On time regularity and related conditions for power‐bounded operators

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 ) T n ‖ ⩽ 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 β ∈ (0, 1) and some power‐bounded operator S ; or if and only if T is power‐bounded and the discrete semigroup ( T n ) 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 < α ⩽ 1 we characterize the condition that T is power‐bounded and ‖( I − T ) T n ‖ ⩽ c n − α for all n , in terms of estimates on the semigroup e − t ( I − T ) .