Fix 0<alpha,r<∞. For every 0<p,q≤∞, determine whether, for every f∈ C^1([-1,1]) satisfying f(0)=0 and |f'(t)|lesssim |t|^alpha, and every real sequence lambda∈ell^r with |lambda|_ell^∞≤ 1, the Schur--Hadamard multiplier S_Psi_f,lambda is bounded from S^q to S^p, where Psi_f,lambda=(fracf(lambda_i)-f(lambda_j)lambda_i-lambda_j)_i,j with diagonal entries Psi_f,lambda(i,i)=f'(lambda_i). Arazy's conjecture asserts that this boundedness holds for every such f and lambda exactly when frac1p≤fracalphar+min1,frac1q.

Arazy’s conjecture concerning Schur multipliers

Fix $0<\alpha,r<\infty$ . For every $0<p,q\leq\infty$ , determine whether, for every $f\in C^1([-1,1])$ satisfying $f(0)=0$ and $|f'(t)|\lesssim |t|^\alpha$ , and every real sequence $\lambda\in\ell^r$ with $\|\lambda\|_{\ell^\infty}\leq 1$ , the Schur--Hadamard multiplier $S_{\Psi_{f,\lambda}}$ is bounded from $\mathcal{S}^q$ to $\mathcal{S}^p$ , where $\Psi_{f,\lambda}=\left(\frac{f(\lambda_i)-f(\lambda_j)}{\lambda_i-\lambda_j}\right)_{i,j}$ with diagonal entries $\Psi_{f,\lambda}(i,i)=f'(\lambda_i)$ . Arazy's conjecture asserts that this boundedness holds for every such $f$ and $\lambda$ exactly when $\frac{1}{p}\leq\frac{\alpha}{r}+\min\left\{1,\frac{1}{q}\right\}$ .

References

Primary source

arXiv

Additional references

Arazy's conjecture concerning Schur multipliers: revisited and resolved — arXiv — Jinghao Huang, Fedor Sukochev

Posted 3 weeks ago · edited 3 weeks ago

Progress summary

Refreshed 3 weeks ago

Claimed solved

A new unrefereed preprint claims to complete the classification of when these matrix transformations work between Schatten classes, but the result has not been independently verified.

Arazy’s conjecture concerns the exact parameter range for bounded Schur multipliers from $S^q$ to $S^p$ . Earlier work settled important same-exponent and divided-difference cases; the new claim addresses the remaining endpoints and quasi-Banach regimes.

Known results

Potapov and Sukochev (2011) established the operator-Lipschitz consequence for divided-difference multipliers on $S^p$ , $1<p<\infty$ .
A paper published online on October 26, 2023, proved a broad criterion for Schur multipliers on $S_p$ and stated that it includes Arazy’s $S_p$ -multiplier conjecture.

September 2026 claimed resolution

Jinghao Huang and Fedor Sukochev’s preprint Arazy's conjecture concerning Schur multipliers: revisited and resolved claims that boundedness from $S^q$ to $S^p$ holds exactly under the conjectured parameter inequality, including previously untreated endpoint and quasi-Banach cases. This is presented as a complete resolution, but it remains unrefereed and unverified.

Current status (as of September 2026): Earlier partial cases are established, while the complete $S^q$ -to- $S^p$ classification is only claimed in the unrefereed preprint and remains unverified.

Sources

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