Functional Analysis, Spectral Theory, and Applications

626 Pages·2017·3.077 MB·English

by  Manfred Einsiedler, Thomas Ward

Graduate Texts in Mathematics
Manfred Einsiedler
Thomas Ward
Functional
Analysis,
Spectral Theory,
and Applications
Graduate Texts in Mathematics 276
Graduate Texts in Mathematics
SeriesEditors:
SheldonAxler
SanFranciscoStateUniversity,SanFrancisco,CA,USA
KennethRibet
UniversityofCalifornia,Berkeley,CA,USA
AdvisoryBoard:
AlejandroAdem,UniversityofBritishColumbia
DavidEisenbud,UniversityofCalifornia,Berkeley&MSRI
IreneM.Gamba,TheUniversityofTexasatAustin
J.F.Jardine,UniversityofWesternOntario
JeffreyC.Lagarias,UniversityofMichigan
KenOno,EmoryUniversity
JeremyQuastel,UniversityofToronto
FadilSantosa,UniversityofMinnesota
BarrySimon,CaliforniaInstituteofTechnology
GraduateTextsinMathematicsbridgethegapbetweenpassivestudyandcreative
understanding, offering graduate-level introductions to advanced topics in mathe-
matics.Thevolumesarecarefullywrittenasteachingaidsandhighlightcharacter-
isticfeaturesofthetheory.Althoughthesebooksarefrequentlyusedastextbooks
ingraduatecourses,theyarealsosuitableforindividualstudy.
Moreinformationaboutthisseriesathttp://www.springer.com/series/136
Manfred Einsiedler • Thomas Ward
Functional Analysis, Spectral
Theory, and Applications
Manfred Einsiedler Thomas Ward
ETH Zürich School of Mathematics
Zürich, Switzerland University of Leeds
Leeds, UK
ISSN 0072-5285 ISSN 2197-5612 (electronic)
Graduate Texts in Mathematics
ISBN 978-3-319-58539-0 ISBN 978-3-319-58540-6 (eB ook)
DOI 10.1007/978-3-319-58540-6

Library of Congress Control Number: 2017946473

Mathematics Subject Classification (2010): 46-01, 47-01, 11N05, 20F69, 22B05, 35J25, 35P10, 35P20,
37A99, 47A60

© Springer International Publishing AG 2017
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,
recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or
information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar
methodology now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this
publication does not imply, even in the absence of a specific statement, that such names are exempt
from the relevant protective laws and regulations and therefore free for general use.
The publisher, the authors and the editors are safe to assume that the advice and information in this
book are believed to be true and accurate at the date of publication. Neither the publisher nor the
authors or the editors give a warranty, express or implied, with respect to the material contained herein
or for any errors or omissions that may have been made. The publisher remains neutral with regard to
jurisdictional claims in published maps and institutional affiliations.

