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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

Let K + and define the set HK(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 K. We prove that HK(𝔻) is an algebra precisely when + K is an abelian semigroup. In particular, we show that if K is a finite set, then K yielding an algebra is equivalent to 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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bounded analytic functions, subalgebras, numerical semigroups
Mathematical Subject Classification
Primary: 30H05, 11B75
Secondary: 46J30
Received: 25 August 2021
Revised: 11 July 2022
Accepted: 25 July 2022
Published: 10 August 2023

Communicated by Kenneth S. Berenhaut
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