Richard M. Aron

Lineability and spaceability of sets of functions on R

We show that there is an infinite-dimensional vector space of differentiable functions on R, every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension 2 c of functions R → R, every non-zero element of which is everywhere surjective.

Weakly continuous mappings on Banach spaces

Abstract It is shown that every n -homogeneous continuous polynomial on a Banach space E which is weakly continuous on the unit ball of E is weakly uniformly continuous on the unit ball of E . Applications of the result to spaces of polynomials and holomorphic mappings on E are given.

ON UNIVERSAL FUNCTIONS

An entire function is called universal with respect to translations if for any

REGULARITY AND ALGEBRAS OF ANALYTIC FUNCTIONS IN INFINITE DIMENSIONS

g\;{\in}\;H(\mathbb{C}),\;R\;>\;0,\;and\;{\epsilon}\;>\;0

The Bishop-Phelps-Bollobás theorem and Asplund operators

, there is such that

Spectra of weighted composition operators on weighted banach spaces of analytic functions

f(z\;+\;n)\;-\;g(z)\;z\;{\leq}\;R

Weakly Uniformly Continuous and Weakly Sequentially Continuous Entire Functions

. Similarly, it is universal with respect to differentiation if for any g, R, and , there is n such that

Zeros of real polynomials

f^{(n)}(z)\;-\;g(z)\;z\;{\leq}\;R

Estimates by polynomials

. In this note, we review G. MacLanes proof of the existence of universal functions with respect to differentiation, and we give a simplified proof of G. D. Birkhoffs theorem showing the existence of universal functions with respect to translation. We also discuss Godefroy and Shapiros extension of these results to convolution operators as well as some new, related results and problems.

Zeros of polynomials on Banach spaces: The real story

A Banach space E is known to be Arens regular if every continuous linear mapping from E to E′ is weakly compact. Let U be an open subset of E, and letHb(U) denote the algebra of analytic functions on U which are bounded on bounded subsets of U lying at a positive distance from the boundary of U. We endow Hb(U) with the usual Frechet topology. Mb(U) denotes the set of continuous homomorphisms φ : Hb(U) → C. We study the relation between the Arens regularity of the space E and the structure of Mb(U).