One-parameter semigroups holomorphic away from zero

Abstract:

Suppose $T$ is a one-parameter semigroup of bounded linear operators on a Banach space, strongly continuous on $[0,\infty )$. It is known that $\lim {\sup _{x \to 0}}|T(x) - I| < 2$ implies $T$ is holomorphic on $(0,\infty )$. Theorem I is a generalization of this as follows: Suppose $M > 0,0 < r < s$, and $\rho$ is in (1,2). If $|{(T(h) - I)^n}| \leq M{\rho ^n}$ whenever $nh$ is in $[r,s],n = 1,2, \cdots ,h > 0$, then there exists $b > 0$ such that $T$ is holomorphic on $[b,\infty )$. Theorem II shows that, in some sense, $b \to 0$ as $r \to 0$. Theorem I is an application of Theorem III: Suppose $M > 0,0 < r < s,\rho$ is in (1,2), and $f$ is continuous on $[ - 4s,4s]$. If $|\sum _{q = 0}^n \binom {n}{q}( - 1)^{n - q} f(t + qh)| \leq M \rho ^n$ whenever $nh$ is in $[r,s]$, $n = 1$, $2$, …, $h > 0$, $[t,t + nh] \subset [ - 4s,4s]$, then $f$ has an analytic extension to an ellipse with center zero. Theorem III is a generalization of a theorem of Beurling in which the inequality on the differences is assumed for all $nh$. An example is given to show the hypothesis of Theorem I does not imply $T$ holomorphic on $(0,\infty )$.