Variation and oscillation inequalities for operator averages on a complex Hilbert space

Abstract: 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 lacunary sequence, 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 $f\in \mathcal{H}$.\ \indent Let $(n_k)$ be a lacunary sequence, and let $\mathbb{N}$ be the set of natural numbers. Then there exists a constant $C_2>0$ such that $$\sum{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\m\in \mathbb{N}}}|A_m(T)f-A_{n_k}(T)f|{\mathcal{H}}\leq C_2|f|{\mathcal{H}}$$ for all $f\in \mathcal{H}$.