The spectral problem (s²I - ϕ(V)*ϕ(V))f = 0 for an arbitrary complex polynomial ϕ of the classical Volterra operator V in L₂(0,1) is considered. An equivalent boundary value problem for a differential equation of order 2n, n = deg(ϕ), is constructed. In the case ϕ(z) = 1 + az the singular numbers are explicitly described in terms of roots of a transcendental equation, their localization and asymptotic behavior is investigated, and an explicit formula for the ||I + aV||₂ is given. For all a ≠ 0 this norm turns out to be greater than 1.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-3,
author = {Yuri Lyubich and Dashdondog Tsedenbayar},
title = {The norms and singular numbers of polynomials of the classical Volterra operator in L2(0,1)},
journal = {Studia Mathematica},
volume = {196},
year = {2010},
pages = {171-184},
zbl = {1216.47029},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-3}
}
Yuri Lyubich; Dashdondog Tsedenbayar. The norms and singular numbers of polynomials of the classical Volterra operator in L₂(0,1). Studia Mathematica, Tome 196 (2010) pp. 171-184. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm199-2-3/