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Proceedings of the American Mathematical Society

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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
DOI: https://doi.org/10.1090/S0002-9939-1970-0264456-0
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 $|\zeta | \geqq 1$.


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