Inner functions as strongly extreme points: stability properties
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arXiv
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| Natura: | Preprint |
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2023
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| _version_ | 1866914431996264448 |
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| author | Dyakonov, Konstantin M. |
| author_facet | Dyakonov, Konstantin M. |
| contents | Given a Banach space $\mathcal X$, let $x$ be a point in $\text{ball}(\mathcal X)$, the closed unit ball of $\mathcal X$. One says that $x$ is a strongly extreme point of $\text{ball}(\mathcal X)$ if it has the following property: for every $\varepsilon>0$ there is $δ>0$ such that the inequalities $\|x\pm y\|<1+δ$ imply, for $y\in\mathcal X$, that $\|y\|<\varepsilon$. We are concerned with certain subspaces of $H^\infty$, the space of bounded holomorphic functions on the disk, that arise upon imposing finitely many linear constraints and can be viewed as finite-dimensional perturbations of $H^\infty$. It is well known that the strongly extreme points of $\text{ball}(H^\infty)$ are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions $f\in H^\infty$ with $\log(1-|f|)$ non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed $H^\infty$-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_11162 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Inner functions as strongly extreme points: stability properties Dyakonov, Konstantin M. Functional Analysis Classical Analysis and ODEs Complex Variables 30H05, 30H10, 30J05, 46A55 Given a Banach space $\mathcal X$, let $x$ be a point in $\text{ball}(\mathcal X)$, the closed unit ball of $\mathcal X$. One says that $x$ is a strongly extreme point of $\text{ball}(\mathcal X)$ if it has the following property: for every $\varepsilon>0$ there is $δ>0$ such that the inequalities $\|x\pm y\|<1+δ$ imply, for $y\in\mathcal X$, that $\|y\|<\varepsilon$. We are concerned with certain subspaces of $H^\infty$, the space of bounded holomorphic functions on the disk, that arise upon imposing finitely many linear constraints and can be viewed as finite-dimensional perturbations of $H^\infty$. It is well known that the strongly extreme points of $\text{ball}(H^\infty)$ are precisely the inner functions, while the (usual) extreme points of this ball are the unit-norm functions $f\in H^\infty$ with $\log(1-|f|)$ non-integrable over the circle. Here we show that similar characterizations remain valid for our perturbed $H^\infty$-type spaces. Also, we investigate to what extent a non-inner function can differ from a strongly extreme point. |
| title | Inner functions as strongly extreme points: stability properties |
| topic | Functional Analysis Classical Analysis and ODEs Complex Variables 30H05, 30H10, 30J05, 46A55 |
| url | https://arxiv.org/abs/2301.11162 |