Double-layer potentials, configuration constants, and applications to numerical ranges

Malman, B., Mashreghi, J., O’Loughlin, R. ORCID: https://orcid.org/0009-0003-7225-654X and Ransford, T. (2025) Double-layer potentials, configuration constants, and applications to numerical ranges. International Mathematics Research Notices, 2025 (8). rnaf084. ISSN 1687-0247 doi: 10.1093/imrn/rnaf084

Abstract/Summary

Given a compact convex planar domain Ω with non-empty interior, the classical Neu- mann’s configuration constant cR(Ω) is the norm of the Neumann-Poincar´e operator KΩ acting on the space of continuous real-valued functions on the boundary ∂Ω, modulo constants. We in- vestigate the related operator norm cC(Ω) of KΩ on the corresponding space of complex-valued functions, and the norm a(Ω) on the subspace of analytic functions. This change requires intro- duction of techniques much different from the ones used in the classical setting. We prove the equality cR(Ω) = cC(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 )∥ ≤ K supz∈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 ≤ 1 + √2 can be improved to K ≤ 1 + p1 + 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.

Item Type	Article
URI	https://centaur.reading.ac.uk/id/eprint/128285
Identification Number/DOI	10.1093/imrn/rnaf084
Refereed	Yes
Divisions	Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
Publisher	Oxford University Press
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