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On the power boundedness of certain Volterra operator pencils

100%
Tsedenbayar D.
Studia Mathematica
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2003
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tom 156
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nr 1
59-66
EN
Let V be the classical Volterra operator on L²(0,1), and let z be a complex number. We prove that I-zV is power bounded if and only if Re z ≥ 0 and Im z = 0, while I-zV² is power bounded if and only if z = 0. The first result yields $||(I-V)ⁿ - (I-V)^{n+1}|| = O(n^{-1/2})$ as n → ∞, an improvement of [Py]. We also study some other related operator pencils.
2
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Polynomials in the Volterra and Ritt operators

63%
Tsedenbayar D., Zemánek J.
Banach Center Publications
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2005
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tom 67
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nr 1
385-390
EN
We continue the paper [Ts] on the boundedness of polynomials in the Volterra operator. This provides new ways of constructing power-bounded operators. It seems interesting to point out that a similar procedure applies to the operators satisfying the Ritt resolvent condition: compare Theorem 5 and Theorem 9 below.
3
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The norms and singular numbers of polynomials of the classical Volterra operator in L₂(0,1)

63%
Lyubich Y., Tsedenbayar D.
Studia Mathematica
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2010
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tom 199
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nr 2
171-184
EN
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.
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