On Hille-type approximation of degenerate semigroups of operators

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• 作者：Adam Bobrowski ^{a.bobrowski@pollub.pl" class="auth_mail" title="E-mail the corresponding author}
• 关键词：47D03 ; 92D25 ; 46N60 ; 37N25
• 刊名：Linear Algebra and its Applications
• 出版年：2016
• 出版时间：15 December 2016
• 年：2016
• 卷：511
• 期：Complete
• 页码：31-53
• 全文大小：461 K

文摘

The result that goes essentially back to Euler [15] says that for any element a   of a unital Banach algebra A$\mathbb{A}$ with unit u  , the limit lim_ε→0+⁡(u+εa)^[ε−1t]${\lim }_{\epsilon \to 0+}{(u+\epsilon a)}^{[{\epsilon }^{-1}t]}$ (where [⋅]$[\cdot ]$ denotes the integral part) exists for all t∈R$t\in \mathbb{R}$ and equals e^ta${\mathrm{e}}^{ta}$. As developed by E. Hille [22, Thm. 12.2.1], in the case where a is replaced by the generator A   of a strongly continuous semigroup {e^tA,t≥0}$\{{\mathrm{e}}^{tA},t\ge 0\}$ in a Banach space X$\mathbb{X}$, a proper counterpart of this formula is e^tA=lim_ε→0+⁡(I_X−εA)^−[ε−1t]${\mathrm{e}}^{tA}={\lim }_{\epsilon \to 0+}{(I_{\mathbb{X}}-\epsilon A)}^{-[{\epsilon }^{-1}t]}$ strongly in X$\mathbb{X}$. Motivated by an example from mathematical biology (related to Rotenberg's model of cell growth [40]) we study convergence of a similar approximation in which u   (resp. I_X$I_{\mathbb{X}}$) is replaced by j∈A$j\in \mathbb{A}$ (resp. J∈L(X)$J\in \mathcal{L}(\mathbb{X})$) such that for some ℓ≥2$\ell \ge 2$, j^ℓ=u$j^{\ell }=u$ (resp. J^ℓ=I_X$J^{\ell }=I_{\mathbb{X}}$).