One parameter semigroups for linear evolution equations, klaus jochen engel, rainer nagel

To Carla and Ursula Preface The theory of one-parameter semigroups of linear operators on Banach spaces started in the first half of this century, acquired its core in 1948 with the Hille–Yosida generation theorem, and attained its first 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 efforts of many different schools, the theory reached a certain state of perfection, which is well represented 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 differential equations or stochastic processes Semigroups have become important tools for integro-differential equations and functional differential equations, in quantum mechanics or in infinite-dimensional control theory Semigroup methods 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 fields, and that as a whole, the relevant functional-analytic toolbox now presents a highly diversified picture At this point we decided to write a new book, reflecting this situation but based on our personal mathematical taste Thus, it is a book on semigroups or, more precisely, on one-parameter semigroups of bounded linear operators In our view, this reflects the basic philosophy, first and strongly emphasized by A Hadamard (see p 152), that an autonomous deterministic system is described by a one-parameter semigroup of transformations Among the many continuity properties of these semigroups that were vii viii Preface already studied by E Hille and R.S Phillips in [HP57], we deliberately concentrate on strong continuity and show that this is the key to a deep and beautiful theory Referring to many concrete equations, one might object that semigroups, and especially strongly continuous semigroups, are of limited value, and that other concepts such as integrated semigroups, regularized semigroups, cosine families, or resolvent families are needed While we not question the good reasons leading to these concepts, we take a very resolute stand in this book insofar as we put strongly continuous semigroups of bounded linear operators into the undisputed center of our attention Around this concept we develop techniques that allow us to obtain • a semigroup on an appropriate Banach space even if at first glance the semigroup property does not hold, and • strong continuity in an appropriate topology where originally only weaker regularity properties are at hand In Chapter VI we then show how these constructions allow the treatment of many different evolution equations that initially not have the form of a homogeneous abstract Cauchy problem and/or are not “well-posed” in a strict sense Structure of the Book This is not a research monograph but an introduction to the theory of semigroups After developing the fundamental results of this theory we emphasize spectral theory, qualitative properties, and the broad range of applications Moreover, our book is written in the spirit of functional analysis This means that we prefer abstract constructions and general arguments in order to underline basic principles and to minimize computations Some of the required tools from functional analysis, operator theory, and vector-valued integration are collected in the appendices In Chapter I, we intentionally take a slow start and lead the reader from the finite-dimensional and uniformly continuous case through multiplication and translation semigroups to the notion of a strongly continuous semigroup To these semigroups we associate a generator in Chapter II and characterize these generators in the Hille–Yosida generation theorem and its variants Semigroups having stronger regularity properties such as analyticity, eventual norm continuity, or compactness are then characterized, whenever possible, in a similar way A special feature of our approach is the use of a rich scale of interpolation and extrapolation spaces associated to a strongly continuous semigroup A comprehensive