Mean ergodic theorems in $L^r(\mu )$ and $H^r(\mathbb{T})$, $0<r<1$

Let $T$ be the Koopman operator of a measure preserving transformation $\theta$ of a probability space $(X,\mathrm{\Sigma },\mu )$. We study the convergence properties of the averages $M_{n}f:=\frac{1}{n}{\sum }_{k=0}^{n-1}{T}^{k}f$ when $f\in L^r(\mu )$, $0<r<1$. We prove that if $\int \mathrm{\mid }{M}_{n}f{\mathrm{\mid }}^{r}d\mu \to 0$, then $f\in \overline{(I-T)L^r}$, and show that the converse fails whenever $\theta$ is ergodic aperiodic. When $\theta$ is invertible ergodic aperiodic, we show that for $0<r<1$ there exists $f_r\in (I-T)L^r$ for which $M_n{f}_r$ does not converge a.e. (although $\int \mathrm{\mid }{M}_{n}f{\mathrm{\mid }}^{r}d\mu \to 0$). We further establish that for $1\le p<\frac{1}{r},$ there is a dense $G_{\delta }$ subset $\mathcal{F}\subset L^p(X,\mu )$ such that ${\mathrm{lim\hspace{0.17em}sup}}_n\frac{\mathrm{\mid }{T}^{n}h\mathrm{\mid }}{n^r}=\mathrm{\infty }$ a.e. for any $h\in \mathcal{F}$.