Y$, there exists a set $E$ of zero density such that $\lim_{n\to\infty,\;n\not\in E} \ell(n)n^{1-\beta}(T_nv)(y) = 
    d_\beta \int_Y v\,d\mu$.

-   If $v\ge0$, then $\liminf_{n\to\infty} \ell(n)n^{1-\beta}T_nv = d_\beta \int_Y v\,d\mu$ pointwise on $Y$.

\[rmk-Gouezel\] In general, Theorem \[thm-conv\] fails for $\beta\le\frac12$. However, there is the possibility of proving the result for all $\beta$ under additional hypotheses. Indeed, Gouëzel [@GouezelPC] informs us that he is able to prove Theorem \[thm-conv\] for all $\beta\in(0,1)$ under the additional assumption that $\mu(\varphi=n)=O(\ell(n)n^{-(\beta+1)}$). In particular, Gouëzel’s result applies to the family .

It is worth recalling the situation from the scalar case (where the $T_n$ are probabilities instead of operators). Under the additional assumption $\mu(\varphi=n)=O(\ell(n)n^{-(\beta+1)})$, Garsia & Lamperti [@GarsiaLamperti62] were able to extend Theorem \[thm-conv\] to the case $\beta\in(\frac14,\frac12)$. This is the only part of [@GarsiaLamperti62] that we are unable to generalise to the operator setting. However, an argument of Doney [@Doney97] applies to all $\beta\in(0,1)$ and according to Gouëzel [@GouezelPC] this argument can be extended to the operator case.

\[rmk-DK\] An immediate consequence of Theorem \[thm-conv\] is that $Y$ is a Darling-Kac set whenever $\beta>\frac12$. We refer to Aaronson [@Aaronson Chapter 3] and Aaronson, Denker & Urbanksi [@ADU93 Section 1] for definitions and numerous consequences. Other 