Sandy Grabiner

Convergent expansions and bounded operators in the umbral calculus

On etudie des proprietes analytiques de certaines suites de fonctions speciales appelees suites de Sheffer

Standard homomorphisms and regulated weights on weighted convolution algebras

Abstract In this paper we introduce a class of homomorphisms between weighted convolution algebras which we call standard homomorphisms and derive various equivalents of standardness. We also introduce the convergence ideal of a homomorphism. We find various descriptions of the convergence ideal together with its relation to standardness. We show that every continuous homomorphism from a weighted convolution algebra into another weighted convolution algebra, with a regulated weight, is standard, and when the algebras have both regulated weights the extension of a homomorphism to the weighted measure algebras satisfies additional continuity properties.

Equivalent weights and standard homomorphisms for convolution algebras on

We take a second look at two basic topics in the study of weighted convolution algebras L 1 (ω) on ℝ + . An early result showed that one could replace the weight

A Radical Approach to Real Analysis, by David Bressoud

\omega

Uniform ascent and descent of bounded operators

with a very well-behaved weight without changing the space L 1 (ω) as long as L 1 (ω) was an algebraffi We prove the analogous result for measure algebras when M (ω) is an algebraffi This allows us to preserve not only the norm topology but also the relative weak* topology on L 1 (ω). A homomorphism between weighted convolution algebras is said to be standard if it preserves generators of dense principal ideals. The original proofs of standardness and its variants are all based on finding the generator of a particular strongly continuous convolution semigroup. In this paper we give much simpler direct proofs of these results. We also improve the statement and proof of the theorem, giving useful properties equivalent to the standardness of a homomorphism.

ASCENT, DESCENT, AND ERGODIC PROPERTIES OF LINEAR OPERATORS

(1995). A Radical Approach to Real Analysis, by David Bressoud. The American Mathematical Monthly: Vol. 102, No. 7, pp. 661-664.