treatment of these “Sobolev towers” is presented by Simon Brendle in Section II.5 In Chapter III we start with the classical Bounded Perturbation Theorem III.1.3, but then present a new simultaneous treatment of unbounded Desch–Schappacher and Miyadera–Voigt perturbations in Section III.3 In the remaining Sections III.4 and it was our goal to discuss a broad range Preface ix of applications of the Trotter–Kato Approximation Theorem III.4.8 Spectral theory is the core of our approach, and in Chapter IV we discuss in great detail under what conditions the so-called spectral mapping theorem is valid A first payoff is the complete description of the structure of periodic groups in Theorem IV.2.27 On the basis of this spectral theory we then discuss in Chapter V qualitative properties of the semigroup such as stability, hyperbolicity, and mean ergodicity Inspired by the classical Liapunov stability theorem we try to describe these properties by the spectrum of the generator It is rewarding to see how a combination of spectral theory with geometric properties of the underlying Banach space can help to overcome many of the typical difficulties encountered in the infinite-dimensional situation Only at the end of Chapter II differential equations and initial value problems appear explicitly in our text This does not mean that we neglect this aspect On the contrary, the many applications of semigroup theory to all kinds of evolution equations elaborated in Chapter VI are the ultimate goal of our efforts However, we postpone this discussion until a powerful and systematic theory is at hand In the final chapter, Chapter VII, Tanja Hahn and Carla Perazzoli try to embed today’s theory into a historical perspective in order to give the reader a feeling for the roots and the raison d’ˆetre of semigroup theory Furthermore, we add to our exposition of the mathematical theory an epilogue by Gregor Nickel, in which he discusses the philosophical question concerning the relationship between semigroups and evolution equations and the philosophical concept of “determinism.” This is certainly a matter worth considering, but regrettably not much discussed in the mathematical community For this reason, we encourage the reader to grapple and come to terms with this genuine philosophical question It is enlightening to see how such questions were formulated and resolved in different epochs of the history of thought Perhaps a deeper understanding will emerge of how one’s own contemporary mathematical concepts and theories are woven into the broad tapestry of metaphysics Guide to the Reader The text is not meant to be read in a linear manner Thus, the reader already familiar with, or not interested in, the finite-dimensional situation and the detailed discussion of examples may start immediately with Section I.5 and then proceed quickly to the Hille–Yosida Generation Theorems II.3.5 and II.3.8 via Section II.1 To indicate other shortcuts, several sections, subsections, and paragraphs are given in small print Such an individual reading style is particularly appropriate with regard to Chapter VI, since our applications of semigroup theory to the various evolution equations are more or less independent of each other The reader should select a section according to his/her interest and then continue with the more specialized literature indicated in the notes Or, he/she may x Preface even start with a suitable section of Chapter VI and then follow the back references in the text in order to understand our arguments The exercises at the end of each section should lead to a better understanding of the theory Occasionally, we state interesting recent results as an exercise marked by ∗ The notes are intended to identify our sources, to integrate the text into a broader picture, and to suggest further reading Inevitably, they also reflect our personal perspective, and we apologize for omissions and inaccuracies Nevertheless, we