Table Of ContentPhilip J. Maher
Operator
Approximant
Problems Arising
from Quantum
Theory
Operator Approximant Problems Arising
from Quantum Theory
Philip J. Maher
Operator Approximant
Problems Arising from
Quantum Theory
PhilipJ.Maher
London,UnitedKingdom
ISBN978-3-319-61169-3 ISBN978-3-319-61170-9 (eBook)
DOI10.1007/978-3-319-61170-9
LibraryofCongressControlNumber:2017945908
MathematicsSubjectClassification(2010):34-XX,46-XX,46N50,41-XX
©SpringerInternationalPublishingAG2017
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To thememoryof mymother,MarjorieRose
(1914–2012);and tothefutureof Shirley
andAnna.
Preface
Thisbookrepresentsanaccountofsomepartsofoperatortheory,developedmainly
since the 1980swhose problemshave their rootsin quantumtheory.The research
presentedisinnon-commutativeoperatorapproximationtheoryor,touseHalmos’
terminology, in operator approximants. The crucial concept of approximant is
explicated in Chap.1 (“What this book is about”) where the range of problems
is outlined. The setting is mainly,but notexclusively,the Von Neumann-Schatten
classesC .
p
Thus, quantum chemistry—approximating a Hamiltonian and the Lowdin
orthogonalization—precipitatesChap.3(“Self-adjointandpositiveapproximants”)
and Chap.6 (“Unitary, isometric and partially isometric approximants”)
respectively.ThecommutationrelationofquantummechanicsprecipitatesChap.4
(“Commutatorapproximants”).
Ihavetried—bysomenecessarysimplification—topresentthequantumtheory
backgroundasself-contained.Thebookthereforeassumesnoscientificknowledge
onthepartofthereader.Inanycase,ifthereaderisinterestedinthemathematics
alonethensheorhecouldskipthequantumtheorymotivationalsections;butthat
wouldbetomisssomeoftheinterest.
Obviously, this book presents the necessary mathematical machinery to tackle
thevariousapproximantproblems.Specifically,Chap.2statestheAiken,Erdosand
Goldsteinresultindifferentiatingthe mapC ! RC givenbyX 7! kXkp, crucial
p p
formostofthiswork;andChap.5developsthematerialonspectralapproximants
requiredinChap.6.
Chapter 2 onward come equipped with a set of exercises whose purpose is to
extend, in various directions, the material presented in the body of the chapter. I
stronglyadvisethereadertotackletheseexercisessincetheywill,asitwere,enable
the readerto activelyparticipatein the contentofthiswork.Solutionsof manyof
theexercisescanbefoundinthevariouspapersdiscussedinthe“Notes”withwhich
eachchapterconcludes.
The reader of this book is expected to have a background in Hilbert space
operatortheoryapproachingthatofHalmos’marvellous“AHilbertSpaceProblem
vii
viii Preface
Book” [23](towhichreferenceisfrequentlymade).Forsuchareaderthisbookis
suitableforstudyatpostgraduatelevel.
IthankDr.RehanaBariforproducingthebookinimmaculateLATEX.
IthankDr.RogerSchafirfordiscussionsonquantumtheory.
I thankDr. ThomasHempflingofBirkhäuserfortheforbearancehehasshown
overthelonggestationofthetypescript.
I thank Dr. Matthew M. Jones and Dr. Thomas Bending for their characteristi-
cally meticulouscheckingof the final manuscriptwhich saved me from makinga
lotoffoolisherrors.
Someofthisworkoriginated,longago,inmyPh.Dthesis.IthankDr.JohnErdos
forthehelphesofreelyandgenerouslygavemeasmyPh.Dsupervisor.
