the unit circle $S^1$, so it is convenient to abuse notation, writing $T(\theta)$ instead of $T(e^{i\theta})$ and so on. For $\beta\in(\frac12,1)$, our treatment follows Garsia & Lamperti [@GarsiaLamperti62] but there are some significant differences in two of the three steps.

The first step, Section \[sec-T\], is to study the singularity for $T(\theta)$ at $\theta=0$. The argument in [@GarsiaLamperti62] is scalar and similar results can be found in [@Feller49; @Hardy31]. In our situation, the key is to use the fact that the return time $\varphi$ lies in the domain of a stable law, and a Nagaev-type argument due to Aaronson & Denker [@AaronsonDenker01] shows that $(1-\lambda(\theta))^{-1}\sim {\rm const}\, \ell(1/\theta)^{-1}\theta^{-\beta}$ when $\beta\in(0,1)$. By Corollary \[cor-H1\] and Proposition \[prop-PQ\], $T(\theta)\sim {\rm const}\, \ell(1/\theta)^{-1}\theta^{-\beta}P$.

In particular $T(\theta)\in L^1$ with Fourier coefficients $\widehat T_n$. The second step is to relate $T_n$ to $\widehat T_n$. In the scalar case, [@GarsiaLamperti62] invokes ideas of Herglotz [@Herglotz11] on analytic functions with positive real part. A different approach is required here since we are working with operators. In Section \[sec-Fourier\], we verify that $T_n=\widehat T_n$.

In the final step, Section \[sec-half\], we investigate the behaviour of $\widehat T_n$ as $n\to\infty$, directly following [@GarsiaLamperti62].

Tower extensions {#sec-tower}
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The following 