关于复 \(C(K)\) 空间的博洛巴斯定理
On the Bollobás theorem for complex $C(K)$-spaces
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中文总结 AI 辅助
研究复 \(C(K)\) 空间中算子的博洛巴斯型定理，结合实情形迭代法与复测度构造法，证明几乎达到范数的算子可被逼近，强化了约翰逊 - 沃尔夫密度定理复版本，紧致算子情形也有相同结论。
中文摘要 AI 辅助
设 \(K\) 和 \(S\) 为紧致豪斯多夫空间。我们证明了关于连续函数复空间之间算子的一个博洛巴斯型定理。具体而言，每个在初始函数处几乎达到其范数的算子都可以由一个达到范数的算子逼近，且达到范数的函数与原函数保持接近。我们的证明将先前在实情形下发展的迭代方法与复测度的最新构造相结合。特别地，我们的结果对约翰逊 - 沃尔夫密度定理的复版本进行了定量强化。作为我们构造的一个副产品，仅考虑紧致算子时也能得到相同结论。
英文摘要
Let $K$ and $S$ be compact Hausdorff spaces. We prove a Bollobás-type theorem for operators between complex spaces of continuous functions. More precisely, every operator that almost attains its norm at an initial function can be approximated by a norm-attaining operator whose norm-attaining function remains close to the original one. Our proof combines the iterative method developed previously in the real case with the recent construction for complex measures. In particular, our result provides a quantitative strengthening of the complex version of the Johnson-Wolfe density theorem. As a byproduct of our construction, the same conclusion can be obtained when considering only compact operators.
