Table Of ContentLecture Notes in Mathematics 2322
Mohammed Hichem Mortad
The Fuglede-Putnam
Theory
Lecture Notes in Mathematics
Volume 2322
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Mohammed Hichem Mortad
The Fuglede-Putnam Theory
MohammedHichemMortad
DepartmentofMathematics
UniversityofOran1
Oran,Algeria
ISSN0075-8434 ISSN1617-9692 (electronic)
LectureNotesinMathematics
ISBN978-3-031-17781-1 ISBN978-3-031-17782-8 (eBook)
https://doi.org/10.1007/978-3-031-17782-8
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Preface
When I took a course on spectraltheory some 25 yearsago, the Fuglede theorem
was not part of the syllabus of that course. Indeed, that course was mainly
directed toward unbounded self-adjoint operators, which is the main application
of unbounded operators in quantum mechanics, say. The first time I heard of
the Fuglede theorem was during a Banach algebra conference in honor of Barry
Johnson,thattookplaceatNewcastleUniversityuponTyne(U.K.)in2001.
Asreadersareprobablyaware,thespectraltheoremisthemostcrucialresultin
operatortheory.We cansay,atleastasfarasnormaloperatorsareconcerned,that
theFuglede-Putnamtheoremisthesecondsalientresult.
Manymathematicianshavewrittenpapersaboutgeneralizationsandapplications
of the Fuglede-Putnam theorem. Most of these contributions are included in this
book.TheFuglede-Putnamtheoremisnotjustsomeveryusefultheoremanymore,
it has evolved into a small theory, and that is how I have chosen the title of this
manuscript.
Thepresentmonographismainlyintendedforpostgraduatestudentsaswellas
researchersinterestedin operatortheory.Readersare thereforeexpectedto have a
goodknowledgeofbothboundedandunboundedoperatortheory,andwhilemost
definitionsandnotationsshouldbeknowntothem,theyshouldreadAppendixA.
AlthoughthemaintopicistheFuglede-Putnamtheoremanditsvariants,appli-
cations, and other related results should interest readers. This book could also be
taughtasaspecializedcourse.Itisdividedintoseveralchapters,eachchapterhaving
its particularity. Also, most chapters have a counterexamples section in which
readersmayfindafairamountofinterestingexamplesdemonstratingtheboundaries
of possible assertions. Overall, I have tried to gather all (?) about the Fuglede-
Putnam theorem as regards historical notes, classical versions with their different
proofs, generalization to Banach spaces, asymptotic versions, generalizations to
non-normal operators, generalizations to unbounded operators, counterexamples,
applications,intertwiningrelations,andconjectures.Myaimhasnotbeentojustlist
versionsandgeneralizationsoftheFuglede-Putnamtheorembuttoalsogiveideas
ofhowtoapplythispowerfultoolinsomesituationsinoperatortheory.Sometimes,
v
vi Preface
wethinkthatapriori,thereisnoroomfortheFuglede-Putnamtheorem,yetwefind
anefficientplacewhereitfitsperfectly.
A comment about applications needs, however, to be singled out. There is a
chapterdevotedtoapplicationsthat,alas,cannotallbeincludedforobviousreasons.
Indeed,Icertainlycannotbeawareofallapplications.Moreover,evenifIpresume
thatIknowthemall,thereisstillanissueregardingthefeasibility.So,thechoiceof
applicationsmightnotbeunanimous,forwantofabetterword.
In fine, I must say that I have enjoyed writing this book. I hope readers will
equallyenjoyreadingit.Theyareverymuchencouragedtopointoutpossibleerrors,
typos,andsuggestionstomye-mailaddress:[email protected].
