The algebraic numerical range as a spectral set in Banach algebras

TL;DR

This paper explores conditions under which the algebraic numerical range forms a spectral set in Banach algebras, presenting counterexamples and positive results for matrix and infinite-dimensional algebras.

Contribution

It provides new insights into when the algebraic numerical range is a spectral set, including explicit constants for 2x2 matrices and results for infinite-dimensional cases.

Findings

Counterexamples based on classical and combinatorial Banach spaces.

Positive results for matrix algebras, including a constant for 2x2 matrices.

Results for infinite-dimensional Banach algebras like the Calkin algebra.

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 $2\times2$-matrices with the induced $1$-norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.