An extension of the univalent condition for a family of integral operators

The main object of this work is to extend the univalent condition for a family of integral operators. Several other closely-related results are also considered. A number of known univalent conditions would follow upon specializing the parameters involved in our main result.     

Hahn-Banach extension operators and spaces of operators

Let be Banach spaces and let be closed operator ideals. Let be a Banach space having the Radon-Nikodým property. The main results are as follows. If is a Hahn-Banach extension operator, then there exists a set of Hahn-Banach extension operators , , such that , where . If is an ideal in for all equivalently renormed versions of , then there exist Hahn-Banach extension operators and such that .

     

A class of Loewner chain preserving extension operators

We consider operators that extend locally univalent mappings of the unit disk Δ in C to locally biholomorphic mappings of the Euclidean unit ball B of Cn. For such an operator Φ, we seek conditions under which etΦ(e−tf(⋅,t)), t?0, is a Loewner chain on B whenever f(⋅,t), t?0, is a Loewner chain on Δ. We primarily study operators of the form , , where β∈[0,1/2] and is holomorphic, finding that, for ΦG,β to preserve Loewner chains, the maximum degree of terms appearing in the expansion of G is a function of β. Further applications involving Bloch mappings and radius of starlikeness are given, as are elementary results concerning extreme points and support points.     

A family of degenerate differential operators

Certain second-order partial differential operators, expressed as sums of squares of complex vector fields, are shown not to beC ^∞ hypoelliptic even at a point, rather than merely in an open set. The proof is based on an asymptotic analysis of a family of ordinary differential operators depending on a complex parameter. Research supported by the National Science Foundation.     

A certain family of summation-integral type operators

In the present paper, the authors introduce and investigate a new sequence of linear positive operators G_n,c, which includes some well-known operators as its special cases. The results obtained here include an estimate on the rate of convergence of G_n,c (f, x) by means of the decomposition technique for functions of bounded variation.     

A family of univariate rational Bernstein operators

We define and study a new family of univariate rational Bernstein operators. They are positive operators exact on linear polynomials. Moreover, like classical polynomial Bernstein operators, they enjoy the traditional shape preserving properties and they are total variation diminishing. Finally, for a specific class of denominators, some convergence results are proved, in particular a Voronovskaja theorem, and some error bounds are given.     

Continuous extension operators and convexity

Given a nonempty closed subset A of a Hilbert space X, we denote by L(A) the space of all bounded Lipschitz mappings from A into X, equipped with the supremum norm. We show that there is a continuous mapping F_c:L(A)?L(X) such that for each g∈L(A), F_c(g)|_A=g, , and . We also prove that the corresponding set-valued extension operator is lower semicontinuous.     

On extension of abstract Urysohn operators

We consider the extension of an orthogonally additive operator from a lateral ideal and a lateral band to the whole space. We prove in particular that every orthogonally additive operator, extended from a lateral band of an order complete vector lattice, preserves lateral continuity, narrowness, compactness, and disjointness preservation. These results involve the strengthening of a recent theorem about narrow orthogonally additive operators in vector lattices.     

Nonstandard extension of not-everywhere-defined positive operators

Translated fromSibirskii Matematicheskii Zhurnal, Vol. 35, No. 4, pp. 719–727, July–August, 1994.     

An extension of invertibility of Hammerstein-type operators

My aim is to show that some properties, proved to be true for the square matrices, are true for some not necessarily linear operators on a linear space, in particular, for Hammerstein-type operators.     

A two-parameter family of an extension of Beatty sequences

Beatty sequences ⌊nα+γ⌋ are nearly linear, also called balanced, namely, the absolute value of the difference D of the number of elements in any two subwords of the same length satisfies D?1. For an extension of Beatty sequences, depending on two parameters s,t∈Z_>0, we prove D?⌊(s-2)/(t-1)⌋+2, and D?2s+1. We show that each value that is assumed, is assumed infinitely often. Under the assumption (s-2)?(t-1)² the first result is optimal, in that the upper bound is attained. This provides information about the gap-structure of (s,t)-sequences, which, for s=1, reduce to Beatty sequences. The (s,t)-sequences were introduced in Fraenkel [Heap games, numeration systems and sequences, Ann. Combin. 2 (1998) 197-210; E. Lodi, L. Pagli, N. Santoro (Eds.), Fun with Algorithms, Proceedings in Informatics, vol. 4, Carleton Scientific, University of Waterloo, Waterloo, Ont., 1999, pp. 99-113], where they were used to give a strategy for a 2-player combinatorial game on two heaps of tokens.     

A universal model for a family of doubly commuting operators

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 43, pp. 3–9, 1985.     

Tame linear extension operators for smooth Whitney functions

For a compact set K⊆Rd we present a rather easy construction of a linear extension operator E:E(K)→C^∞(Rd) for the space of Whitney jets E(K) which satisfies linear tame continuity estimates , where ‖⋅s_‖ denotes the s-th Whitney norm. The construction turns out to be possible if and only if the local Markov inequality LMI(s) introduced by Bos and Milman holds for every s>r on K. In particular, E(K) admits a tame linear extension operator if and only if the local Markov inequality LMI(s) holds on K for some s?1.