Oran,Algeria MohammedHichemMortad
July5,2022
Contents
1 ClassicalVersionsandSomeHistoricalNotes............................ 1
1.1 Finite-DimensionalVersions ......................................... 1
1.2 TheClassicalFuglede-PutnamTheorem ............................ 3
1.3 Weiss’TheoremEtal. ................................................ 10
1.4 Counterexamples ...................................................... 11
References .................................................................... 14
2 GeneralizationstoBoundedNonnormalOperators ..................... 17
2.1 Barría’sLemmaEtal. ................................................. 17
2.2 GeneralizationsModulotheHilbert-SchmidtClass ................. 18
2.3 GeneralizationstoSubnormalorHyponormalOperators ........... 25
2.4 Generalizationstop-Hyponormalorlog-Hyponormal
Operators .............................................................. 27
2.5 AGeneralizationBasedontheJuliaOperator ....................... 29
2.6 FurtherCommentsonOtherGeneralizations ........................ 32
2.7 Counterexamples ...................................................... 32
References .................................................................... 36
3 AsymptoticVersions......................................................... 39
References .................................................................... 42
4 GeneralizationsoftheFuglede-PutnamTheoremtoBanach
AlgebrasandSpaces......................................................... 43
4.1 Generalizations of the Fuglede-Putnam Theorem
toBanachAlgebras ................................................... 43
4.2 Generalizations of the Fuglede-Putnam Theorem
toBanachSpaces...................................................... 44
References .................................................................... 48
5 GeneralizationstoUnboundedOperators ................................ 51
5.1 All-Unbounded-OperatorVersions ................................... 51
5.2 OtherGeneralizationstoUnboundedOperators ..................... 58
vii
viii Contents
5.3 Stochel’s Version InvolvingUnboundedSubnormal
andHyponormalOperators ........................................... 59
5.4 Counterexamples ...................................................... 62
References .................................................................... 65
6 SomeApplications ........................................................... 67
6.1 Putnam’sApplications ................................................ 67
6.2 Kaplansky’sTheorem ................................................. 69
6.3 Devinatz-Nussbaum’sTheorem ...................................... 72
6.4 Embry’sThereom ..................................................... 73
6.5 NormalityoftheProductofNormalOperators...................... 74
6.6 NormalityoftheSumofNormalOperators ......................... 78
6.7 AbsoluteValuesofProductsandSums .............................. 82
6.8 Self-AdjointnessoftheNormalProductofSelf-Adjoint
Operators .............................................................. 89
6.8.1 CommutativityUptoaFactor ............................... 97
6.9 AlgebrasofNormalOperators ....................................... 102
6.10 AResultbyF.Kittaneh ............................................... 103
6.11 RootsofNormalOperators ........................................... 104
6.12 Radjavi-Rosenthal’sTheorem ........................................ 107
6.13 CertainMaximalityResults .......................................... 108
6.14 Counterexamples ...................................................... 111
References .................................................................... 114
7 SomeOtherIntertwiningRelations........................................ 119
7.1 Beck-Putnam’sTheoremandItsGeneralizations ................... 119
7.2 Berberian’sTheorem:GeneralizationsandApplications ........... 121
7.3 Sheth-Williams’TheoremandItsGeneralization ................... 129
7.4 OperatorEquationsInvolvingtheShiftOperator ................... 131
7.5 Counterexamples ...................................................... 132
References .................................................................... 136
8 Conjectures................................................................... 139
8.1 Conjectures ............................................................ 139
References .................................................................... 141
A BoundedandUnboundedLinearOperators ............................. 143
A.1 BoundedLinearOperators............................................ 143
A.2 UnboundedLinearOperators ......................................... 150
References .................................................................... 155
Index............................................................................... 157
Chapter 1
Classical Versions and Some
Historical Notes
1.1 Finite-Dimensional Versions
Itallstartedin1942whenthelegendaryJ.vonNeumannestablishedthefollowing
result:
Theorem1.1.1([23]) LetAandB betwosquarematricesofordern.Then
(cid:2)AB−BA(cid:2)2 −(cid:2)AB∗−B∗A(cid:2)2 =−tr[(A∗A−AA∗)(B∗B−BB∗)].
F F
√
where(cid:2)·(cid:2) istheFrobeniusnorm,i.e.,(cid:2)A(cid:2) = tr(A∗A).
F F
Inparticular,ifeitherAorB isnormal,then
AB =BA⇐⇒AB∗ =B∗A.
Proof Usingbasicpropertiesofthetraceofmatrices,weeasilyobtain:
(cid:2)AB−BA(cid:2)2 =tr[(AB−BA)∗(AB−BA)]
F
=tr(B∗A∗AB−B∗A∗BA−A∗B∗AB+A∗B∗BA)
=tr(B∗A∗AB)−tr(B∗A∗BA)−tr(A∗B∗AB)+tr(A∗B∗BA)
=tr(A∗ABB∗)−tr(AB∗A∗B)−tr(BA∗B∗A)+tr(AA∗B∗B)
=tr(A∗ABB∗)−tr(AB∗A∗B)−tr(ABA∗B∗)+tr(AA∗B∗B)
=tr(A∗ABB∗−AB∗A∗B−ABA∗B∗+AA∗B∗B).
∗
ByreplacingeachB byB ,weget
(cid:2)AB∗−B∗A(cid:2)2 =tr(A∗AB∗B−ABA∗B∗−AB∗A∗B+AA∗BB∗).
F
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