On Hille-type approximation of degenerate semigroups of operators

详细信息    查看全文

• 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 379516303883&_mathId=si1.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=db98e793236828b350178206f70df4c9" title="Click to view the MathML source">A$\mathbb{A}$ with unit u  , the limit 379516303883&_mathId=si2.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=ed0b18d502548a5633cd6d459c48ae81" title="Click to view the MathML source">lim_ε→0+⁡(u+εa)^[ε−1t]${\lim }_{\epsilon \to 0+}{(u+\epsilon a)}^{[{\epsilon }^{-1}t]}$ (where 379516303883&_mathId=si3.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=bc157ee15cf580f16e0d873fe355f486" title="Click to view the MathML source">[⋅]$[\cdot ]$ denotes the integral part) exists for all 379516303883&_mathId=si4.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=b1e14fbd54004aef5905b687c4ebec82" title="Click to view the MathML source">t∈R$t\in \mathbb{R}$ and equals 379516303883&_mathId=si5.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=25ab5ae5d9b4bc88c2db11f620306e18" title="Click to view the MathML source">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 379516303883&_mathId=si6.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=1876ee358e25d585efb3dd4d8c398062" title="Click to view the MathML source">{e^tA,t≥0}$\{{\mathrm{e}}^{tA},t\ge 0\}$ in a Banach space 379516303883&_mathId=si7.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=979611233079b46ce481140e7cb77aca" title="Click to view the MathML source">X$\mathbb{X}$, a proper counterpart of this formula is 379516303883&_mathId=si8.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=7084447a8a2f3af9bbd37740d3ccb58c" title="Click to view the MathML source">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 379516303883&_mathId=si7.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=979611233079b46ce481140e7cb77aca" title="Click to view the MathML source">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. 379516303883&_mathId=si9.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=521cac1da8e5f9f7d66184eabe56a068" title="Click to view the MathML source">I_X$I_{\mathbb{X}}$) is replaced by 379516303883&_mathId=si10.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=9c43a1017e4bd0a5acaeb7f31f772d4e" title="Click to view the MathML source">j∈A$j\in \mathbb{A}$ (resp. 379516303883&_mathId=si11.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=a7daf4c829a57c302a60d9b7fbc514f0" title="Click to view the MathML source">J∈L(X)$J\in \mathcal{L}(\mathbb{X})$) such that for some 379516303883&_mathId=si12.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=ca56b4d2f3eafeb9178b9db17bffa826" title="Click to view the MathML source">ℓ≥2$\ell \ge 2$, 379516303883&_mathId=si13.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=7ae7d3f8f5b423d2bba441569fd1b89e" title="Click to view the MathML source">j^ℓ=u$j^{\ell }=u$ (resp. 379516303883&_mathId=si14.gif&_user=111111111&_pii=S0024379516303883&_rdoc=1&_issn=00243795&md5=a735d14d148d93fad651cfeb42a9d1d7" title="Click to view the MathML source">J^ℓ=I_X$J^{\ell }=I_{\mathbb{X}}$).