The norms and singular numbers of polynomials of the classical Volterra operator in $L_2(0,1)$
Volume 199 / 2010
Studia Mathematica 199 (2010), 171-184
MSC: 47A10, 47A35, 47G10.
DOI: 10.4064/sm199-2-3
Abstract
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.
