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Talagrand's Operator Cotype Problem

Talagrand's operator cotype problem (Talagrand 2021) concerns a bounded operator U:X->Y between Banach spaces. Its Gaussian cotype- C_q^g(U) and Rademacher cotype- C_q^r(U) are the least constants for which

(sum_(i)||Ux_i||^q)^(1/q)<=C_q^g(U)E||sum_(i)g_ix_i||
(1)
(sum_(i)||Ux_i||^q)^(1/q)<=C_q^r(U)E||sum_(i)epsilon_ix_i||,
(2)

where E denotes the expectation value, the g_i are independent standard Gaussian random variables, and the epsilon_i are independent Rademacher random variables. The (q,1)-summing norm ||U||_(q,1) is the least constant for which

(sum_(i)||Ux_i||^q)^(1/q)<=||U||_(q,1)max_(eta_i=+/-1)||sum_(i)eta_ix_i||.
(3)

Talagrand asked whether there is a universal constant L such that C_q^r(U)<=Lmax{C_q^g(U),||U||_(q,1)}.

Wu (2026) constructed counterexamples already for q=2, with

(C_2^r(U_n))/(max{C_2^g(U_n),||U_n||_(2,1)})>=csqrt(ln(n+1))
(4)

for an absolute constant c>0. Wu (2026) credits ChatGPT GPT-5.6 with discovering the counterexample and reports checking the proof. As of Sep. 22, 2026, independent specialist review had not been reported.

See also

Banach Space, Gaussian Cotype, Normal Distribution, Linear Operator, Operator Cotype, Rademacher Cotype

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References

Talagrand, M. Upper and Lower Bounds for Stochastic Processes: Decomposition Theorems, 2nd ed. Cham, Switzerland: Springer, 2021. https://doi.org/10.1007/978-3-030-82595-9.Wu, X. "A Counterexample to Talagrand's Operator Cotype Problem." 17 Sep 2026. https://arxiv.org/abs/2609.19731.

Cite this as:

Weisstein, Eric W. "Talagrand's Operator Cotype Problem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TalagrandsOperatorCotypeProblem.html

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