The algebraic numerical range as a spectral set in Banach algebras

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

doi:10.4153/S0008414X25000124

The algebraic numerical range as a spectral set in Banach algebras

Hanna Blazhko, Daniil Homza, Felix L. Schwenninger, Jens De Vries^*, Michał Wojtylak

^*Corresponding author for this work

Research output: Contribution to journal › Article › Academic › peer-review

4 Downloads (Pure)

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
Journal	Canadian journal of mathematics
DOIs	https://doi.org/10.4153/S0008414X25000124
Publication status	E-pub ahead of print/First online - 17 Feb 2025

Keywords

UT-Hybrid-D
Banach algebra
Crouzeix's conjecture
functional calculus
spectral constant
Algebraic numerical range

Access to Document

10.4153/S0008414X25000124Licence: CC BY

the-algebraic-numerical-range-as-a-spectral-set-in-banach-algebrasFinal published version, 554 KBLicence: CC BY

Cite this

@article{5b2d605af2874b7cb653d84be40f6ea9,

title = "The algebraic numerical range as a spectral set in Banach algebras",

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.",

keywords = "UT-Hybrid-D, Banach algebra, Crouzeix's conjecture, functional calculus, spectral constant, Algebraic numerical range",

author = "Hanna Blazhko and Daniil Homza and Schwenninger, {Felix L.} and {De Vries}, Jens and Micha{\l} Wojtylak",

note = "Publisher Copyright: {\textcopyright} CanadianMathematical Society 2020.",

year = "2025",

month = feb,

day = "17",

doi = "10.4153/S0008414X25000124",

language = "English",

journal = "Canadian journal of mathematics",

issn = "0008-414X",

publisher = "Canadian Mathematical Society",

}

TY - JOUR

T1 - The algebraic numerical range as a spectral set in Banach algebras

AU - Blazhko, Hanna

AU - Homza, Daniil

AU - Schwenninger, Felix L.

AU - De Vries, Jens

AU - Wojtylak, Michał

N1 - Publisher Copyright: © CanadianMathematical Society 2020.

PY - 2025/2/17

Y1 - 2025/2/17

N2 - 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.

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.

KW - UT-Hybrid-D

KW - Banach algebra

KW - Crouzeix's conjecture

KW - functional calculus

KW - spectral constant

KW - Algebraic numerical range

UR - http://www.scopus.com/inward/record.url?scp=85219689009&partnerID=8YFLogxK

U2 - 10.4153/S0008414X25000124

DO - 10.4153/S0008414X25000124

M3 - Article

AN - SCOPUS:85219689009

SN - 0008-414X

JO - Canadian journal of mathematics

JF - Canadian journal of mathematics

ER -

The algebraic numerical range as a spectral set in Banach algebras

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

The algebraic numerical range as a spectral set in Banach algebras

Cite this