—泊松图，并通过实例分析，说明诊断过程。     

Hardy空间到加权Hardy空间上的复合算子

本文研究了从Hardy空间到加权Hardy空间上的复合算子,并给出此复合算子为有界和紧的条件。