OP-00235 Solution claimed

A dimension-independent polynomial norm bound from super-identical pseudospectra

For $A \in \mathbb{C}^{N \times N}$ let $s_1(A) \ge \dots \ge s_N(A)$ be its singular values, and let $\|\cdot\|_2$ be the spectral norm. Matrices $A, B \in \mathbb{C}^{N \times N}$ have super-identical pseudospectra, written $A \sim_{\mathrm{sip}} B$, if $s_j(A - zI) = s_j(B - zI)$ for all $z \in \mathbb{C}$ and $1 \le j \le N$. Such matrices are similar (Armentia, Gracia and Velasco), so $p(A) = 0$ exactly when $p(B) = 0$, for every polynomial $p$. Let $$C_N = \sup\left\{\frac{\|p(A)\|_2}{\|p(B)\|_2} : A, B \in \mathbb{C}^{N \times N},\ A \sim_{\mathrm{sip}} B,\ p \in \mathbb{C}[z],\ p(B) \ne 0\right\}.$$ Fortier Bourque and Ransford proved $C_N \le \sqrt N$, and $C_N = 1$ for $N \le 3$ (see Ransford's survey). Ransford and Walsh proved $C_N \le \sqrt{N-2}$ for $N \ge 4$, with equality for $N = 4$.

Problem. Determine whether $\sup_{N \ge 1} C_N < \infty$, that is, whether there is an absolute constant $C$ such that $\|p(A)\|_2 \le C\|p(B)\|_2$ for every $N \ge 1$, all $A, B \in \mathbb{C}^{N \times N}$ with $A \sim_{\mathrm{sip}} B$ and every polynomial $p$.

Source: Fortier Bourque and Ransford (2009), after Theorem 1.3 (as reported by OpenProblemsInNLA); also Pal and Yakubovich (2017), Introduction. Stated with the constants C_N by OpenProblemsInNLA.

Imported from github.com/ajt60gaibb/OpenProblemsInNLA.

Classification

Primary: NA — Numerical Analysis
Secondary: FA, SP
Keywords: matrix polynomials, OpenProblemsInNLA, pseudospectra, singular values
Literature checked: Oct. 8, 2026 by clemens-admin

References

@article{fortierbourque2009,
  author  = {Fortier Bourque, Maxime and Ransford, Thomas},
  title   = {Super-identical pseudospectra},
  journal = {J. London Math. Soc. (2)},
  volume  = {79},
  number  = {2},
  year    = {2009},
  pages   = {511--528},
  doi     = {10.1112/jlms/jdn085},
}
@article{pal2017,
  author        = {Pal, Avijit and Yakubovich, Dmitry V.},
  title         = {Infinite-dimensional features of matrices and pseudospectra},
  journal       = {J. Math. Anal. Appl.},
  volume        = {447},
  number        = {1},
  year          = {2017},
  pages         = {109--127},
  doi           = {10.1016/j.jmaa.2016.09.064},
  eprint        = {1609.08325},
  archivePrefix = {arXiv},
}
@article{ransford2022,
  author        = {Ransford, Thomas and Walsh, Nathan},
  title         = {A four-mean theorem and its application to pseudospectra},
  journal       = {Positivity},
  volume        = {26},
  number        = {4},
  year          = {2022},
  pages         = {Paper No. 70},
  doi           = {10.1007/s11117-022-00928-8},
  eprint        = {2109.14472},
  archivePrefix = {arXiv},
}
@article{ransford2010,
  author  = {Ransford, Thomas},
  title   = {Pseudospectra and matrix behaviour},
  journal = {Banach Center Publ.},
  volume  = {91},
  year    = {2010},
  pages   = {327--338},
  doi     = {10.4064/bc91-0-19},
}
@article{armentia2012,
  author  = {Armentia, Gorka and Gracia, Juan-Miguel and Velasco, Francisco E.},
  title   = {Identical pseudospectra of any geometric multiplicity},
  journal = {Linear Algebra Appl.},
  volume  = {436},
  number  = {6},
  year    = {2012},
  pages   = {1683--1688},
  doi     = {10.1016/j.laa.2011.01.014},
}
@misc{openproblemsinnla-mf24,
  author       = {Townsend, Alex},
  title        = {{MF-24}: A dimension-independent polynomial norm bound from super-identical pseudospectra},
  howpublished = {Open Problems in Numerical Linear Algebra, GitHub repository},
  year         = {2026},
  url          = {https://github.com/ajt60gaibb/OpenProblemsInNLA/blob/main/matrix-functions-and-stability/MF-24/README.md},
  note         = {Read at commit eb7c9d6, 2026-10-07},
}

Resolution claims

Unbounded polynomial norm ratios for super-identical pseudospectra v1 GitHub ↗

Georg Maierhofer

We give explicit nonnegative nilpotent weighted shifts of order $(m+1)^2$ whose singular values agree after every complex scalar shift, but whose norms under a common polynomial have ratio at least $(4/5)\sqrt m$. Thus no dimension-independent comparison constant exists. A two-by-two transfer identity proves equality of the shifted singular values; a residue-class decomposition gives the norm bounds. The proof uses only …

Disproof · not verified · fidelity not established · posted on GitHub 15 Sep 2026

Status history

When	Change	By	Reason
2026-10-08	Needs verification → Solution claimed	clemens-admin	Listed with the record of its solution, unreviewed.

Citing this problem

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OP-00235, "A dimension-independent polynomial norm bound from super-identical pseudospectra", clopen.solutions.