Composition Semigroups on the Besov Spaces

Abstract

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, $[\ddot{I}{†}_t,X]$, exists for every semigroup $\{\ddot{I}{†}_t\}$ of analytic self-maps of the disk, and the question whether $[\ddot{I}{†}_t,X]$ equals $X$ itself has an answer independent of $\{\ddot{I}{†}_t\}$. Such is the case for the Hardy and Bergman spaces, Bloch, BMOA, and $H^{\infty }$. For the disk algebra $A$, $[\ddot{I}{†}_t,A]=A$ precisely when $\{\ddot{I}{†}_t\}\subset A$. For $B_p$ with $p\ge 2$, every $\{\ddot{I}{†}_t\}\subset B^p$ and always $[\ddot{I}{†}_t,B_p]=B_p$, but this fails when $1<p<2$. We give an example where $\{\ddot{I}{†}_t\}\subset B_p$ and yet the induced composition operators $\{C_t\}$ are not bounded on $B_p$ and we do not know if $[\ddot{I}{†}_t,B_p]$ 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 $\{C_t\}$, $0\le t\le 1$, we characterize the semigroups $\{\ddot{I}{†}_t\}$ such that $[\ddot{I}{†}_t,B_p]=B_p$.

References in corpus (1)

• Composition Semigroups on BMOA and $H^{\infty }$