Table Of ContentInfosys Science Foundation Series in Mathematical Sciences
Pintu Bhunia
Silvestru Sever Dragomir
Mohammad Sal Moslehian
Kallol Paul
Lectures
on Numerical
Radius
Inequalities
Infosys Science Foundation Series
Infosys Science Foundation Series
in Mathematical Sciences
SeriesEditors
GopalPrasad,UniversityofMichigan,AnnArbor,USA
IreneFonseca,CarnegieMellonUniversity,Pittsburgh,PA,USA
EditorialBoard
ChandrashekharKhare,UniversityofCalifornia,LosAngeles,USA
MahanMj,TataInstituteofFundamentalResearch,Mumbai,India
ManindraAgrawal,IndianInstituteofTechnologyKanpur,Kanpur,India
RitabrataMunshi,TataInstituteofFundamentalResearch,Mumbai,India
S.R.S.Varadhan,NewYorkUniversity,NewYork,USA
WeinanE,PrincetonUniversity,Princeton,USA
The Infosys Science Foundation Series in Mathematical Sciences, a Scopus-
indexedbookseries,isasub-seriesoftheInfosysScienceFoundationSeries.This
sub-seriesfocusesonhigh-qualitycontentinthedomainofmathematicalsciences
andvariousdisciplinesofmathematics,statistics,bio-mathematics,financialmathe-
matics, applied mathematics, operations research, applied statistics and computer
science. With this series, Springer and the Infosys Science Foundation hope to
provide readers with monographs, handbooks, professional books, edited volume,
andtextbooksofthehighestacademicqualityoncurrenttopicsinrelevantdisciplines.
Literatureinthissub-serieswillappealtoawideaudienceofresearchers,students,
educators,andprofessionalsacrossmathematics,appliedmathematics,statisticsand
computersciencedisciplines.
· ·
Pintu Bhunia Silvestru Sever Dragomir
·
Mohammad Sal Moslehian Kallol Paul
Lectures on Numerical
Radius Inequalities
PintuBhunia SilvestruSeverDragomir
DepartmentofMathematics CollegeofEngineeringandScience
JadavpurUniversity VictoriaUniversity
Kolkata,India Melbourne,VIC,Australia
MohammadSalMoslehian KallolPaul
DepartmentofPureMathematics DepartmentofMathematics
FerdowsiUniversityofMashhad JadavpurUniversity
Mashhad,Iran Kolkata,India
ISSN 2363-6149 ISSN 2363-6157 (electronic)
InfosysScienceFoundationSeries
ISSN 2364-4036 ISSN 2364-4044 (electronic)
InfosysScienceFoundationSeriesinMathematicalSciences
ISBN 978-3-031-13669-6 ISBN 978-3-031-13670-2 (eBook)
https://doi.org/10.1007/978-3-031-13670-2
©TheEditor(s)(ifapplicable)andTheAuthor(s),underexclusivelicensetoSpringerNature
SwitzerlandAG2022
Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether
thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse
ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and
transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar
ordissimilarmethodologynowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthors,andtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressedorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.Thepublisherremainsneutralwithregardtojurisdictional
claimsinpublishedmapsandinstitutionalaffiliations.
ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG
Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland
To
JyotsanaBhunia
AudreyandSienna
MaryamAmyari
Dipanwita PaulGhosh
Preface
This book is a self-contained advanced monograph on inequalities involving the
numericalradiusofboundedlinearoperatorsactingoncomplexHilbertspaces.The
studyofvariouskindsofnumericalrangeandnumericalradiushasalonganddistin-
guishedhistorystartingfromtheRayleighquotientsusedinthenineteenthcentury
tonowadaysapplicationsinnumerousfieldssuchasfunctionalanalysis(estimates
andboundsofnorms),operatortheory(differentialoperators),matrixtheory(loca-
tionoftheeigenvalues),numericalanalysis(convergencerateofalgorithms),system
theory(singularvalues),quantumcomputing(quantumerrorcorrection),quantum
informationtheory(quantumchannels),andquantumcontrol(optimizingwitnesses).
Theconceptofanumericalrangeisanaturalextensionofquadraticformsstudied
inlinearalgebra.Indeed,thenumericalrangeofamatrixAmayberegardedasan
estimateoftheeigenvaluesofA.Theconvexityofthenumericalrangeisoneofits
fundamentalproperties.Inparticular,ifAisanormalmatrix,thenthenumericalrange
is just the convex hull of the eigenvalues of A. This fact leads to many interesting
consequencesandhelpfulapplications.Theideaofthenumericalrangegoesback
totheworkofToeplitz[166]in1918,whereheintroducedthefieldofvaluesofa
bilinearform.ThetermnumericalrangewasfirstintroducedbyM.Stonein1932
[165].
Thestudyofnumericalrangeassistsinunderstandingthebehaviorofabounded
linear operator, and the major role in this connection is played by the well-known
notion“numericalradius”associatedwiththenumericalrange.Inequalitiesinvolving
the numerical radius are one of the most important topics of research as far as the
studyofinequalitiesinoperatortheoryandmatrixanalysisisconcerned.
InChap.1,wecollectsomebasicfactsoffunctionalanalysisthatareessentialto
studyingthenumericalrangeandthenumericalradiusofboundedlinearoperators
definedoncomplexHilbertspaces.Inthischapter,webrieflydiscussfundamental
results like the Cauchy–Schwarz inequality, the Riesz representation theorem, and
thespectralrepresentationtheorem.WealsomentiontheGrüssinequalityforinner
productsandtheHölder–McCarthyinequalityforpositiveoperators.
