Approximations of the Identity Operator by Semigroups of Linear Operators

Let <!-- MATH $T(t),t \geqq 0$ --> , be a strongly continuous semigroup of linear operators on a Banach space X. It is proved that if for every 0$"> there exists a <!-- MATH ${\delta _c} > 0$ --> 0$"> such that <!-- MATH $\left\| {I - T(t)} \right\| \leqq 2 - Ct\log (1/t)$ --> for <!-- MATH $0 < t < {\delta _c}$ --> <img width="93" height="39" align="MIDDLE" border="0" src="images/img5.gif" alt="$ 0 < t < {\delta _c}$"> then is bounded for every 0$">. It is shown by means of an example that <!-- MATH $\left\| {I - T(t)} \right\| \leqq 2 - Ct$ --> for a fixed C and all <!-- MATH $0 < t < \delta$ --> <img width="86" height="39" align="MIDDLE" border="0" src="images/img9.gif" alt="$ 0 < t < \delta $"> is not sufficient to assure the boundedness of for any .