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1

The power boundedness and resolvent conditions for functions of the classical Volterra operator

100%

Lyubich Y.

Studia Mathematica

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2010

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tom 196

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nr 1

41-63

EN

Let ϕ(z) be an analytic function in a disk |z| < ρ (in particular, a polynomial) such that ϕ(0) = 1, ϕ(z)≢ 1. Let V be the operator of integration in $L_{p}(0,1)$, 1 ≤ p ≤ ∞. Then ϕ(V) is power bounded if and only if ϕ'(0) < 0 and p = 2. In this case some explicit upper bounds are given for the norms of ϕ(V)ⁿ and subsequent differences between the powers. It is shown that ϕ(V) never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if ϕ'(0) < 0, at least in the polynomial case.

2

The boundary spectrum of linear operators in finite-dimensional spaces

100%

Lyubich Y.

Banach Center Publications

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1997

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tom 38

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nr 1

217-226

3

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.

Ograniczanie wyników

2 Studia Mathematica

1 Banach Center Publications

3 Lyubich Y.

1 Tsedenbayar D.

2 2010

1 1997

The power boundedness and resolvent conditions for functions of the classical Volterra operator

The boundary spectrum of linear operators in finite-dimensional spaces

The norms and singular numbers of polynomials of the classical Volterra operator in L₂(0,1)