On the power boundedness of certain Volterra operator pencils

Dashdondog Tsedenbayar ¹

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 P.O. Box 21 00-956 Warszawa, Poland and Department of Mathematics Mongolian Teacher's University Ulan-Bator, Mongolia

Studia Mathematica, Tome 156 (2003) no. 1, pp. 59-66

Résumé

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.

Détail

DOI : 10.4064/sm156-1-4

Keywords: classical volterra operator complex number prove i zv power bounded only mathop mathop while i zv power bounded only first result yields i v n i v quad rightarrow infty improvement study other related operator pencils