Summability of Hankel matrix entries

DefinitionScottish Book Problem 88 (Mazur)#✓fully proved

The claim that every real sequence \((a_n)_{n \ge 0}\) with the following property satisfies \(\sum_n n |a_n| < \infty\): for every real sequence \((x_n)_{n \ge 0}\) with \(|x_n| \le 1\) for all \(n\), each series \(\sum_n a_{m+n} x_n\) with \(m \ge 0\) converges, as the limit of its partial sums, to some \(L_m\), and \(\sum_m |L_m| < \infty\).

The inner series is required to converge as a limit of partial sums, not absolutely, as in Mazur's original question.

def ScottishBookProblem88Claim : Prop :=
  ∀ a : ℕ → ℝ,
    (∀ x : ℕ → ℝ, (∀ n, |x n| ≤ 1) →
      ∃ L : ℕ → ℝ,
        (∀ m, Filter.Tendsto
          (fun N => ∑ n ∈ Finset.range N, a (m + n) * x n)
          Filter.atTop (nhds (L m))) ∧
        Summable (fun m => |L m|)) →
    Summable (fun n : ℕ => (n : ℝ) * |a n|)

Used by 1

Used byMazur's Scottish Book Problem 88 fails

TheoremMazur's Scottish Book Problem 88 fails#●ready to prove

The Scottish Book Problem 88 claim is false: there is a real sequence \((a_n)_{n \ge 0}\) such that for every real sequence \((x_n)_{n \ge 0}\) with \(|x_n| \le 1\) for all \(n\), each series \(\sum_n a_{m+n} x_n\) with \(m \ge 0\) converges, as the limit of its partial sums, to some \(L_m\), and \(\sum_m |L_m| < \infty\), but \(\sum_n n |a_n|\) diverges.

theorem_wanted mazur88_refuted : ¬ ScottishBookProblem88Claim

Sources

• S. Mazur, Problem 88 of the Scottish Book; negative solution as an immediate consequence of Peller (1982), J. Operator Theory 7, 341-372, §5, via V. V. Peller, "On S. Mazur's Problems 8 and 88 from the Scottish Book", arXiv:math/0703716: a Hankel matrix in the injective tensor product need not have finite sum of moduli of its entries.

Uses 1

Statement usesScottish Book Problem 88 (Mazur)

1 fully proved1 ready to prove

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