## A Short Course on Operator SemigroupsThetheoryofstronglycontinuoussemigroupsoflinearoperatorsonBanach spaces, operator semigroups for short, has become an indispensable tool in a great number of areas of modern mathematical analysis. In our Springer Graduate Text [EN00] we presented this beautiful theory, together with many applications, and tried to show the progress made since the pub- cation in 1957 of the now classical monograph [HP57] by E. Hille and R. Phillips. However, the wealth of results exhibited in our Graduate Text seems to have discouraged some of the potentially interested readers. With the present text we o?er a streamlined version that strictly sticks to the essentials. We have skipped certain parts, avoided the use of sophisticated arguments,and,occasionally,weakenedtheformulationofresultsandm- i?ed the proofs. However, to a large extent this book consists of excerpts taken from our Graduate Text, with some new material on positive se- groups added in Chapter VI. We hope that the present text will help students take their ?rst step into this interesting and lively research ?eld. On the other side, it should provide useful tools for the working mathematician. Acknowledgments This book is dedicated to our students. Without them we would not be able to do and to enjoy mathematics. Many of them read previous versions when it served as the text of our Seventh Internet Seminar 2003/04. Here Genni Fragnelli, Marc Preunkert and Mark C. Veraar were among the most active readers. Particular thanks go to Tanja Eisner, Vera Keicher, Agnes Radl for proposing considerable improvements in the ?nal version. |

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### Contents

Examples | 11 |

MoreSemigroups | 19 |

Semigroups Generators and Resolvents | 34 |

GenerationTheorems | 63 |

PerturbationofSemigroups | 115 |

ApproximationofSemigroups | 136 |

Spectral Theory and Asymptotics for Semigroups | 156 |

Positive Semigroups | 205 |

Appendix | 222 |

231 | |

Selected References to Recent Research | 237 |

243 | |

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A-bounded abstract Cauchy problem adjoint algebraic analytic semigroup approximate assertions are equivalent assume Banach space bounded operator characterization closed operator continuous function continuous semigroup T(t))tzo contraction semigroup convergence Corollary Deﬁnition densely defined differential operator dissipative dissipative operator domain D(A eigenvalue equation estimate Example Exercise exists ﬁrst ﬁxed following assertions formula function q growth bound hence Hilbert space Hint holds implies integral inverse Lemma Let T(t))tzo linear operator Lp(R matrix Moreover multiplication operator multiplication semigroup norm-continuous obtain operator Mq operator topology operators T(t Paragraph perturbation PROOF Proposition prove quasi-compact rescaling satisﬁes satisfying Section semi sequence Sobolev spaces spectral bound spectral mapping theorem spectral radius spectral theory strictly positive strong operator topology strongly continuous semigroup subspace sufﬁces Tn(t translation semigroup unbounded uniform boundedness principle uniformly continuous uniformly exponentially stable WSMT yields