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Proceedings of the London Mathematical Society Advance Access published online on March 12, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdm058
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© 2007 London Mathematical Society

On time regularity and related conditions for power-bounded operators

Nick Dungey

Department of Mathematics
Macquarie University
NSW 2109
Australia

Received 2 April 2007. Revision received 14 November 2007.

Let T be a bounded linear operator in a complex Banach space. Our main result gives various characterizations of the condition: T is power-bounded and an estimate |(I–T)Tn |≤cn–1/2 holds for all positive integers n. In particular, this condition holds if and only if T=β S+(1–β)I, for some βisin(0, 1) and some power-bounded operator S; or if and only if T is power-bounded and the discrete semigroup (Tn) is dominated by the continuous semigroup (e–t(I–T))t≥0 in a natural sense. As a consequence of our main results, for 1/2<{alpha}≤1 we characterize the condition that T is power-bounded and |(I–T)Tn |≤c n–{alpha} for all n, in terms of estimates on the semigroup e–t(I–T).

2000 Mathematics Subject Classification 47A30, 47A10 (primary), 47D06 (secondary).


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