Approximations of the identity operator by semigroups of linear operators

Pazy, A.

doi:10.1090/S0002-9939-1971-0287362-5

Approximations of the identity operator by semigroups of linear operators
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Proc. Amer. Math. Soc. 30 (1971), 147-150 Request permission

Abstract:

Let $T(t),t \geqq 0$, be a strongly continuous semigroup of linear operators on a Banach space X. It is proved that if for every $C > 0$ there exists a ${\delta _c} > 0$ such that $\left \| {I - T(t)} \right \| \leqq 2 - Ct\log (1/t)$ for $0 < t < {\delta _c}$ then $AT(t)$ is bounded for every $t > 0$. It is shown by means of an example that $\left \| {I - T(t)} \right \| \leqq 2 - Ct$ for a fixed C and all $0 < t < \delta$ is not sufficient to assure the boundedness of $AT(t)$ for any $t \geqq 0$.

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