London,UK PhilipJ.Maher
March2016
Contents
1 WhatThisBookIsAbout:Approximants ................................ 1
2 Preliminaries ................................................................. 5
2.1 Operators(inGeneral).................................................. 5
2.2 TheSpectralTheoremandthePolarDecomposition ................. 6
2.3 CompactOperators...................................................... 8
2.4 TheVonNeumann-SchattenClassesC ................................ 9
p
Exercises....................................................................... 13
Notes........................................................................... 13
3 Self-AdjointandPositiveApproximants .................................. 15
3.1 QuantumChemicalBackground:Approximating
aHamiltonian ........................................................... 15
3.2 Self-AdjointApproximants............................................. 16
3.3 PositiveApproximants.................................................. 20
Exercises....................................................................... 24
Notes........................................................................... 25
4 CommutatorApproximants ................................................ 27
4.1 The Commutation Relation of Quantum Mechanics
andtheHeisenbergUncertaintyPrinciple ............................. 27
4.2 Wielandt–WintnerTheorem............................................ 31
4.3 CommutatorApproximantsinL.H/................................... 33
4.4 CommutatorApproximantsinC ...................................... 41
p
4.5 GeneralizedCommutatorApproximantsinC ........................ 46
p
4.6 Self-CommutatorApproximantsinC ................................. 48
p
Exercises....................................................................... 55
Notes........................................................................... 56
5 Spectral,andNumericalRange,Approximants.......................... 57
5.1 SpectralApproximantsinL.H/........................................ 57
5.2 SpectralApproximantsinC ........................................... 61
p
ix
x Contents
5.3 NumericalRangeApproximants....................................... 68
5.4 Proximality.............................................................. 69
Exercises....................................................................... 71
Notes........................................................................... 73
6 Unitary, Isometric and PartiallyIsometric Approximation
ofPositiveOperators ........................................................ 75
6.1 Quantum Chemical Background: The Lowdin
Orthogonalization ...................................................... 75
6.2 IsometricApproximationofPositiveOperators....................... 79
6.3 PartiallyIsometricApproximationofPositiveOperators............. 87
Exercises....................................................................... 100
Notes........................................................................... 101
Bibliography...................................................................... 103
Index............................................................................... 105
Chapter 1
What This Book Is About: Approximants
Thekeyconceptofthisbookisthatofanapproximant(thecharacteristicallysnappy
term is due to Halmos [21]). Let L, say, be a space of mathematical objects
(complexnumbersorsquarematrices,say);letN bea subsetofLeachofwhose
elementshavesome“nice”propertyp(ofbeingrealorbeingself-adjoint,say);and
let A be some given, not nice element of L; then a p-approximantof A is a nice
mathematical object that is nearest, with respect to some norm, to A. In the first
examplejustmentioned,agivencomplexnumberzhasitsrealpartRz.D zCzN/as
2
its(unique)realapproximant.Inthe secondexample,a givensquarematrixA has
(byTheorem3.2.1)itsrealpartRA.D ACA(cid:2)/asitsuniqueself-adjointapproximant.
2
An approximantthen minimizes the distance between the set N of nice math-
ematical objects in L and the given object A. Thus, with jjj(cid:2)jjj the norm on L, an
elementA0inN isap-approximantofAif,forallXinN,
jjjA(cid:3)A0jjj(cid:4)jjjA(cid:3)Xjjj:
Theconceptofanapproximantisillustratedbythefollowingdiagrams:
Figure 1.1 shows A0 as the unique approximant of A; Fig.1.2 shows every point
ontheinnercircleasanapproximantofA.Thereare,thus,ineachcontextseveral
problems:
(I) findanapproximantofthegivenobjectA;
(II) decideifitisunique.
Figures1.1and1.2suggestthattheuniquenessofanapproximantdepends—in
part, at least—on the “shape” of the set N of nice objects, as will be confirmed
in the contextof the Von Neumann-Schattenclasses C and normsk(cid:2)k , for 1 <
p p
p < 1 (See Theorem 2.4.1). Notice that in the two examples mentioned earlier
(ofrealapproximationandofself-adjointapproximation),wherethereisa unique
approximantthe set N (ofrealnumbersandof self-adjointmatrices)is convexin
eachcase.
©SpringerInternationalPublishingAG2017 1
P.J.Maher,OperatorApproximantProblemsArisingfromQuantumTheory,
DOI10.1007/978-3-319-61170-9_1
Description:This book offers an account of a number of aspects of operator theory, mainly developed since the 1980s, whose problems have their roots in quantum theory. The research presented is in non-commutative operator approximation theory or, to use Halmos' terminology, in operator approximants. Focusing on
