The power boundedness and resolvent conditions for functions of the classical Volterra operator

Yuri Lyubich. "The power boundedness and resolvent conditions for functions of the classical Volterra operator." Studia Mathematica 196.1 (2010): 41-63. <http://eudml.org/doc/284671>.

@article{YuriLyubich2010,
abstract = {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 $L_\{p\}(0,1)$, 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.},
author = {Yuri Lyubich},
journal = {Studia Mathematica},
keywords = {power bounded operator; Volterra operator; Ritt condition},
language = {eng},
number = {1},
pages = {41-63},
title = {The power boundedness and resolvent conditions for functions of the classical Volterra operator},
url = {http://eudml.org/doc/284671},
volume = {196},
year = {2010},
}

TY - JOUR
AU - Yuri Lyubich
TI - The power boundedness and resolvent conditions for functions of the classical Volterra operator
JO - Studia Mathematica
PY - 2010
VL - 196
IS - 1
SP - 41
EP - 63
AB - 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 $L_{p}(0,1)$, 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.
LA - eng
KW - power bounded operator; Volterra operator; Ritt condition
UR - http://eudml.org/doc/284671
ER -