Decay estimates for Cayley transforms and inverses of semigroup generators via the $\mathcal{B}$-calculus
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arXiv
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866929340013346816 |
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| author | Wakaiki, Masashi |
| author_facet | Wakaiki, Masashi |
| contents | Let $-A$ be the generator of a bounded $C_0$-semigroup $(e^{-tA})_{t \geq 0}$ on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform $V_ω(A) := (A-ωI) (A+ωI)^{-1}$ with $ω>0$. We give a decay estimate for $\|V_ω(A)^nA^{-1}\|$ when $(e^{-tA})_{t \geq 0}$ is polynomially stable. Considering the case where the parameter $ω$ varies, we estimate $\|(\prod_{k=1}^n V_{ω_k}(A))A^{-1}\|$ for exponentially stable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$. Next we show that if the generator $-A$ of the bounded $C_0$-semigroup has a bounded inverse, then $\sup_{t \geq 0} \|e^{-tA^{-1}} A^{-α} \| < \infty$ for all $α>0$. We also present an estimate for the rate of decay of $\|e^{-tA^{-1}} A^{-1} \|$, assuming that $(e^{-tA})_{t \geq 0}$ is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the $\mathcal{B}$-calculus. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_05692 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Decay estimates for Cayley transforms and inverses of semigroup generators via the $\mathcal{B}$-calculus Wakaiki, Masashi Functional Analysis Numerical Analysis Let $-A$ be the generator of a bounded $C_0$-semigroup $(e^{-tA})_{t \geq 0}$ on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform $V_ω(A) := (A-ωI) (A+ωI)^{-1}$ with $ω>0$. We give a decay estimate for $\|V_ω(A)^nA^{-1}\|$ when $(e^{-tA})_{t \geq 0}$ is polynomially stable. Considering the case where the parameter $ω$ varies, we estimate $\|(\prod_{k=1}^n V_{ω_k}(A))A^{-1}\|$ for exponentially stable $C_0$-semigroups $(e^{-tA})_{t \geq 0}$. Next we show that if the generator $-A$ of the bounded $C_0$-semigroup has a bounded inverse, then $\sup_{t \geq 0} \|e^{-tA^{-1}} A^{-α} \| < \infty$ for all $α>0$. We also present an estimate for the rate of decay of $\|e^{-tA^{-1}} A^{-1} \|$, assuming that $(e^{-tA})_{t \geq 0}$ is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the $\mathcal{B}$-calculus. |
| title | Decay estimates for Cayley transforms and inverses of semigroup generators via the $\mathcal{B}$-calculus |
| topic | Functional Analysis Numerical Analysis |
| url | https://arxiv.org/abs/2312.05692 |