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PreprintCC BY 4.0arXiv

Spectral constants for algebraic numerical ranges

  • Tomasz Kania
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Abstract

We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If Γn denotes the universal constant for algebraic elements of degree at most n, then Γ1=1, 2n−1leqslantΓn<∞ (ngeqslant2). Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain 3leqslantΓ2 leqslant√(1+(2e−1)2)<4.55. For unital C*-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang--Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.

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Cite

BibTeXRIS
BibTeX
@misc{kania2026spectral,
  title = {{Spectral constants for algebraic numerical ranges}},
  author = {Tomasz Kania},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.18963},
  eprint = {2609.18963},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.18963},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Spectral constants for algebraic numerical ranges
AU  - Tomasz Kania
PY  - 2026
DA  - 2026/09/01
PB  - arXiv
DO  - 10.48550/arxiv.2609.18963
UR  - https://arxiv.org/abs/2609.18963
AB  - We study spectral-set constants associated with the algebraic numerical range in unital Banach algebras. If Γ_n denotes the universal constant for algebraic elements of degree at most n, then Γ_1=1, 2n−1leqslantΓ_n<∞ (ngeqslant2). Thus the algebraic numerical range is a spectral set with a constant depending only on the algebraic degree. In degree two we obtain 3leqslantΓ_2 leqslant√(1+(2e−1)^2)<4.55. For unital C^*-algebras we prove a gap theorem: the algebra-level constant is one precisely in the commutative case, whereas every non-commutative algebra has constant at least two. Moreover, the constant of the Jiang--Su algebra is exactly the universal Crouzeix constant. In the opposite direction, we construct a norm-one operator with a contractive polynomial calculus on the unit disc but infinite numerical-range spectral constant, and we show that the canonical left shift on every spreading combinatorial space in a broad class has infinite constant. These results settle the three questions posed by Blazhko, Homza, Schwenninger, de Vries, and Wojtylak.
ER  -