贝索夫空间上的复合半群
Composition Semigroups on the Besov Spaces
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中文总结 AI 辅助
研究贝索夫空间\(B_p\)上复合算子半群，对比其与经典空间特性差异，给出\(p \geq 2\)及\(1 < p < 2\)时不同情况，给出反例，在算子范数有界假设下刻画使\([\varphi_t, B_p] = B_p\)的半群\(\{\varphi_t\}\)。
中文摘要 AI 辅助
我们研究作用于贝索夫空间\(B_p\)的复合算子半群，它们相对于许多经典空间呈现出一些新特性。对于单位圆盘上解析函数的巴拿赫空间\(X\)，通常对圆盘的每个解析自映射半群\(\{\varphi_t\}\)，强连续性的最大闭空间\([\varphi_t, X]\)存在，且\([\varphi_t, X]\)是否等于\(X\)本身的答案与\(\{\varphi_t\}\)无关，哈代空间、伯格曼空间等就是如此。对于圆盘代数\(A\)，当\(\{\varphi_t\} \subset A\)时\([\varphi_t, A] = A\)。对于\(p \geq 2\)的\(B_p\)，每个\(\{\varphi_t\} \subset B^p\)时总有\([\varphi_t, B_p] = B_p\)，但\(1 < p < 2\)时不成立。我们给出一个例子，其中\(\{\varphi_t\} \subset B_p\)但诱导的复合算子\(\{C_t\}\)在\(B_p\)上无界，且不知\([\varphi_t,B_p]\)是否存在，若存在也不等于\(B_p\)。在\(\{C_t\}\)的算子范数有一致界的假设下，我们刻画了使得\([\varphi_t, B_p] = B_p\)的半群\(\{\varphi_t\}\)。
英文摘要
We study semigroups of composition operators acting on the Besov spaces $B_p$, where they exhibit some new behaviors relative to many classical spaces. Often for a Banach space $X$ of analytic functions on the unit disk, the maximal closed space of strong continuity, $[ φ_t, X ]$, exists for every semigroup $\{ φ_t \}$ of analytic self-maps of the disk, and the question whether $[φ_t , X ]$ equals $X$ itself has an answer independent of $\{φ_t\}$. Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and $H^{\infty}$. For the disk algebra $A$, $[φ_t , A ] = A$ precisely when $\{φ_t\} \subset A$. For $B_p$ with $p \geq 2$, every $\{φ_t\} \subset B_p$ and always $[ φ_t, B_p ] = B_p$, but this fails when $1 < p < 2$. We give an example where $\{φ_t\} \subset B_p$ and yet the induced composition operators $\{C_t\}$ are not bounded on $B_p$ and we do not know if $[φ_t,B_p]$ exists. If it does exist, it cannot be equal to $B_p$. Under the hypothesis that there is a uniform bound for the operator norms of the $\{C_t\}$, $0 \leq t \leq 1$, we characterize the semigroups $\{ φ_t \}$ such that $[ φ_t, B_p ] = B_p$.
发表机构
- University of Hawaii（夏威夷大学）
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