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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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A characterization of holomorphic semigroups

Author: Tosio Kato
Journal: Proc. Amer. Math. Soc. 25 (1970), 495-498
MSC: Primary 47.50
MathSciNet review: 0264456
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Abstract | References | Similar Articles | Additional Information

Abstract: A necessary and sufficient condition is given for a one-parameter semigroup $\{ U(t)\} ,\;0 \leqq t < \infty$, of class ${C_0}$ on a Banach space to be holomorphic (of class $H({\Phi _1},\;{\Phi _2})$ for some ${\Phi _1} < 0 < {\Phi _2}$). The condition is expressed in terms of the spectral properties of $U(t) - \zeta$ for small $t > 0$ and for a complex number $\zeta$ with $|\zeta | \geqq 1$.

References [Enhancements On Off] (What's this?)

  • Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
  • J. W. Neuberger Analyticity and quasi-analyticity for one-parameter semi-groups, (to appear).
  • Kôsaku Yosida, On the differentiability of semigroups of linear operators, Proc. Japan Acad. 34 (1958), 337–340. MR 98990
  • Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473

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Keywords: Semigroup of class <IMG WIDTH="29" HEIGHT="38" ALIGN="MIDDLE" BORDER="0" SRC="images/img1.gif" ALT="${C_0}$">, holomorphic semigroup, resolvent set, spectral radius, operator calculus, semi-Fredholm operator, index
Article copyright: © Copyright 1970 American Mathematical Society