Let ϕ(z) be an analytic function in a disk |z| < ρ (in particular, a polynomial) such that ϕ(0) = 1, ϕ(z)≢ 1. Let V be the operator of integration in , 1 ≤ p ≤ ∞. Then ϕ(V) is power bounded if and only if ϕ’(0) < 0 and p = 2. In this case some explicit upper bounds are given for the norms of ϕ(V)ⁿ and subsequent differences between the powers. It is shown that ϕ(V) never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if ϕ’(0) < 0, at least in the polynomial case.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-1-4,
author = {Yuri Lyubich},
title = {The power boundedness and resolvent conditions for functions of the classical Volterra operator},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {41-63},
zbl = {1191.47004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-1-4}
}
Yuri Lyubich. The power boundedness and resolvent conditions for functions of the classical Volterra operator. Studia Mathematica, Tome 196 (2010) pp. 41-63. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm196-1-4/