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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

One-parameter semigroups holomorphic away from zero


Author: Melinda W. Certain
Journal: Trans. Amer. Math. Soc. 187 (1974), 377-389
MSC: Primary 47D05
DOI: https://doi.org/10.1090/S0002-9947-1974-0336442-1
MathSciNet review: 0336442
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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}}\vert T(x) - I\vert < 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 $ \vert{(T(h) - I)^n}\vert \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 $ \vert\sum\nolimits_{q = 0}^n {(\mathop n\limits_q ){{( - 1)}^{n - q}}f(t + qh)\vert \leq M{\rho ^n}} $ whenever nh is in $ [r,s],n = 1,2, \cdots ,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 )$.


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DOI: https://doi.org/10.1090/S0002-9947-1974-0336442-1
Keywords: Semigroup of operators, holomorphic semigroup, analytic extension of functions, finite differences, quasianalytic classes of functions
Article copyright: © Copyright 1974 American Mathematical Society

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