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

### Tom 199 / 2010

Studia Mathematica 199 (2010), 171-184 MSC: 47A10, 47A35, 47G10. DOI: 10.4064/sm199-2-3

#### Streszczenie

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.

#### Autorzy

• Yuri LyubichTechnion
Haifa 32000, Israel
e-mail
• Dashdondog TsedenbayarDepartment of Mathematics
Mongolian University of Science and Technology
P.O. Box 46/520
Ulaanbaatar, Mongolia
e-mail

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