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The algebraic numerical range as a spectral set in Banach algebras

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

TL;DR

The paper investigates when the algebraic numerical range $V(T)$ forms a $C$-spectral set in Banach algebras by introducing the spectral constant $\Psi(T)$ and its algebraic counterpart $\Psi_{\mathcal A}$. It establishes finiteness of $\Psi(T)$ for algebraic elements in any unital Banach algebra, and presents concrete positive results for matrix algebras, including a finite constant for complex $2\times2$ matrices with the induced $1$-norm, as well as infinite-dimensional algebras such as $\mathcal B(H)$ and the Calkin algebra where $\Psi$ is bounded by the Crouzeix constant. The work also constructs explicit counterexamples showing $\Psi(A)=\infty$ for shifts on $\ell^p$ and certain combinatorial Banach spaces, highlighting fundamental limitations beyond the Hilbert-space setting. It concludes with open questions about uniform boundedness of $\Psi$ in broader classes, the possibility of polynomially bounded operators with infinite $\Psi$, and the behavior on combinatorial spaces like Schreier, pointing toward rich avenues for further study.

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.

The algebraic numerical range as a spectral set in Banach algebras

TL;DR

The paper investigates when the algebraic numerical range forms a -spectral set in Banach algebras by introducing the spectral constant and its algebraic counterpart . It establishes finiteness of for algebraic elements in any unital Banach algebra, and presents concrete positive results for matrix algebras, including a finite constant for complex matrices with the induced -norm, as well as infinite-dimensional algebras such as and the Calkin algebra where is bounded by the Crouzeix constant. The work also constructs explicit counterexamples showing for shifts on and certain combinatorial Banach spaces, highlighting fundamental limitations beyond the Hilbert-space setting. It concludes with open questions about uniform boundedness of in broader classes, the possibility of polynomially bounded operators with infinite , and the behavior on combinatorial spaces like Schreier, pointing toward rich avenues for further study.

Abstract

We investigate when the algebraic numerical range is a -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 -matrices with the induced -norm. Furthermore, we discuss positive results for infinite-dimensional Banach algebras, including the Calkin algebra.

Paper Structure

This paper contains 8 sections, 16 theorems, 113 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Spectral constants
  4. Finite dimensional algebras
  5. Infinite-dimensional algebras with finite constant Ψ(A)
  6. Algebras with infinite constant Ψ(A)
  7. l1-induced norm
  8. Conclusions

Key Result

Lemma 2.1

Let $\mathcal{A}$ be a unital Banach algebra. For all $S,T\in\mathcal{A}$ it holds that In particular, the mapping $T\mapsto V(T)$ is uniformly continuous.

Figures (3)

  • Figure 1: Illustration of the Gershgorin disk $D(t'_{j,j}, \sum_{k=1, k\neq j}^{\infty} | t_{k,j}'|)$ together the tangent line $l_{j,\theta}$. Its radius and the distance from 0 to its center is highlighted. The left picture shows an example when $\text{Re}(t_{j,j}')$ is positive and the right one corresponds to the negative case.
  • Figure 2: Illustration of the two disks $D(-ac,|c|^2)$ and $D(ac,|a|^2)$ with highlighted trapezoid formed by their common tangent and radii drawn to this tangent. The circle with dotted line illustrates the disk $D\left(0,\frac{|a|^2 +|c|^2}{2}\right)$ which is contained in the closure of the convex hull of $D(-ac,|c|^2)$ and $D(ac,|a|^2)$.
  • Figure 3: Illustration of the disks $D_1:=D(ad, |cd|)$ and $D_2:=D(-bc,|ab|)$ with highlighted trapezoid formed by their common tangent $l$ closest to 0 and radii drawn to it.

Theorems & Definitions (35)

  • Lemma 2.1
  • proof
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • proof
  • Proposition 3.3
  • proof
  • Remark 3.4
  • Theorem 4.1
  • ...and 25 more