Double-layer potentials, configuration constants and applications to numerical ranges
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917968865132544 |
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| author | Malman, Bartosz Mashreghi, Javad O'Loughlin, Ryan Ransford, Thomas |
| author_facet | Malman, Bartosz Mashreghi, Javad O'Loughlin, Ryan Ransford, Thomas |
| contents | Given a compact convex planar domain $Ω$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(Ω)$ is the norm of the Neumann-Poincaré operator $K_Ω$ acting on the space of continuous real-valued functions on the boundary $\partial Ω$, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(Ω)$ of $K_Ω$ on the corresponding space of complex-valued functions, and the norm $a(Ω)$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(Ω) = c_{\mathbb{C}}(Ω)$, the analytic Neumann-type inequality $a(Ω) < 1$, and provide various estimates for these quantities expressed in terms of the geometry of $Ω$. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\|p(T)\| \leq K \sup_{z \in Ω} |p(z)|$, where $p$ is a polynomial and $Ω$ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix-Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(Ω)}$. In the case that $Ω$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_19049 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Double-layer potentials, configuration constants and applications to numerical ranges Malman, Bartosz Mashreghi, Javad O'Loughlin, Ryan Ransford, Thomas Functional Analysis Complex Variables Given a compact convex planar domain $Ω$ with non-empty interior, the classical Neumann's configuration constant $c_{\mathbb{R}}(Ω)$ is the norm of the Neumann-Poincaré operator $K_Ω$ acting on the space of continuous real-valued functions on the boundary $\partial Ω$, modulo constants. We investigate the related operator norm $c_{\mathbb{C}}(Ω)$ of $K_Ω$ on the corresponding space of complex-valued functions, and the norm $a(Ω)$ on the subspace of analytic functions. This change requires introduction of techniques much different from the ones used in the classical setting. We prove the equality $c_{\mathbb{R}}(Ω) = c_{\mathbb{C}}(Ω)$, the analytic Neumann-type inequality $a(Ω) < 1$, and provide various estimates for these quantities expressed in terms of the geometry of $Ω$. We apply our results to estimates for the holomorphic functional calculus of operators on Hilbert space of the type $\|p(T)\| \leq K \sup_{z \in Ω} |p(z)|$, where $p$ is a polynomial and $Ω$ is a domain containing the numerical range of the operator $T$. Among other results, we show that the well-known Crouzeix-Palencia bound $K \leq 1 + \sqrt{2}$ can be improved to $K \leq 1 + \sqrt{1 + a(Ω)}$. In the case that $Ω$ is an ellipse, this leads to an estimate of $K$ in terms of the eccentricity of the ellipse. |
| title | Double-layer potentials, configuration constants and applications to numerical ranges |
| topic | Functional Analysis Complex Variables |
| url | https://arxiv.org/abs/2407.19049 |