hope that the interested reader will be put on the track to uncover additional information Acknowledgments Our research would not have been possible without the invaluable help from many colleagues and friends We are particularly grateful to Jerome A Goldstein, Frank Neubrander, Ulf Schlotterbeck, and Eugenio Sinestrari, who accompanied our work for many years in the spirit of friendship and constructive criticism Wolfgang Arendt, Mark Blake, Donald Cartwright, Radu Cascaval, Ralph Chill, Andreas Fischer, Helmut Fischer, Gisele Goldstein, John Haddock, Matthias Hieber, Sen-Zhong Huang, Niels Jacob, Yuri Latushkin, Axel Markert, Lahcen Maniar, Martin Mathieu, Mark McKibben, Jan van Neerven, John Neuberger, Martin Newell, Tony OFarrell, Frank Ră abiger, Tim Randolph, Werner Ricker, Fukiko Takeo, and Jă urgen Voigt were among the many colleagues who read parts of the manuscript and helped to improve it by their comments Our students Andr´ as Batkai, Benjamin Bă ohm, Gerrit Bungeroth, Gabriele Gă uhring, Markus Haase, Jens Hahn, Georg Hengstberger, Ralf Hofmann, Walter Hutter, Stefan Immervoll, Tobias Jahnke, Michael Kă olle, Franziska Kă uhnemund, Nguyen Thanh Lan, Martina Morlok, Almut Obermeyer, Susanna Piazzera, Jan Poland, Matthias Reichert, Achim Schă adle, Gertraud Stuhlmacher, and Markus Wacker at the Arbeitsgemeinschaft Funktionalanalysis (AGFA) in Tă ubingen were an inexhaustible source of motivation and inspiration during the years of our teaching on semigroups and while we were writing this book We thank them all for their enthusiasm, their candid criticism, and their personal interest Our coauthors Simon Brendle (Tă ubingen), Michele Campiti (Bari), Tanja Hahn (Frankfurt), Giorgio Metafune (Lecce), Gregor Nickel (Tă ubingen), Diego Pallara (Lecce), Carla Perazzoli (Rome), Abdelaziz Rhandi (Marrakesh), Silvia Romanelli (Bari), and Roland Schnaubelt (Tă ubingen) made important contributions to expanding the range of our themes considerably It was a rewarding experience and always a pleasure to collaborate with them References [Ves96a] [Ves96b] [Ves97] [Vid70] [Voi77] [Voi80] [Voi84] [Voi85] [Voi88] [Voi92] [Voi94] [Vol13] [Vor32] [V˜ u92] [V˜ u97] [Web85] [Web87] [Wei80] [Wei89] [Wei90] [Wei91] [Wei95] 575 E Vesentini, Introduction to Continuous Semigroups, Scuola Normale Superiore Pisa, 1996 E Vesentini, Semiflows and semigroups, Rend Mat Acc Lincei (1996), 75–82 E Vesentini, Spectral properties of weakly asymptotically almost periodic semigroups, Adv Math 128 (1997), 217–241 I Vidav, Spectra of perturbed semigroups with applications to transport theory, J Math Anal Appl 30 (1970), 264–279 J Voigt, On the perturbation theory for strongly continuous semigroups, Math Ann 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Xα · n ·, · (·|·) Abstract Cauchy Problem 17, 84, 145, 150, 151, 295, 347, 368, 419 Abstract Second-Order Cauchy Problem 367 extended Abstract Second-Order Cauchy Problem 374 controlled Abstract Cauchy Problem 452 controlled Abstract Second-Order Cauchy Problem 455 inhomogeneous Abstract Cauchy Problem 436 nonautonomous Abstract Cauchy Problem 227 nonautonomous Abstract Cauchy Problem 477 Abstract Delay Differential Equation 420 Differential Equation 3, 11, 497 Functional Equation 2, 6, 8, 14, 36, 497 Integro-Differential Equation 449 Integral Equation/Variation of Parameters Formula 161 Integral Equation/Variation of Parameters Formula 161 Partial Differential Equation with constant coefficients 404 Partial Differential Equation with variable coefficients 404 Spectral Bound equal Growth Bound Condition 281, 282 Spectral Mapping Theorem 271, 280, 281, 483 Weak Spectral Mapping Theorem 32, 271, 281–283, 301 constant one function 26 characteristic function of the set J 231 graph norm for A 52, 515 essential norm 249 Favard norm of order α 129 Hă older norm of order 130 Sobolev norm of order n 124 canonical bilinear form 511 inner product 517 577 578 