Printed on acid-free paper
This Springer imprint is published by Springer Nature
The registered company is Springer International Publishing AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Preface
Believe us, we also asked ourselves what could be the rationale for ‘Yet an-
other book on functional analysis’.(1) Little indeed can justify this beyond
our own enjoyment of the beauty and power of the topics introduced here.
Functional analysis might be described as a part of mathematics where
analysis,topology,measuretheory,linearalgebra,andalgebracometogether
to create a rich and fascinating theory. The applications of this theory are
then equally spread throughout mathematics (and beyond).
We follow some fairly conventional journeys, and have of course been in-
fluencedbyotherbooks,mostnotablythatofLax[59].While developingthe
theory we include reminders of the various areas that we build on (in the
appendices andthroughoutthe text) but we alsoreachsome fairly advanced
anddiverseapplicationsofthematerialusuallycalledfunctionalanalysisthat
often do not find their place in a course on that topic.
Theassembledmaterialprobablycannotbe coveredinayear-longcourse,
but has grown out of several such introductory courses taught at the Eid-
geno¨ssische Technische Hochschule Zu¨rich by the first named author, with
a slightly different emphasis on each occasion. Both the student and (espe-
cially) the lecturer should be brave enough to jump over topics and pick the
material of most interest, but we hope that the student will eventually be
sufficiently interested to find out what happens in the material that was not
covered initially. The motivation for the topics discussed may by found in
Chapter 1.
Notation and Conventions
The symbols N = 1,2,... , N = N 0 , and Z denote the natural
0
{ } ∪{ }
numbers, non-negative integers and integers; Q, R, C denote the rational
numbers, real numbers and complex numbers. The real and imaginary parts
of a complex number are denoted by x= (x+iy) and y = (x+iy).
ℜ ℑ
For functions f,g defined on a set X we write f =O(g) or f g if there
≪
is a constant A>0 with f(x) 6A g(x) for all x X. When the implied
k k k k ∈
constantAdependsonasetofparameters ,wewritef =O (g)orf g
A A ≪A
v
vi Preface
(but we may also forget the index if the set of parameters will not vary at
allinthediscussion).Asequencea ,a ,...inanyspacewillbe denoted(a )
1 2 n
(or (a ) if we wish to emphasize the index variable of the sequence). For
n n
two C-valued functions f,g defined on Xr x for a topological space X
0
{ }
containingx wewritef =o(g)asx x iflim f(x) =0.Thisdefinition
0 → 0 x→x0 g(x)
includes the case of sequences by letting X = N and x = with
0
∪{∞} ∞
the topology of the one-point compactification. Additional specific notation
introducedthroughoutthetextiscollectedinanindexofnotationonp.600.
Prerequisites
We will assume throughout that the reader is familiar with linear algebra
and quite frequently that she is also familiar with finite-dimensional real
analysis and complex analysis in one variable. Further background and con-
ventionsintopologyandmeasuretheoryarecollectedintwoappendices,but
letusnotethatthroughoutcompactandlocallycompactspacesareimplicitly
assumed to be Hausdorff.
Organisation
Thereare402exercisesinthetext,221ofthesewithhintsinanappendix,
allofwhichcontributetothereader’sunderstandingofthematerial.Asmall
numberareessentialtothedevelopment(oftheideasinthesectionoroflater
theories);thesearedenoted‘EssentialExercise’tohighlighttheirsignificance.
WeindicatethedependenciesbetweenthevariouschaptersintheLeitfaden
overleaf and in the guide to the chapters that follows it.
Acknowledgements
We are thankful for various discussions with Menny Aka, Uri Bader,
Michael Bj¨orklund, Marc Burger, Elon Lindenstrauss, Shahar Mozes, Ren´e
Ru¨hr, Akshay Venkatesh, and Benjamin Weiss on some of the topics presen-
ted here. We also thank Emmanuel Kowalski for making available his notes
on spectral theory and allowing us to raid them. We are grateful to sev-
eral people for their comments on drafts of sections, including Menny Aka,
ManuelCavegn,RexCheung,AnthonyFlatters,MaximGerspach,Tommaso
Goldhirsch, Thomas Hille, Guido Lob, Manuel Lu¨thi, Clemens Macho, Alex
Maier, Andrea Riva, Ren´e Ru¨hr, Lukas Ruosch, Georg Schildbach, Samuel
Stark, Andreas Wieser, Philipp Wirth, and Gao Yunting. Special thanks are
due to Roland Prohaska, who proofread the whole volume in four months.
Needless to say, despite these many helpful eyes, some typographical and
other errorswill remain— these are ofcourse solelythe responsibility of the
authors.
The second named author also thanks Grete for her repeated hospitality
whichsignificantlyaidedthisbook’scompletion,andthanksSaskiaandToby
for doing their utmost to prevent it.
Manfred Einsiedler, Zu¨rich
Thomas Ward, Leeds
2nd April 2017
Leitfaden
BanachSpaces 2
HilbertSpaces&FourierSeries
3
4 5 SobolevSpaces
&DirichletProblem
Completeness
6
DualSpaces 7
CompactOperators
&Weyl’sLaw
Weak*Compactness
&LocallyConvexSpaces 8
UnitaryOperators 9 BanachAlgebras
&FourierTransform 10 &PontryaginDuals
HaarMeasure,
Amenability, 11
&Property(T)
SpectralTheorems
12
&PontryaginDuality
14 13
PrimeNumberTheorem UnboundedOperators
vii
viii Preface
Guide to Chapters
Chapter 1 is mostly motivational in character and can be skipped for the
theoretical discussions later.
Chapter 4 has a somewhat odd role in this volume. On the one hand
it presents quite central theorems for functional analysis that also influence
manyofthedefinitionslaterinthevolume,butontheotherhand,bychance,
the theorems are not crucial for our later discussions.
ThedottedarrowsintheLeitfadenindicatepartialdependencies.Chapter6
consists of two parts; the discussion of compact groups depends only on
Chapter3whilethematerialonLaplaceeigenfunctionsalsobuildsonmater-
ial from Chapter 5. The discussionof the adjoint operatorand its properties
in Chapter 6 is crucial for the spectral theory in Chapters 11, 12, and 13.
Moreover, one section in Chapter 8 builds on and finishes our discussion of
Sobolev spaces in Chapters 5 and 6. Finally, some of Chapter 11 needs the
discussion of Haar measures in the first section of Chapter 10.
With these comments and the Leitfaden it should be easy to designmany
different courses of different lengths focused around the topic of Functional
Analysis.
Contents
1 Motivation ............................................... 1
1.1 From Even and Odd Functions to Group Representations.... 1
1.2 Partial Differential Equations and the Laplace Operator..... 5
1.2.1 The Heat Equation ............................... 7
1.2.2 The Wave Equation .............................. 10
1.2.3 The Mantegna Fresco ............................. 11
1.3 What is Spectral Theory? ............................... 12
1.4 The Prime Number Theorem ............................ 13
1.5 Further Topics ......................................... 14
2 Norms and Banach Spaces................................ 15
2.1 Norms and Semi-Norms ................................. 15
2.1.1 Normed Vector Spaces ............................ 16
2.1.2 Semi-Norms and Quotient Norms................... 21
2.1.3 Isometries are Affine .............................. 23
2.1.4 A Comment on Notation .......................... 26
2.2 Banach Spaces ......................................... 26
2.2.1 Proofs of Completeness ........................... 29
2.2.2 The Completion of a Normed Vector Space .......... 36
2.2.3 Non-Compactness of the Unit Ball.................. 38
2.3 The Space of Continuous Functions ....................... 39
2.3.1 The Arzela–Ascoli Theorem ....................... 40
2.3.2 The Stone–Weierstrass Theorem.................... 42
2.3.3 Equidistribution of a Sequence ..................... 48
2.3.4 Continuous Functions in Lp Spaces ................. 51
2.4 Bounded Operators and Functionals ...................... 55
2.4.1 The Norm of Continuous Functionals on C (X) ...... 60
0
2.4.2 Banach Algebras ................................. 61
2.5 Ordinary Differential Equations .......................... 62
2.5.1 The Volterra Equation ............................ 63
2.5.2 The Sturm–Liouville Equation ..................... 66
ix