The algebraic numerical range as a spectral set in Banach algebras

Hanna Blazhko; Daniil Homza; Felix L. Schwenninger; Jens de Vries; Michał Wojtylak

doi:10.48550/arXiv.2410.10678

The algebraic numerical range as a spectral set in Banach algebras

Hanna Blazhko, Daniil Homza, Felix L. Schwenninger, Jens de Vries, Michał Wojtylak

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Abstract

We investigate when the algebraic numerical range is a C-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex 2×2-matrices with the induced 1-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.

Original language	English
Publisher	ArXiv.org
Number of pages	22
DOIs	https://doi.org/10.48550/arXiv.2410.10678
Publication status	Published - 14 Oct 2024

Keywords

math.FA
cs.NA
math.NA
47A25, 47A12, 47B48, 15A60

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ArticleFinal published version, 873 KBLicence: CC BY

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AB - We investigate when the algebraic numerical range is a C-spectral set in a Banach algebra. While providing several counterexamples based on classical ideas as well as combinatorial Banach spaces, we discuss positive results for matrix algebras and provide an absolute constant in the case of complex 2×2-matrices with the induced 1-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.

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The algebraic numerical range as a spectral set in Banach algebras

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Keywords

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