An oscillation inequality on a complex Hilbert space

Abstract

Let $T$ be a contraction on a complex Hilbert space $\mathcal{H}$, and for $f\in \mathcal{H}$ define $$A_n(T)f=\frac{1}{n}\sum_{j=1}^nT^jf.$$ Let $(n_k)$ be an increasing sequence and $M$ be any sequence. We prove that there exists a positive constant $C$ such that $$\ang(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\m\in M}}\|A_m(T)f-A_{n_k}(T)f\|_{\mathcal{H}}^2\ang)^{1/2}\leq C\|f\|_{\mathcal{H}}$$ for all $f\in \mathcal{H}$.