Composition Semigroups on the Besov Spaces

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