绝对Cesàro有界算子的自改进
Self-improvement for absolutely Cesáro bounded operators
浏览论文内容
中文总结 AI 辅助
本文证明了绝对Cesàro有界算子的自改进性质，给出了最优的指数范围，并在Banach空间和$L^p$空间上推广了相关结论，同时构造了满足特定谱条件但具有无界增长的反例。
中文摘要 AI 辅助
我们证明，每个具有常数$K>1$的$p$-绝对Cesàro有界算子对于$1\leq q<pK/(K-1)$是$q$-绝对Cesàro有界的。这个范围是最优的，即使对于正加权移位也是如此。在类型为$s\in (1,2]$的Banach空间上，我们证明每个绝对Cesàro有界算子对于每个$1\leq q<s$都是$q$-绝对Cesàro有界的，并且在$q=s$处具有对数估计。我们还证明了多项式增长界$\\| T^n \\| =O(n^{1/s-\varepsilon})$对于某个$\varepsilon>0$成立。在$L^p$-空间上，$1<p<\infty$，我们将从绝对Cesàro有界性到$p$-绝对Cesàro有界性的蕴含关系从正算子扩展到单个最终正算子。最后，对于每个$1\leq p<\infty$，我们在$\ell^1$上构造一个$p$-绝对Cesàro有界算子$T$，使得$\sigma(T)\subseteq\mathbb D\cup\{1\}$但$\limsup_{n\to\infty}\\|T^n(I-T)\\|=\infty$。我们还观察到该构造产生一个单一的算子，它对每个$1\leq p<\infty$都是$p$-绝对Cesàro有界的。
英文摘要
We prove that every $p$-absolutely Cesàro bounded operator with constant $K>1$ is $q$-absolutely Cesàro bounded for $1\leq q<pK/(K-1)$. This range is optimal, even for positive weighted shifts. On Banach spaces of type $s\in (1,2]$, we prove that every absolutely Cesàro bounded operator is $q$-absolutely Cesàro bounded for every $1\leq q<s$, with a logarithmic estimate at $q=s$. We also prove the polynomial growth bound $\| T^n \| =O(n^{1/s-\varepsilon})$ for some $\varepsilon>0$. On $L^p$-spaces, $1<p<\infty$, we extend the implication from absolute Cesàro boundedness to $p$-absolute Cesàro boundedness from positive to individually eventually positive operators. Finally, for every $1\leq p<\infty$, we construct a $p$-absolutely Cesàro bounded operator $T$ on $\ell^1$ such that $σ(T)\subseteq\mathbb D\cup\{1\}$ but $\limsup_{n\to\infty}\|T^n(I-T)\|=\infty$. We also observe that the construction yields a single operator which is $p$-absolutely Cesàro bounded for every $1\leq p<\infty$.
