2024/03/25 by D, Deepak K., Sarkar, Jaydeb, Siju, Sreejith
#30H10 #32A35 #46H10 #46J15 #47L20 #58C10 #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA)
paper · doi:10.48550/arxiv.2403.16947
This article intends to initiate an investigation into the structure of M-ideals in H^∞(\mathbbD), where H^∞(\mathbbD) denotes the Banach algebra of all bounded analytic functions on the open unit disc \mathbbD in ℂ. We introduce the notion of analytic primes and prove that M-ideals in H^∞(\mathbbD) are analytic primes. From Hilbert function space perspective, we additionally prove that M-ideals in H^∞(\mathbbD) are dense in the Hardy space. We show that outer functions play a key role in representing singly generated closed ideals in H^∞(\mathbbD) that are M-ideals. This is also relevant to M-ideals in H^∞(\mathbbD) that are finitely generated closed ideals in H^∞(\mathbbD). We analyze p-sets of H^∞(\mathbbD) and their connection to the Šilov boundary of the maximal ideal space of H^∞(\mathbbD). Some of our results apply to the polydisc. In addition to addressing questions regarding M-ideals, the results presented in this paper offer some new perspectives on bounded analytic functions.