Symbols and Abbreviations ⊕2n∈N Xn f ⊗y f ⊗T x⊗x A A∗ A An An Aα A⊂B A|Y AC(J) Aσ(A) c c0 , c0 (X) Cα (J) C∞ (J) Ck (J) C(Ω) C0 (Ω) Cb (Ω) Cc (Ω) Cub (Ω) co(K) D(A) D(A∞ ) D(An ) ess sup F Fα f ∗g f fix(T (t))t≥0 G(A) Γ hα (J) Hk (Ω) Hk0 (J) J(x) ker(Φ) K(X) L ∞ , ∞ (X) p Lipu (J) L∞ (J, X) Lp (J, X) L∞ (Ω, μ) Lp (Ω, μ) L(X), L(X, Y ) Mb (R) Hilbert direct sum of the Hilbert spaces Xn 511 element of a space of vector-valued functions 520 operator on a space of vector-valued functions 520 rank-one operator 521 (Banach space) adjoint of A 517 (Hilbert space) adjoint of A 517 closure of A 516 nth power of A 519 part/extension of A in Xn 124, 126 αth power of A 140 A is a restriction of B 516 part of A in Y 60 space of absolutely continuous functions 64, 510 approximate point spectrum of A 242 space of convergent sequences 509 space of null sequences 509 classical Hă older space of order α 136, 510 space of infinitely many times differentiable functions 510 space of k-times continuously differentiable functions 510 space of continuous functions 510 space of continuous functions vanishing at infinity 25, 510 space of bounded continuous functions 510 space of continuous functions having compact support 25, 510 space of bounded, uniformly continuous functions 510 closed convex hull of K 511 domain of A 49 intersection of the domains of all powers of A 53, 519 domain of An 53, 519 essential supremum 32, 524 Fourier transform 406, 526 Favard space of order α 130 convolution of f with g 164, 527, 530 Fourier transform of f 406, 526 fixed space of (T (t))t≥0 337 graph of A 515 unit circle in C 13 classical little Hă older space of order 137, 510 classical Sobolev space of order (k, 2) 407, 510 classical Sobolev space of order (k, 2) 510 duality set for x ∈ X 87 kernel of Φ 516 space of all compact linear operators on X 248 Laplace transform 530 space of bounded sequences 509 space of p-summable sequences 510 space of uniformly Lipschitz continuous functions 510 space of X-valued measurable, essentially bounded functions 510 space of X-valued p-Bochner integrable functions 510 space of measurable, essentially bounded functions 510 space of p-integrable functions 510 space of bounded linear operators 511, 517 space of regular (signed or complex) Borel measures 510 Symbols and Abbreviations Mq N0 ω0 (U ) ω0 (T) ωess (T) P σ(A) qess (Ω) r(A) ress (T ) R(λ, A) Rσ(A) rg(A) ρ(A) ρF (T ) S (RN ) s(A) StDS StMV Σδ σ(A) Σ(A, B, C) σess (T ) σ(X, X ) σ(X , X) σ+ (A) Sp(T) Sp(U ) supp f (T (t))t≥0 (T (t))t∈R T (t)/Y t≥0 T (t)|Y t≥0 (T (t) ) (Tn (t))t≥0 (Tl (t))t≥0 (Tr (t))t≥0 (T (z))z∈Σδ ∪{0} UBV(R) (U (t, s))t≥s W1,p (J, X) Wk,p (Ω, μ) Xα Xn X Y →X 579 multiplication operator associated to q 25 nonnegative natural numbers 36 growth bound of the evolution family (U (t, s))t≥s 479 growth bound of the semigroup T 40, 299 essential growth bound of the semigroup T 258 point spectrum of A 241 essential range of the function q 31, 524 spectral radius of A 241 essential spectral radius of T 249 resolvent of A in λ 239 residual spectrum of A 243 range of A 517 resolvent set of A 239 Fredholm domain of T 248 Schwartz space of rapidly decreasing functions 405, 511 spectral bound of A 57, 250 class of Desch–Schappacher perturbations 183 class of Miyadera–Voigt perturbations 196 sector in C of angle δ 96 spectrum of A 239 control system 452 essential spectrum of T 248 weak topology 511 weak∗ topology 511 boundary spectrum of A 116 Arveson spectrum of the group T 285 Arveson spectrum of the operator U 287 support of f 25 one-parameter semigroup of linear operators 14 one-parameter group of linear operators 14 quotient semigroup of (T (t))t≥0 in X/Y 61 subspace semigroup of (T (t))t≥0 in Y 60 sun dual semigroup of (T (t))t≥0 62 restricted/extrapolated semigroup of (T (t))t≥0 in Xn 124, 126 left translation semigroup 33 right translation semigroup 33 analytic semigroup of angle δ 101 space of functions with uniformly