Abstract
Suppose that is a strongly continuous (even at 0) one-parameter semigroup of bounded linear transformations on a real Banach space and has generator . Theorem A. If <!-- MATH $\lim {\sup _{x \to 0}}|T(x) - I| < 2$ --> <img width="224" height="41" align="MIDDLE" border="0" src="images/img5.gif" alt="$ \lim {\sup _{x \to 0}}\vert T(x) - I\vert < 2$"> then is bounded for all 0$">. Suppose <!-- MATH $\{ {\delta _q}\} _{q = 1}^\infty$ --> is a sequence of positive numbers convergent to 0 and each of <!-- MATH $N(q),\;q = 1,\;2, \cdots$ --> is an increasing sequence of positive integers. Denote by the collection consisting of (1) all real analytic functions on <!-- MATH $(0,\;\infty )$ --> and (2) all on <!-- MATH $(0,\;\infty )$ --> for which there is a Banach space , a member of , a member of <!-- MATH ${S^{\ast}}$ --> and a strongly continuous semigroup of bounded linear transformations so that <!-- MATH $h(x) = f[L(x)p]$ --> for all 0$"> where satisfies <!-- MATH $\lim {\sup _{n \to \infty \,(n \in N(q))}}|L({\delta _q}/n) - I| < 2,\;q = 1,\;2,\; \cdots$ --> <img width="456" height="41" align="MIDDLE" border="0" src="images/img23.gif" alt="$ \lim {\sup _{n \to \infty \,(n \in N(q))}}\vert L({\delta _q}/n) - I\vert < 2,\;q = 1,\;2,\; \cdots $">. Theorem B. No two members of agree on an open subset of <!-- MATH $(0,\;\infty )$ --> .