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Spectraloid operator polynomials, the approximate numerical range and an Eneström-Kakeya theorem in Hilbert space

Jan Swoboda;  Harald K. Wimmer — 2010

Studia Mathematica

We study a class of operator polynomials in Hilbert space which are spectraloid in the sense that spectral radius and numerical radius coincide. The focus is on the spectrum in the boundary of the numerical range. As an application, the Eneström-Kakeya-Hurwitz theorem on zeros of real polynomials is generalized to Hilbert space.

Limiting configurations for solutions of Hitchin’s equation

Rafe Mazzeo;  Jan Swoboda;  Hartmut Weiß;  Frederik Witt

Séminaire de théorie spectrale et géométrie

We review recent work on the compactification of the moduli space of Hitchin’s self-duality equation. We study the degeneration behavior near the ends of this moduli space in a set of generic directions by showing how limiting configurations can be desingularized. Following ideas of Hitchin, we can relate the top boundary stratum of this space of limiting configurations to a Prym variety. A key role is played by the family of rotationally symmetric solutions to the self-duality equation on ℂ , which...

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