Let A∈ M_n be a complex matrix, and let |·| denote the operator norm. A matrix is an idempotent if E^2=E, and a scalar is unimodular if its absolute value is 1. Direct-summand conjecture. If |A^k|=|A| qquadfor all k≥ 1, then some power of A has a direct summand of the form lambda E, where E is an idempotent and |lambda|=1. The paper states that this conjecture is disproved later in the work, so the claim is refuted rather than open.

The direct-summand conjecture for norm-preserving matrix powers

Less than 1 year old · traced to 2026

Let $A\in M_n$ be a complex matrix, and let $\|\cdot\|$ denote the operator norm. A matrix is an idempotent if $E^2=E$ , and a scalar is unimodular if its absolute value is $1$ .

Direct-summand conjecture. If

\|A^k\|=\|A\| \qquad\text{for all } k\geq 1,

then some power of $A$ has a direct summand of the form $\lambda E$ , where $E$ is an idempotent and $|\lambda|=1$ .

The paper states that this conjecture is disproved later in the work, so the claim is refuted rather than open.

References

Primary source

Hwa-Long Gau, Jia-Huo Hong, Chi-Kwong Li and Kuo-Zhong Wang, “Numerical radius of certain two-by-two block matrices”, arXiv:2606.08576 (2026).

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

A paper claims the conjecture is false by giving a counterexample, but that refutation has not been independently verified.

The conjecture asserts that a complex matrix whose powers all preserve its operator norm must eventually contain a unimodular scalar multiple of an idempotent as a direct summand. The cited paper says this assertion is disproved later in the work.

Claimed counterexample, June 2026

The cited paper claims a counterexample and therefore a refutation of the conjecture. The available scan does not establish independent verification of the construction or proof.

Current status (as of September 2026): The conjecture is claimed refuted by the cited paper, but the counterexample remains unverified in the retrieved record.

Sources

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