An oscillation inequality on a complex Hilbert space
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2024), pp. 16-21
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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 $$\left(\sum_{k=1}^\infty\sup_{\substack{n_k\leq m< n_{k+1}\\ m\in M}}\|A_m(T)f-A_{n_k}(T)f\|_{\mathcal{H}}^2\right)^{1/2}\leq C\|f\|_{\mathcal{H}}$$ for all $f\in \mathcal{H}$.
Keywords:
Hilbert space, contraction, oscillation inequality.
@article{IVM_2024_9_a1,
author = {S. Demir},
title = {An oscillation inequality on a complex {Hilbert} space},
journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
pages = {16--21},
year = {2024},
number = {9},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/IVM_2024_9_a1/}
}
S. Demir. An oscillation inequality on a complex Hilbert space. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2024), pp. 16-21. http://geodesic.mathdoc.fr/item/IVM_2024_9_a1/
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