bounded variation 511 evolution family of linear operators 478 Sobolev space of order (1, p) of Bochner p-integrable functions 510 classical Sobolev space of order (k, p) 413, 510 abstract Hă older space of order α 130 abstract Sobolev space of order n 124, 126, 515 sun dual of X 62 Y continuously embedded in X 60 Index A a-adic solenoid 322 absolutely continuous function 64 absorption operator 362 abstract Cauchy problem See Cauchy problem control system See control system delay differential equation See delay differential equation Dyson–Phillips series 198 extrapolation space 125 Favard space 130 Hă older space 130 second-order Cauchy problem See second-order Cauchy problem Volterra operator 162, 182, 195 adjoint operator 517 algebra homomorphism 58 approximate eigenvalue 242 eigenvector 242 point spectrum 242 Arveson spectrum 285, 287 autonomous system 535 B balanced exponential growth 358 Banach lattice Bernstein operators Bochner integral Bohr compactification Boltzmann, operator boundary condition Dirichlet elastic barrier Neumann Ventcel boundary point boundary spectrum 116, 282, 356 cyclic 336, 353 224 523 322 362 256 397 397 397 398 396 324, 357 C Calkin algebra 248 Cauchy problem 17, 84, 145, 150, 295, 347, 368, 419 controlled 452 inhomogeneous 436 mild solution of 146 nonautonomous 227, 477 second-order See second-order Cauchy problem 580 Index solution of 145 well-posed 151 Ces` aro means 337 chaos paradigm 538 chaos theory 536 character group 313, 320 characteristic equation 256, 335, 352, 388, 427, 451 generalized 431 characteristic function 231, 256, 352 Chernoff product formula 220 classical solution 145, 368, 420, 436, 477 closed convex hull 511 closure of an operator 516 collisionless transport operator 362 compact operator group 319 resolvent 117 semigroup 308 complex inversion formula 233 control Banach space 452 feedback 453 function 452 operator 452 control system approximately p-controllable 457 approximately q-observable 467 exactly p-controllable 456 exactly p-null controllable 457 exactly q-observable 467 exponentially detectable 471 exponentially stabilizable 468 external stability of 475 input–output stability of 475 internal stability of 475 similar 466 strongly stabilizable 472 transfer function of 474 controllability 453 map 456 controlled abstract wave system 455 heat equation 453, 459, 465, 468, 471, 474, 476 wave equation 454, 460, 465, 468, 473, 474 convex compactness property 525 convolution of functions 164, 527, 530 of sequences 527 core of an operator 52, 58, 517 581 D damping operator 367 delay differential equation 93, 255, 420 operator 420 derivation 58 determinism 531 deterministic system 535 diffusion semigroup 68, 69, 177, 224, 384 domain of an operator 49, 53, 519 dominant eigenvalue 13, 331, 360 duality set 87 dynamical system Dyson–Phillips series 163, 186 E eigenvalue 241 dominant 13, 331, 360 simple 13 eigenvector 241 elastic operator 367 entire vector 81 essential growth bound norm range 31, spectral radius spectrum supremum 32, 258 249 524 249 248 524 evolution equation 531 evolution family exponentially bounded exponentially stable hyperbolic with exponential dichotomy 478 479 479 480 480 evolution semigroup 481 extended transfer function 474 extension of an operator 516 external stability 475 extrapolation space 125, 126 582 Index group generator of 79 induced by flow 91 one-parameter 14, 535 periodic 35 rotation 35 solenoidal 320 strongly continuous 37, 78 growth bound of a semigroup 40, 250, 299 of an evolution family 479 F Favard space 130 extrapolated 190 feedback control 453 operator 453 feedback operator 468 flow 91 formula Chernoff product 220 complex inversion 233 Fourier inversion 406, 527 Phragm´en inversion 236 Post–Widder 223, 231, 236 Trotter product 227 variation of parameters 162, 183, 186, 196, 198, 425, 436 Fourier transform 406, 526 inversion theorem 406, 527 uniqueness theorem 527 fractional power 137 Fredholm