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 $$A_{n}f=\frac{1}{n}\sum _{j=1}^n{T}^{j}f$$ for $f\in \mathcal{H}$. Let $(n_k)$ be a lacunary sequence, then there exists a constant $C_1>0$ such that $$\sum _{k=1}^{\infty }\Vert A_{n_{k+1}}f-A_{n_k}f{\Vert }_{\mathcal{H}}\le C_1\Vert f{\Vert }_{\mathcal{H}}$$ 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 $$\sum _{k=1}^{\infty }\underset{\begin{array}{c}{n}_k\le m<n_{k+1}\\ m\in \mathbb{N}\end{array}}{\sup }\Vert A_m(T)f-A_{n_k}(T)f{\Vert }_{\mathcal{H}}\le C_2\Vert f{\Vert }_{\mathcal{H}}$$ for all $f\in \mathcal{H}$.