On the power boundedness of certain Volterra operator pencils

Volume 156 / 2003

Dashdondog Tsedenbayar Studia Mathematica 156 (2003), 59-66 MSC: Primary 47A10. DOI: 10.4064/sm156-1-4


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.


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

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