domain 248 operator 248 function (strongly) measurable 523 absolutely continuous 64 additive almost periodic 317 Bochner integrable 523 control 452 essentially bounded 524 observation 452 rapidly decreasing 405 simple 522 subadditive 251 functional calculus 231, 284, 285 functional equation 2, 6, 497 G Gaussian semigroup 69 generator 17, 47, 49, 79 graph 515 norm 52, 515 Green’s function 391, 485 H Hă older space abstract 130 classical 510 heat semigroup 69 Hilbert direct sum 511 I ideal impulse response function inhomogeneous abstract Cauchy problem initial state initial value problem input input–output stability integral Bochner integral solution integrated semigroup internal stability isolated singularities isometric limit semigroup 309 474 436 452 145 452 475 523 156 153 475 246 263 L Landau–Kolmogorov inequality 59 Laplace transform 530 convolution theorem 530 uniqueness theorem 530 left multiplicative perturbation 201 translation 33 Liapunov stability theorem 12 linear dynamical system 6, 14 Index M mean ergodic projection 338 mild solution 146, 436 moment inequality 141 motion 533 multiplication operator 25, 31, 218 semigroup 27, 32, 65, 120, 217 N non-quasi-analytic weight 289 nonautonomous abstract Cauchy problem 227, 477 hyperbolic case 479 parabolic case 479 solution of 477, 478 well-posed 477 nonautonomous Cauchy problem mild solution of 488 nonobservable subspace 468 O observability 453 map 466 observation Banach space 452 function 452 operator 452 one-parameter group semigroup operator adjoint 61, 517 approximate eigenvalue of 242 approximate eigenvector of 242 approximate point spectrum of 242 bounded 517 central 46 closable 82, 516 closed 515 closure 516 with compact resolvent 117 control 452 core of 52 damping 367 degenerate 383, 390 delay 420 delay differential 92, 255, 420 differential 91 dissipative 82 583 domain of 49 doubly power bounded 285 eigenvalue of 241 elastic 367 elliptic 409 feedback 453 fractional power of 137 Fredholm 248 Hille–Yosida 87 homogeneous 409 implemented 20 infinitely divisible 23 kernel of 516 local 33, 46 multiplication 25, 31 nondegenerate 383 normal 30, 105, 282 observation 452 part of 60, 124, 245, 261 point spectrum of 241 range of 517 relatively bounded 169 relatively compact 179 residual spectrum of 243 resolvent of 15, 55, 239 resolvent set of 15, 55, 239 second-order differential 93, 383 sectorial 96 self-adjoint 30, 90, 106, 324, 459, 460, 506 skew-adjoint 89 spectral bound of 57, 250 spectral decomposition of 244 spectrum of 15, 55, 239 system 452 uniformly elliptic 411, 489 Volterra 162, 182, 195 weakly closed 515 optimal regularity 442 space regularity 447 orbit map 36, 48 reversible 315 stable 315 output 452 output injection operator 471 584 Index P period 266 perturbation additive 157, 169, 201 Desch–Schappacher 182 Miyadera–Voigt 195 multiplicative 201 Phragm´en inversion formula 236 Plancherel’s equation 407 point spectrum 241 pole 246 algebraic multiplicity of 246 algebraically simple 246 first-order 246 geometric multiplicity of 246 population equation 348, 419, 434 Post–Widder inversion formula 223, 231, 236 principal part 409 principle of uniform boundedness 512 pseudoresolvent 206 Q quantum mechanics 536 R reachability space 466 reactor problem 361 relativity 536 residual spectrum 243 residue 246 resolvent 15, 239 compact 117 equation 239 integral representation 55 operator 47 pseudo 206 set 15, 47, 239 resolvent map 239 isolated singularities of 246 poles of 246 residue of 246 Riemann–Lebesgue lemma 406, 526 right multiplicative perturbation 201 translation 33 rotation group 35, 320 S scattering operator 362 Schwartz space 405 second-order Cauchy problem 367 controlled 455 extended 374 overdamped 370 solution of 368 semiflow 95 uniquely ergodic 345 semigroup 308 abstract Sobolev space of 124, 126 adjoint 43 analytic 101 asymptotically norm-continuous 282 with balanced exponential growth 336, 346, 358, 364 bounded 