AI 中文总结
本文在 $L^1(0,1)$ 上构造了一个新的真的大闭理想，回答了 Johnson 等人关于存在性的问题，并证明了 Dunford-Pettis 理想无右逼近单位，其证明结合了平移平均、Rajchman 测度与无原子分解。
AI 中文摘要
我们证明了在 $L^1(0,1)$ 上两个 Dunford-Pettis 算子乘积的范数闭线性张成严格位于可表示算子与 Dunford-Pettis 算子之间。这给出了 $\mathcal{B}(L^1(0,1))$ 中一个额外的真的大闭理想，回答了 Johnson、Pisier 和 Schechtman 的存在性问题。Dunford-Pettis 理想对此平方理想的商包含一个等距同构于 $L^1(0,1)$ 的合同补余的等距拷贝。我们还证明了 Dunford-Pettis 理想没有右逼近单位，回答了 Johnson 和 Schechtman 的一个问题。更精确地，一个正范数一的卷积算子 $R$ 满足对每个 Dunford-Pettis 算子 $T$ 有 $\\|R-RT\\|\geq1$ 和 $\\|R-TR\\|\geq1$。证明结合了平移平均、支撑在强独立集上的 Rajchman 测度，以及正算子的两个无原子分解。逼近单位障碍推广到每个具有非零无原子部分的有限测度空间。最后，我们证明一个理想的定量乘法估计传递到其所有闭幂理想。
英文摘要
We show that the norm-closed linear span of products of two Dunford-Pettis operators on $L^1(0,1)$ lies strictly between the representable and the Dunford-Pettis operators. This gives an additional proper large closed ideal in $\mathcal{B}(L^1(0,1))$, answering the existence question of Johnson, Pisier and Schechtman. The quotient of the Dunford-Pettis ideal by this square ideal contains a contractively complemented isometric copy of $L^1(0,1)$. We also prove that the Dunford-Pettis ideal has no right approximate identity, answering a question of Johnson and Schechtman. More precisely, a positive norm-one convolution operator $R$ satisfies $\|R-RT\|\geq1$ and $\|R-TR\|\geq1$ for every Dunford-Pettis operator $T$. The proofs combine translation averaging, Rajchman measures supported on a strongly independent set, and two atomless disintegrations of positive operators. The approximate-identity obstruction extends to every finite measure space with a nonzero atomless part. Finally, we prove that quantitative multiplication estimates for an ideal pass to all its closed power ideals.
Comments11 pages
