Möbius transformations of the disc and one-parameter groups of isometries of $H^{p}$

Abstract:

Let $\{ {T_t}\}$ be a strongly continuous one-parameter group of isometries in ${H^p}(1 \leqslant p < \infty ,p \ne 2)$ with unbounded generator. There is a uniquely determined one-parameter group $\{ {\phi _t}\} ,t \in {\mathbf {R}}$, of Möbius transformations of the (open) disc $D$ corresponding to $\{ {T_t}\}$ by way of Forelli’s theorem. The interplay between $\{ {T_t}\}$ and $\{ {\phi _t}\}$ is studied, and the spectral properties of the generator $A$ of $\{ {T_t}\}$ are analyzed in this context. The nature of the set $S$ of common fixed points of the functions ${\phi _t}$ plays a crucial role in determining the behavior of $A$. The spectrum of $A$, which is a subset of $i{\mathbf {R}}$, can be a discrete set, a translate of $i{{\mathbf {R}}_ + }$ or of $i{{\mathbf {R}}_ - }$, or all of $i{\mathbf {R}}$. If $S$ is not a doubleton subset of the unit circle, $\{ {T_t}\}$ can be extended to a holomorphic semigroup of ${H^p}$-operators, the semigroup being defined on a half-plane. The treatment of $\{ {T_t}\}$ is facilitated by developing appropriate properties of one-parameter groups of Möbius transformations of $D$. In particular, such groups are in one-to-one correspondence (via an initial-value problem) with the nonzero polynomials $q$, of degree at most 2, such that $\text {Re} [\bar zq(z)] = 0$ for all unimodular $z$. A has an explicit description (in terms of the polynomial corresponding to $\{ {\phi _t}\}$) as a differential operator.