$M$-ideals in $H^\infty(\mathbb{D})$
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| Format: | Preprint |
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2024
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| author | D, Deepak K. Sarkar, Jaydeb Siju, Sreejith |
| author_facet | D, Deepak K. Sarkar, Jaydeb Siju, Sreejith |
| contents | This article intends to initiate an investigation into the structure of $M$-ideals in $H^\infty(\mathbb{D})$, where $H^\infty(\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. We introduce the notion of analytic primes and prove that $M$-ideals in $H^\infty(\mathbb{D})$ are analytic primes. From Hilbert function space perspective, we additionally prove that $M$-ideals in $H^\infty(\mathbb{D})$ are dense in the Hardy space. We show that outer functions play a key role in representing singly generated closed ideals in $H^\infty(\mathbb{D})$ that are $M$-ideals. This is also relevant to $M$-ideals in $H^\infty(\mathbb{D})$ that are finitely generated closed ideals in $H^\infty(\mathbb{D})$. We analyze $p$-sets of $H^\infty(\mathbb{D})$ and their connection to the Šilov boundary of the maximal ideal space of $H^\infty(\mathbb{D})$. 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_16947 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $M$-ideals in $H^\infty(\mathbb{D})$ D, Deepak K. Sarkar, Jaydeb Siju, Sreejith Functional Analysis Complex Variables Operator Algebras 46H10, 30H10, 32A35, 47L20, 46J15, 58C10 This article intends to initiate an investigation into the structure of $M$-ideals in $H^\infty(\mathbb{D})$, where $H^\infty(\mathbb{D})$ denotes the Banach algebra of all bounded analytic functions on the open unit disc $\mathbb{D}$ in $\mathbb{C}$. We introduce the notion of analytic primes and prove that $M$-ideals in $H^\infty(\mathbb{D})$ are analytic primes. From Hilbert function space perspective, we additionally prove that $M$-ideals in $H^\infty(\mathbb{D})$ are dense in the Hardy space. We show that outer functions play a key role in representing singly generated closed ideals in $H^\infty(\mathbb{D})$ that are $M$-ideals. This is also relevant to $M$-ideals in $H^\infty(\mathbb{D})$ that are finitely generated closed ideals in $H^\infty(\mathbb{D})$. We analyze $p$-sets of $H^\infty(\mathbb{D})$ and their connection to the Šilov boundary of the maximal ideal space of $H^\infty(\mathbb{D})$. 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. |
| title | $M$-ideals in $H^\infty(\mathbb{D})$ |
| topic | Functional Analysis Complex Variables Operator Algebras 46H10, 30H10, 32A35, 47L20, 46J15, 58C10 |
| url | https://arxiv.org/abs/2403.16947 |