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

TL;DR

This paper establishes variation and oscillation inequalities for averages of a contraction operator on a complex Hilbert space, demonstrating boundedness properties for sequences with lacunary gaps.

Contribution

It proves new boundedness results for operator averages involving lacunary sequences, extending classical ergodic theorems to oscillation and variation inequalities.

Findings

Bounded variation inequalities for lacunary operator averages

Oscillation inequalities for operator averages on Hilbert spaces

Constants controlling the inequalities are independent of the function

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 $f\in \mathcal{H}$.