Characterization of decay rates for discrete operator semigroups
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911351205527552 |
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| author | Wakaiki, Masashi |
| author_facet | Wakaiki, Masashi |
| contents | Let $T$ be a power-bounded linear operator on a Hilbert space $X$, and let $S$ be a bounded linear operator from another Hilbert space $Y$ to $X$. We investigate the non-exponential rate of decay of $\|T^nS\|$ as $n \to \infty$. First, when $X = Y$ and $S$ commutes with $T$, we characterize the decay rate of $\|T^nS\|$ in terms of the growth rate of $\|(λI - T)^{-k}S\|$ as $|λ| \downarrow 1$ for some $k \in \mathbb{N}$. Next, we provide another characterization by means of an integral estimate of $\|(λI - T)^{-k}S\|$. The second characterization is then applied to asymptotic estimates for perturbed discrete operator semigroups. Finally, we present some results on the relation between the decay rate of $\|T^nS\|$ and the boundedness of the sum $\sum_{n=1}^{\infty} f(n)\|T^nSy\|^p$ for all $y \in Y$ in the Banach space setting, where $f \colon \mathbb{N} \to(0,\infty)$ and $p \geq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_19534 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Characterization of decay rates for discrete operator semigroups Wakaiki, Masashi Functional Analysis Optimization and Control Let $T$ be a power-bounded linear operator on a Hilbert space $X$, and let $S$ be a bounded linear operator from another Hilbert space $Y$ to $X$. We investigate the non-exponential rate of decay of $\|T^nS\|$ as $n \to \infty$. First, when $X = Y$ and $S$ commutes with $T$, we characterize the decay rate of $\|T^nS\|$ in terms of the growth rate of $\|(λI - T)^{-k}S\|$ as $|λ| \downarrow 1$ for some $k \in \mathbb{N}$. Next, we provide another characterization by means of an integral estimate of $\|(λI - T)^{-k}S\|$. The second characterization is then applied to asymptotic estimates for perturbed discrete operator semigroups. Finally, we present some results on the relation between the decay rate of $\|T^nS\|$ and the boundedness of the sum $\sum_{n=1}^{\infty} f(n)\|T^nSy\|^p$ for all $y \in Y$ in the Banach space setting, where $f \colon \mathbb{N} \to(0,\infty)$ and $p \geq 1$. |
| title | Characterization of decay rates for discrete operator semigroups |
| topic | Functional Analysis Optimization and Control |
| url | https://arxiv.org/abs/2412.19534 |