Characterization of sets K for which HK∞(𝔻) is an algebra

Banjade, Debendra P.; Dunivin, Jeremiah

doi:10.2140/involve.2023.16.421

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Characterization of sets $K$ for which $H^{\infty}_K(\mathbb{D})$ is an algebra

Debendra P. Banjade and Jeremiah Dunivin

Vol. 16 (2023), No. 3, 421–429

DOI: 10.2140/involve.2023.16.421

Abstract

Let $K \subset ℤ_{+}$ and define the set $H_{K}^{\infty} (𝔻)$ to be the collection of all bounded analytic functions on the unit disk $𝔻$ in the complex plane whose $k$ -th derivative vanishes at zero for all $k \in K$ . We prove that $H_{K}^{\infty} (𝔻)$ is an algebra precisely when $ℤ_{+} ∖ K$ is an abelian semigroup. In particular, we show that if $K$ is a finite set, then $K$ yielding an algebra is equivalent to $ℕ ∖ K$ being a numerical semigroup. Moreover, an algorithm for constructing $K$ when $K$ is finite is provided. These results answer the questions posed by Ryle (2009). We then classify some of these sets and give a list of their exact representations.

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Keywords

bounded analytic functions, subalgebras, numerical semigroups

Mathematical Subject Classification

Primary: 30H05, 11B75

Secondary: 46J30

Milestones

Received: 25 August 2021

Revised: 11 July 2022

Accepted: 25 July 2022

Published: 10 August 2023

Communicated by Kenneth S. Berenhaut

Authors

Debendra P. Banjade
Department of Mathematics and Statistics Coastal Carolina University Conway, SC United States

Jeremiah Dunivin
Department of Mathematical and Statistical Sciences Clemson University Clemson, SC United States

Vol. 16, No. 3, 2023