40 C0 36 compact 308 contractive 40 differentiable 109 diffusion 68, 69, 177, 224, 384 dual 62 essential growth bound of 258 essentially compact 336 eventually compact 117 eventually differentiable 109 eventually norm-continuous 112 exponentially stable 298 extrapolation space of 126 fixed space 337 generator of 47 growth bound of 40, 250 hyperbolic 13, 24, 305 immediately compact 117 immediately differentiable 109 immediately norm-continuous 112 irreducible 357 isometric 40 isometric limit 263 law 14, 536 matrix mean ergodic 338 multiplication 27, 122, 217 nilpotent 35, 120, 123 norm-continuous 16 one-parameter 14, 535 orbit of periodic 266 positive 322, 353 product 44, 64 Index quasi-compact 332 quasicontractive 76 quotient 43, 61 rescaled 43, 60 semitopological 308 similar 10, 59 stable 12 strongly continuous 36 strongly stable 296, 323, 326 subspace 43, 60 sun dual 62, 134 totally ergodic 345 translation 34, 35 type of 40 uniformly continuous 16 uniformly exponentially stable 18, 296 uniformly mean ergodic 342 uniformly stable 296 weak∗ generator of 61 weak∗ -continuous 43 weak∗ -stable 298 weakly continuous 40 weakly stable 296 semigroups isomorphic 43 similar 43 semitopological semigroup 308 similar control systems 466 Sobolev embedding theorem 408 tower 126 Sobolev space abstract 124, 126, 129 classical 407 of vector-valued functions 525 solution classical 145, 420, 436 integral 156 mild 146, 436 space of strong continuity 132 spectral decomposition 244 inclusion theorem 276 projection 244 radius 241 theorem 30 spectral bound 57, 250 equal growth bound condition 281 function 430 585 spectral mapping theorem 19, 243, 270, 271 for eventually norm-continuous semigroups 280 for evolution semigroups 483 for the point and residual spectrum 277 for the resolvent 243 weak 32, 271, 283 spectrum 15 approximate point 242 essential 248 point 241 residual 243 stability condition 212, 227 exponential 298 external 475 input–output 475 internal 475 Liapunov 12 strong 296, 323, 326 uniform 296 uniform exponential 18, 296 weak 296 weak∗ 298 stabilizability 453 state space 368, 452, 533 streaming semigroup 363 strong operator topology 512 structural damping 372 subspace of strong continuity 62 sun dual 62 support of a function 25 system 533 autonomous 535 deterministic 535 system operator 452 T tensor product 520 theorem Arendt–Batty–Lyubich–V˜ u 326 Banach–Alaoglu 511 Datko–Pazy 300 Desch–Schappacher 182 GearhartGreinerPră uss 302 Gelfand 521 Greiner 366 Halmos–von Neumann 322 Hille–Yosida 73 Jacobs–Glicksberg–DeLeeuw 314 Kreˇın 511 586 Index Liapunov 12, 19, 302, 346 Lumer–Phillips 83 Miyadera–Voigt 195 Perron–Frobenius 356 Plancherel 407, 529 Sobolev’s embedding 408 Stone 89 Trotter–Kato 209, 212 time 533 topology strong operator 512 uniform operator 512 weak 511 weak operator 512 weak∗ 511 transfer function 474 translation group 34 property 95, 422 semigroup 33–35 transport equation 361 operator 362 semigroup 363 Trotter product formula 227 V variation of parameters formula 162, 183, 186, 196, 198, 425, 436 vector lattice 353 Volterra integro-differential equation 449 operator 162, 182, 195 W weak derivative 407, 413 operator topology 512 spectral mapping theorem 32, 271, 283 topology 511 weak∗ generator 61 topology 511 Wronskian 391 Y Yosida approximation 74, 205, 214 U uniform operator topology 512 ... result we have achieved a theory for uniformly continuous semigroups that is completely parallel to the one for matrix semigroups In particular, assuming uniform continuity, Cauchy’s problem... (2)) Therefore, the following question is natural and leads directly to the objects forming the main objects of this book 3.8 Problem Do there exist “natural” one- parameter semigroups of linear. .. concrete equations, one might object that semigroups, and especially strongly continuous semigroups, are of limited value, and that other concepts such as integrated semigroups, regularized semigroups,