Remarks on rates of convergence of powers of contractions

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• 作者：Guy Cohen^a ; ¹ ; ^{guycohen@bgu.ac.il" class="auth_mail" title="E-mail the corresponding author} ; Michael Lin^b ; ^{lin@math.bgu.ac.il" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Numerical range ; Ritt operators ; Quasi-sectorial operators ; Rate of convergence of powers ; Resolvent estimates ; Contractions on L2L2
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 April 2016
• 年：2016
• 卷：436
• 期：2
• 页码：1196-1213
• 全文大小：507 K

文摘

We prove that if the numerical range of a Hilbert space contraction T   is in a certain closed convex set of the unit disk which touches the unit circle only at 1, then ‖Tⁿ(I−T)‖=O(1/n^β)$\Vert T^n(I-T)\Vert =\mathcal{O}(1/n^{\beta })$ with . For normal contractions the condition is also necessary. Another sufficient condition for , necessary for T normal, is that the numerical range of T   be in a disk {z:|z−δ|≤1−δ}$\{z:|z-\delta |\le 1-\delta \}$ for some δ∈(0,1)$\delta \in (0,1)$. As a consequence of results of Seifert, we obtain that a power-bounded T   on a Hilbert space satisfies ‖Tⁿ(I−T)‖=O(1/n^β)$\Vert T^n(I-T)\Vert =\mathcal{O}(1/n^{\beta })$ with β∈(0,1]$\beta \in (0,1]$ if and only if sup_1<|λ|<2⁡|λ−1|^1/β‖R(λ,T)‖<∞${\sup }_{1<|\lambda |<2}{|\lambda -1|}^{1/\beta }\Vert R(\lambda ,T)\Vert <\infty$. When T   is a contraction on ac" title="Click to view the MathML source">L₂$L_2$ satisfying the numerical range condition, it is shown that Tⁿf/n^1−β$T^{n}f/n^{1-\beta }$ converges to 0 a.e. with a maximal inequality, for every f∈L₂$f\in L_2$. An example shows that in general a positive contraction T   on ac" title="Click to view the MathML source">L₂$L_2$ may have an f≥0$f\ge 0$ with a.e.