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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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: 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 $ \vert\zeta \vert \geqq 1$.


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

  • [1] 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 (19,664d)
  • [2] J. W. Neuberger Analyticity and quasi-analyticity for one-parameter semi-groups, (to appear).
  • [3] Kôsaku Yosida, On the differentiability of semigroups of linear operators, Proc. Japan Acad. 34 (1958), 337–340. MR 0098990 (20 #5435)
  • [4] Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473 (34 #3324)

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1970-0264456-0
PII: S 0002-9939(1970)0264456-0
Keywords: Semigroup of class $ {C_0}$, holomorphic semigroup, resolvent set, spectral radius, operator calculus, semi-Fredholm operator, index
Article copyright: © Copyright 1970 American Mathematical Society