Table Of ContentProgress in Mathematics
Volume 281
Series Editors
Hyman Bass
Joseph Oesterlé
Alan Weinstein
Topics in Operator Semigroups
Shmuel Kantorovitz
Birkhäuser
Boston • Basel • Berlin
Shmuel Kantorovitz
Department of Mathematics
Bar Ilan University
52900 Ramat Gan
Israel
kantor@macs.biu.ac.il
ISBN 978-0-8176-4931-9 e-ISBN 978-0-8176-4932-6
DOI 10.1007/978-0-8176-4932-6
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009938183
Mathematics Subject Classification (2000): 47D03, 47D06, 47D60, 47D62
© Birkhäuser Boston, a part of Springer Science+Business Media, LLC 2010
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To Ita, Bracha, Pnina, Pinchas, and Ruth
Contents
Preface ........................................................ xi
Part I General Theory
A. Basic Theory .............................................. 3
A.1 Overview............................................... 3
A.2 The Generator .......................................... 5
A.3 Type and Spectrum...................................... 9
A.4 Uniform Continuity...................................... 10
A.5 Core for the Generator ................................... 11
A.6 The Resolvent .......................................... 13
A.7 Pseudo-Resolvents....................................... 15
A.8 The Laplace Transform................................... 17
A.9 Abstract Potentials ...................................... 18
A.10The Hille–Yosida Theorem................................ 20
A.11The Hille–Yosida Space .................................. 22
A.12Dissipative Operators .................................... 25
A.13The Trotter–Kato Convergence Theorem ................... 28
A.14Exponential Formulas.................................... 32
A.15Perturbation of Generators ............................... 36
A.16Groups of Operators ..................................... 42
A.17Bounded Groups of Operators............................. 43
A.18Stone’s Theorem ........................................ 44
A.19Bochner’s Theorem ...................................... 47
B. The Semi-Simplicity Space for Groups..................... 49
B.1 The Bochner Norm ...................................... 49
B.2 The Semi-Simplicity Space................................ 53
B.3 Scalar-Type Spectral Operators ........................... 59
viii Contents
C. Analyticity ................................................ 63
C.1 Analytic Semigroups..................................... 63
C.2 The Generator of an Analytic Semigroup ................... 65
D. The Semigroup as a Function of its Generator ............. 71
D.1 Noncommutative Taylor Formula .......................... 71
D.2 Analytic Families of Semigroups........................... 79
E. Large Parameter........................................... 87
E.1 Analytic Semigroups..................................... 87
E.2 Resolvent Iterates ....................................... 90
E.3 Mean Stability .......................................... 94
E.4 The Asymptotic Space ...................................103
E.5 Semigroups of Isometries .................................107
E.6 The ABLV Stability Theorem.............................109
F. Boundary Values ..........................................113
F.1 Regular Semigroups and Boundary Values ..................113
F.2 The Generator of a Regular Semigroup.....................118
F.3 Examples of Regular Semigroups ..........................121
G. Pre-Semigroups............................................131
G.1 The Abstract Cauchy Problem ............................132
G.2 The Exponentially Tamed Case ...........................136
Part II Integral Representations
A. The Semi-Simplicity Space.................................141
A.1 The Real Spectrum Case .................................141
A.2 The Case R+ ⊂ρ(−A) ...................................154
B. The Laplace–Stieltjes Space ...............................161
B.1 The Laplace–Stieltjes Space...............................161
B.2 Semigroups of Closed Operators...........................166
B.3 The Integrated Laplace Space.............................169
B.4 Integrated Semigroups ...................................173
C. Families of Unbounded Symmetric Operators..............177
C.1 Local Symmetric Semigroups..............................177
C.2 Nelson’s Analytic Vectors Theorem ........................181
C.3 Local Bounded Below Cosine Families......................183
C.4 Local Symmetric Cosine Families ..........................187
Contents ix
Part III A Taste of Applications
A. Analytic Families of Evolution Systems ....................195
A.1 Coefficients Analyticity and Solutions Analyticity............195
A.2 Kato’s Conditions .......................................196
A.3 Tanabe’s Conditions .....................................198
B. Similarity..................................................203
B.1 Overview...............................................203
B.2 Similarity Within the Family S+ζV ......................203
B.3 Similarity of Certain Perturbations ........................217
Miscellaneous Exercises........................................219
Notes and References..........................................249
Bibliography...................................................253
Index..........................................................263
Preface
This book is based on lecture notes from a second-year graduate course, and
is a greatly expanded version of our previous monograph [K8]. We expose
some aspects ofthe theory ofsemigroups of linear operators,mostly (but not
only) from the point of view of its meeting with that part of spectral theory
which is concerned with the integral representation of families of operators.
