  Let $\mathcal{H}$ be a complex Hilbert space and $T:\mathcal{H}\to
\mathcal{H}$ be a contraction. Let $$A_nf=\frac{1}{n}\sum_{j=1}^nT^jf$$ for
$f\in \mathcal{H}$. Let $(n_k)$ be a sequence satisfying $\beta \geq
n_{k+1}/n_k\geq \alpha >1$ for all $k\geq 1$, then there exists a constant
$C_1>0$ such that
$$\sum_{k=1}^\infty\|A_{n_{k+1}}f-A_{n_k}f\|_{\mathcal{H}}\leq
C_1\|f\|_{\mathcal{H}}$$ for all $f\in \mathcal{H}$. Let $(n_k)$ be a sequence
satisfying $\beta \geq n_{k+1}/n_k\geq \alpha >1$ for all $k\geq 1$, and let
$M$ be any sequence. Then there exists a constant $C_2>0$ such that
$$\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in
M}}\|A_m(T)f-A_{n_k}(T)f\|_{H}\leq C_2\|f\|_{\mathcal{H}}$$ for all $f\in
\mathcal{H}$.

Classical Analysis and ODEs