The norms and singular numbers of polynomials of the classical Volterra operator in $L_2(0,1)$

Yuri Lyubich¹ ; Dashdondog Tsedenbayar²

¹ Technion Haifa 32000, Israel
² Department of Mathematics Mongolian University of Science and Technology P.O. Box 46/520 Ulaanbaatar, Mongolia

Studia Mathematica, Tome 199 (2010) no. 2, pp. 171-184

Résumé

The spectral problem $(s^2I-\phi(V)^{*}\phi(V))f=0$ for an arbitrary complex polynomial $\phi$ of the classical Volterra operator $V$ in $L_2(0,1)$ is considered. An equivalent boundary value problem for a differential equation of order $2n$, $n=\deg(\phi)$, is constructed. In the case $\phi(z)=1+az$ 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 $a\neq 0$ this norm turns out to be greater than 1.

Détail

DOI : 10.4064/sm199-2-3

Keywords: spectral problem i phi * phi arbitrary complex polynomial phi classical volterra operator considered equivalent boundary value problem differential equation order deg phi constructed phi singular numbers explicitly described terms roots transcendental equation their localization asymptotic behavior investigated explicit formula given neq norm turns out greater