This approach and selection of topics differentiate this book from others in
the generalarea,and reflectthe author’s ownresearchdirections.There is no
attempt therefore to cover thoroughly the theory of semigroups of operators.
This theoryandits applicationsareextensivelyexposedinmany books,from
theclassicHille–Phillipsmonograph[HP]tothemostrecenttextbookofEngel
and Nagel [EN2] (see [A], [BB], [Cl], [D3], [EN1], [EN2], [Fat], [G], [HP], [P],
[Vr], and others), as well as in chapters in more general texts on Functional
Analysis and the theory of linear operators (cf. [D5], [DS I–III], [Kat1], [RS],
[Y], and many others). Nevertheless, because the book is based on a course,
andbecause we intendedto make itreasonablyself-containedand convenient
both for independent study and for a graduate course or seminar, we have
included in Section A of Part I (making it thereby the longest section of the
book!) an exposition of the basic theory: the classical Hille–Yosida theory on
theinterplaybetweenasemigroupanditsgeneratoruptothecharacterization
of the generator of a (strongly continuous) semigroup by means of estimates
on the resolvent iterates, the Lumer–Phillips theory of dissipative operators
with its “resolvent-free” characterization of the generator, the Trotter–Kato
convergencetheoremonthe equivalenceof“graphconvergence”ofgenerators
and “strong convergence” of the corresponding semigroups, the Kato unified
treatment of the “exponential formula” and the “Trotter product formula,”
and the Hille–Phillips perturbation theorem for generatorsofCo-semigroups.
Asatransitiontothe“integralrepresentations”mentionedabove,weconclude
this section with Stone’s theorem for (semi)groups of unitary operators and
Sz.-Nagy’s spectral integral representationfor bounded groups of operatorsin
Hilbert space.
xii Preface
In Section B of Part I, we construct the semi-simplicity space for a given
Co-group of operators in Banach space. It is a Banach subspace which is
maximal for the existence of a spectral integral representation of the group
on it.
In Section C, we are concerned with analytic semigroups, that is, semi-
groups that possess an analytic continuation to some sector in the complex
plane. We present an approach independent of contour integrals, that yields
easily to characterizations of the generators of such semigroups.
The semigroup is considered as a function of its generator in Section D.
We provea “noncommutativeTaylorformula,” andconsider families of semi-
groups whose generators depend analytically on a complex parameter in a
naturalsense.The conceptualmeaningofthe latteranalysisis the hereditary
property of analyticity from the coefficients of an Abstract Cauchy Problem
to its solution.
Theasymptotic behavior of(one-parameter)semigroupsforlargevaluesof
theparameteristakenupinSectionE.Wefirstconsidertherelativelysimple
caseofanalyticsemigroupsandofvariouskindsof“averages”ofasemigroup,
which include as special cases its Cesaro,Abel, and Gauss averages.We then
prove the Arendt–Batty–Lyubich–Vu (“ABLV”) stability theorem, using the
technique of the so-called “asymptotic space.” Adequate conditions on the
spectrum of the generator insure the (strong) “stability” of the semigroup,
that is, the latter’s strong convergence to zero when the parameter tends
to infinity. Additional results on stability are included in the “Miscellaneous
Exercises” section at the end of the book.
In Section F, we obtain a characterization of generators of regular semi-
groups,thatis,analyticsemigroupsintherighthalfplanethatpossessbound-
ary values on the imaginary axis. We then proceed with the analysis of some
classical examples.
Abriefdiscussionofpre-semigroups, alsocalled“C-semigroups”or“regu-
larizedsemigroups”intheliterature,concludesPartIofthebook(SectionG).
Pre-semigroups were introduced in germinal form in [DaP], and their exten-
sivestudywasstartedin[DP].Theyplayaroleinthesolutionoftheabstract
Cauchy problem for an operator which is not necessarily the generator of a
semigroup, and is not even densely defined. (The monograph [DL4] presents
the theory in great detail, as well as many applications to partial differential
equations.)
In PartII, we turn to a more detailed study of integral representationsin
the spirit of Section B of Part I.
In Section A, the semi-simplicity space is constructed for (generally
unbounded)operatorsthatarenotnecessarily semigroupgenerators,provided
they have real spectrum, or at least have a half-line in their resolvent set.
A spectralintegralrepresentationis obtained for the part of the givenopera-
tor in its semi-simplicity space, and the latter is a maximal Banachsubspace
with this property.
