On the power boundedness of certain Volterra operator pencils
Volume 156 / 2003
Studia Mathematica 156 (2003), 59-66
MSC: Primary 47A10.
DOI: 10.4064/sm156-1-4
Abstract
Let $V$ be the classical Volterra operator on $L^2(0,1)$, and let $z$ be a complex number. We prove that $I-zV$ is power bounded if and only if $\mathop{\rm Re} z \ge 0$ and $\mathop{\rm Im} z=0$, while $I-zV^2$ is power bounded if and only if $z=0$. The first result yields $$\|(I-V)^n-(I-V)^{n+1}\|=O(n^{-{1 / 2}})\quad\ {\rm as}\ n\rightarrow\infty ,$$ an improvement of [Py]. We also study some other related operator pencils.
