AI 中文总结
本文研究巴拿赫空间上均值L稳定算子的幂增长，在不同空间与算子类中得到幂增长界，构造了满足特定性质的算子，回答了相关学者提出的问题。
AI 中文摘要
我们研究巴拿赫空间上均值L稳定算子的幂增长。对于线性算子，均值L稳定性等价于密度一致有界性，这在任意巴拿赫空间上给出\\(\\|T^n\\|=O(n)\\)。在希尔伯特空间上，我们证明均值L稳定性等价于绝对切萨罗有界性，并得到\\(\\|T^n\\|=O(n^{1/2-c_T})\\)，其中\\(c_T>0\\)；对于抽象\\(L^p\\)空间上的正算子，我们类似地得到\\(\\|T^n\\|=O(n^{1/p-c_T})\\)，且两种情形下正的\\(c_T\\)均无法对所有此类算子统一选取。对于p凸巴拿赫格上的正算子，我们证明其界为\\(O(n^{1/p})\\)，并构造正混合算子\\(T_p\\)使得\\(\\|T_p^n\\|\asymp n^{1/p}\\)；这些算子满足一致弱\\((p,p)\\)估计，同时\\(\\|T_p^nx\\|^s\\)的平均值：当\\(s<p\\)时为有界，\\(s=p\\)时阶为\\(\log N\\)，\\(s>p\\)时阶为\\(N^{s/p-1}\\)。算子\\(T_1\\)是一致Kreiss有界的，且具有线性幂增长，回答了Montes-Rodríguez、Sánchez-Álvarez与Zemánek（2005）提出的问题；此外，\\(T_1\\)是均值L稳定且均值Li-Yorke混沌的，但并非分布混沌，回答了Bernardes、Bonilla与Peris（2020）提出的问题。
英文摘要
We study the growth of powers of mean-L-stable operators on Banach spaces. We show that, for linear operators, mean-L-stability is equivalent to uniform boundedness in density; this yields $\|T^n\|=O(n)$ on every Banach space. On Hilbert spaces we prove that mean-L-stability is equivalent to absolute Cesàro boundedness and obtain $\|T^n\|=O(n^{1/2-c_T})$ for some $c_T>0$. For positive mean-L-stable operators on abstract $L^p$-spaces, $1\le p<\infty$, we similarly obtain $\|T^n\|=O(n^{1/p-c_T})$, where in both cases the positive constant $c_T$ cannot be chosen uniformly over all such operators. For positive mean-L-stable operators on $p$-convex Banach lattices, we prove the bound $O(n^{1/p})$ and construct positive topologically mixing operators $T_p$ for which $\|T_p^n\|\asymp n^{1/p}$, where $1\leq p<\infty$. These operators satisfy a uniform weak $(p,p)$ orbit estimate, while the averages of $\|T_p^nx\|^s$ are bounded for $s<p$, of order $\log N$ for $s=p$, and of order $N^{s/p-1}$ for $s>p$. The operator $T_1$ is uniformly Kreiss bounded and has linear power growth, answering a question of Montes-Rodríguez, Sánchez-Álvarez and Zemánek (2005). Moreover, $T_1$ is mean-L-stable and mean Li--Yorke chaotic, while it is not distributionally chaotic. This answers a question of Bernardes, Bonilla and Peris (2020).
Comments29 pages
