Abstract
We prove bounds for a class of homomorphisms arising in the study of spectral sets, by involving extremal functions and vectors. These are used to recover three celebrated results on spectral constants by Crouzeix–Palencia, Okubo–Ando and von Neumann in a unified way and to refine a recent result by Crouzeix–Greenbaum.
The second named author has been supported by the Dutch Research Council (NWO) grant OCENW.M20.292.
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Notes
- 1.
In the literature a \(\kappa \)-spectral set is typically instead defined by requiring that (1) holds for all rational functions p with poles off \(W^{-}\) and such that the spectrum of M is contained in \(W^{-}\). For the applications we have in mind, considering polynomials is however sufficient.
- 2.
This mapping, however, fails to be multiplicative in general.
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Acknowledgements
We would like to thank Georgios Tsikalas (Washington University, St. Louis) for the enriching communication. We are also grateful for the careful reading of the anonymous reviewer.
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Schwenninger, F.L., de Vries, J. (2024). On Abstract Spectral Constants. In: Ptak, M., Woerdeman, H.J., Wojtylak, M. (eds) Operator and Matrix Theory, Function Spaces, and Applications. IWOTA 2022. Operator Theory: Advances and Applications, vol 295. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-50613-0_15
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