## One-Parameter Semigroups for Linear Evolution EquationsThe theory of one-parameter semigroups of linear operators on Banach spaces started in the ?rst half of this century, acquired its core in 1948 with the Hille–Yosida generation theorem, and attained its ?rst apex with the 1957 edition of Semigroups and Functional Analysis by E. Hille and R.S. Phillips. In the 1970s and 80s, thanks to the e?orts of many di?erent schools,thetheoryreachedacertainstateofperfection,whichiswellrep- sented in the monographs by E.B. Davies [Dav80], J.A. Goldstein [Gol85], A. Pazy [Paz83], and others. Today, the situation is characterized by manifold applications of this theory not only to the traditional areas such as partial di?erential eq- tions or stochastic processes. Semigroups have become important tools for integro-di?erentialequationsandfunctionaldi?erentialequations,inqu- tum mechanics or in in?nite-dimensional control theory. Semigroup me- ods are also applied with great success to concrete equations arising, e.g., in population dynamics or transport theory. It is quite natural, however, that semigroup theory is in competition with alternative approaches in all of these ?elds, and that as a whole, the relevant functional-analytic toolbox now presents a highly diversi?ed picture. At this point we decided to write a new book, re?ecting this situation but based on our personal mathematical taste. Thus, it is a book on se- groups or, more precisely, on one-parameter semigroups of bounded linear operators. In our view, this re?ects the basic philosophy, ?rst and strongly emphasized by A. Hadamard (see p. 152), that an autonomous determin- tic system is described by a one-parameter semigroup of transformations. |

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### Contents

III | 1 |

IV | 2 |

V | 6 |

VI | 14 |

VII | 24 |

IX | 30 |

X | 33 |

XI | 36 |

LXXIV | 345 |

LXXV | 347 |

LXXVI | 348 |

LXXVII | 349 |

LXXVIII | 353 |

LXXIX | 358 |

LXXX | 361 |

LXXXIII | 364 |

XII | 37 |

XIII | 42 |

XIV | 46 |

XV | 47 |

XVI | 48 |

XVII | 59 |

XIX | 65 |

XX | 70 |

XXI | 71 |

XXII | 82 |

XXIII | 89 |

XXIV | 96 |

XXVI | 109 |

XXVII | 112 |

XXVIII | 117 |

XXIX | 120 |

XXX | 123 |

XXXI | 124 |

XXXII | 129 |

XXXIII | 137 |

XXXIV | 145 |

XXXV | 154 |

XXXVI | 157 |

XXXVII | 169 |

XXXVIII | 182 |

XL | 192 |

XLI | 195 |

XLII | 201 |

XLIII | 205 |

XLIV | 206 |

XLV | 209 |

XLVI | 214 |

XLVII | 219 |

XLIX | 231 |

L | 236 |

LI | 238 |

LII | 239 |

LIII | 250 |

LV | 259 |

LVI | 266 |

LVII | 270 |

LIX | 275 |

LX | 283 |

LXI | 289 |

LXII | 293 |

LXIII | 295 |

LXIV | 296 |

LXVI | 299 |

LXVII | 305 |

LXVIII | 308 |

LXX | 312 |

LXXI | 315 |

LXXII | 329 |

LXXIII | 337 |

LXXXIV | 367 |

LXXXV | 369 |

LXXXVI | 372 |

LXXXVII | 374 |

LXXXVIII | 382 |

LXXXIX | 383 |

XC | 384 |

XCI | 388 |

XCII | 390 |

XCIII | 400 |

XCIV | 403 |

XCV | 404 |

XCVI | 405 |

XCVII | 408 |

XCVIII | 411 |

XCIX | 419 |

CI | 420 |

CII | 424 |

CIII | 428 |

CIV | 435 |

CVI | 436 |

CVII | 442 |

CVIII | 447 |

CIX | 452 |

CXI | 456 |

CXII | 466 |

CXIII | 468 |

CXIV | 473 |

CXV | 476 |

CXVI | 477 |

CXVIII | 481 |

CXIX | 487 |

CXX | 492 |

CXXI | 496 |

CXXII | 497 |

CXXIII | 500 |

CXXIV | 502 |

CXXV | 506 |

CXXVI | 509 |

CXXVII | 515 |

CXXVIII | 522 |

CXXIX | 526 |

CXXX | 530 |

CXXXI | 531 |

CXXXII | 533 |

CXXXIII | 536 |

CXXXIV | 538 |

CXXXV | 546 |

CXXXVI | 549 |

553 | |

555 | |

CXXXIX | 577 |

580 | |

### Common terms and phrases

abstract Cauchy problem ACP2 adjoint algebra analytic semigroup approximate assertions are equivalent assume assumption Banach space bounded operator C D(A characterization compact operator consider constant continuous functions continuous semigroup T(t))t>o contraction semigroup converges Corollary Definition denote densely defined differential equation differential operator domain D(A E D(A eigenvalue estimate Example Exercise exists exponential function Favard following assertions formula function q G D(A given growth bound hence Hilbert space Hint holds implies initial value integral inverse isometries Lemma linear operator Moreover multiplication operator multiplication semigroup norm continuous obtain one-parameter operator topology operators T(t Paragraph perturbation proof Proposition prove satisfying Section semi sequence Sobolev space solution of ACP spectral bound spectral mapping theorem spectral radius spectral theory spectrum strong operator topology strongly continuous semigroup subset subspace theory translation semigroup unbounded uniformly continuous unique WSMT yields

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Page 553 - the creation of this world is the combined work of necessity and mind. Mind, the ruling power, persuaded necessity to bring the greater part of created things to perfection (...). But if a person will truly tell of the way in which the work was accomplished. he must include the variable cause as well