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 . It is known that implies *T* is holomorphic on . Theorem I is a generalization of this as follows: Suppose , and is in (1,2). If whenever *nh* is in , then there exists such that *T* is holomorphic on . Theorem II shows that, in some sense, as . Theorem I is an application of Theorem III: Suppose is in (1,2), and *f* is continuous on .

If whenever *nh* is in , 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 .

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Additional Information

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