Variation and oscillation inequalities for operator averages on a complex Hilbert space
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2021
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| _version_ | 1866909654663036928 |
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| author | Demir, Sakin |
| author_facet | Demir, Sakin |
| contents | 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 $f\in \mathcal{H}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2107_14030 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Variation and oscillation inequalities for operator averages on a complex Hilbert space Demir, Sakin Classical Analysis and ODEs 47B02, 37A30 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 $f\in \mathcal{H}$. |
| title | Variation and oscillation inequalities for operator averages on a complex Hilbert space |
| topic | Classical Analysis and ODEs 47B02, 37A30 |
| url | https://arxiv.org/abs/2107.14030 |