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On the power boundedness of certain Volterra operator pencils

Dashdondog Tsedenbayar (2003)

Studia Mathematica

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 ) - ( I - V ) n + 1 | | = O ( n - 1 / 2 ) as n → ∞, an improvement of [Py]. We also study some other related operator pencils.

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