Kadets–Martín–Merí square-Daugavet question

Let $X$ be a real Banach space with $\dim X>1$ . The Daugavet property is the condition that $\|I+S\|=1+\|S\|$ for every rank-one operator $S\colon X\to X$ . The square-Daugavet problem asks whether each of the following conditions is equivalent to the Daugavet property: $\|I+T^{2}\|=1+\|T^{2}\|$ for every rank-one operator $T\colon X\to X$ , and $\|I-T^{2}\|=1+\|T^{2}\|$ for every rank-one operator $T\colon X\to X$ .

References

Primary source

arXiv

Additional references

Characterizing the Daugavet property by squares of rank-one operators — arXiv — Johann Langemets

banach-space-theory functional-analysis

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed solved

A September 2026 unrefereed preprint claims to settle the long-standing square-Daugavet question and a related question about extremely non-complex Banach spaces.

The question, posed in 2007, asks for a characterization involving squares of rank-one operators. The new paper claims an operator-theoretic characterization that answers it and the related question.

September 2026 preprint

Johann Langemets’s article Characterizing the Daugavet property by squares of rank-one operators claims both resolutions. The result is presented as new progress, but the preprint is unrefereed and no independent verification or objection was retrieved.

Current status (as of September 2026): The preprint claims the $2007$ square-Daugavet question and its linked question are settled, but the claims remain unverified.

Sources

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