In Chap. 2, we begin with important properties of the numerical range and the
numerical radius of bounded linear operators defined on complex Hilbert spaces.
vii
viii Preface
Amongmanyimportanttheorems,theToeplitz–Hausdorfftheoremandthespectral
inclusiontheoremarediscussedinthischapter.Wedevelopsomeinterestingbounds
of the numerical radius of bounded linear operators by employing their Cartesian
decomposition. Furthermore, we show that the numerical radius attains its lower
bounds if and only if the numerical range is a circular disk with a center at 0. We
alsoexhibitboundsforthenumericalradiusoftheproductoftwoormorebounded
linearoperators.
In Chap. 3, we discuss several numerical radius inequalities using Buzano’s
inequality that is a generalization of the Cauchy–Schwarz inequality. In addition,
we study a generalization of Buzano’s inequality and related inequalities for the
numerical radius of n-tuples of bounded linear operators. We also present some
vectorinequalitiesrelatedtotheCauchy–SchwarzandBuzanoinequalities.
InChap.4,westudytheEuclideanoperatorradiusofn-tuplesofboundedlinear
operators defined on a complex Hilbert space, which leads to the case of classical
numerical radius for n = 1. We establish inequalities related to the p-numerical
radius of an n-tuple of bounded linear operators, which is a generalization of the
Euclidean operator radius. Furthermore, we develop some inequalities involving
hypo-p-numericalradiusandhypo-Euclideannumericalradius.Amonginteresting
inequalitiesrelatedtothenumericalradius,wewouldrefertothepowerinequality
andtheinequalitiesconcerningthenumericalradiusoftheproductoftwobounded
linear operators. In general, the numerical radius of the product of two operators
is not submultiplicative and so it is interesting to search for better bounds of the
numericalradiusoftheproductoftwooperators.
InChap.5,wefocusonthefurtherdevelopmentofthepowerinequalityaswell
asinequalitiesofthenumericalradiusoftheproductoftwooperators.Here,wealso
developinequalitiesofpowersofthenumericalradiusoftheproductofmorethan
twooperators.
Thestudyofnumericalradiiofoperatormatricesisanimportantareaofstudyin
itsownright.InChap.6,wedevelopboundsforthenumericalradiiof2×2operator
matricesandusedthemtoobtainboundsforthenumericalradiiofboundedlinear
operators.Asanapplication,weestimateboundsforthezerosofapolynomialusing
thenotionoftheFrobeniuscompanionmatrix.
InChap.7,weinvestigatetherelationbetweentheoperatorspacenormandthe
operatorspacenumericalradiusonthematrixspaceM (X),whereX isanumerical
n
radiusoperatorspace.Inaddition,wepresentsomeinequalitiesinvolvingtheoperator
spacenumericalradiusandthemaximalnumericalradiusnormofa2×2operator
matrixanditsoff-diagonalpart.
The notion of the numerical radius, due to its importance and applications, has
beenextendedinvariousdirections.Oneoftherecentgeneralizationsisknownas
theA-numericalradius,whereAisapositivelinearoperatoronacomplexHilbert
space.InChap.8,westudythebasicpropertiesoftheA-numericalradiusofbounded
linearoperatorsanddevelopseveralinequalitiesrelatedtotheA-numericalradiusof
operatorsandoperatormatrices.
In Chap. 9, we give several challenging problems for researchers together with
referencestothebackgroundswhichmaybeusefultowellunderstandthem.
Preface ix
This monograph is intended for use by both researchers and graduate students
of mathematics, physics, and engineering who have a basic background in linear
algebraandarefamiliarwiththefundamentalsoffunctionalanalysisandoperator
theory.Thebookprovidesdetailedargumentsforthemajorityoftheresults.Each
chapterisendedwithsomenotesabouthistoricalviewsorfurtherextensionsofthe
topicsdiscussedinthechapter.
Itcontainsabibliographyofmorethan170items,soitcanbeusedasareference
bookincludingmanyclassicalandmodernnumericalradiusinequalities.
Kolkata,India PintuBhunia
Melbourne,Australia SilvestruSeverDragomir
Mashhad,Iran MohammadSalMoslehian
Kolkata,India KallolPaul
Contents
1 Preliminaries .................................................. 1
1.1 GeneralNotationandTerminology ............................ 1
1.2 ElementaryFunctionalAnalysis .............................. 2
1.3 FunctionalCalculusforSelfadjointOperators .................. 7
1.4 SpectralRepresentationofSelfadjointOperators ................ 10
1.5 BlockMatricesofOperators ................................. 12
2 FundamentalNumericalRadiusInequalities ...................... 15
2.1 BasicPropertiesofNumericalRange .......................... 16
2.2 InequalitiesConcerningOneOperator ......................... 18
2.3 InequalitiesRelatedtoDavis–WielandtRadius ................. 36
2.4 InequalitiesInvolvingMoreThanOneOperator ................ 39
2.5 MoreApplicationsofCartesianDecomposition ................. 46
3 BoundsoftheNumericalRadiusUsingBuzano’sInequality ........ 61
3.1 UpperBoundsoftheNumericalRadius ........................ 62
3.2 GeneralizedBuzano’sInequality .............................. 68
3.3 VectorInequalitiesforProjections ............................ 75
4 p-NumericalRadiusInequalitiesofann-TupleofOperators ....... 85
4.1 BasicPropertiesof p-NumericalRadius ....................... 86
4.2 Inequalitiesofw forPairsofOperators ....................... 87
p
4.3 Inequalitiesofw forn-TuplesofOperators .................... 92
p
4.4 Hypo-p-NumericalRadius ................................... 100
4.5 ReverseInequalitiesforHypo-EuclideanNumericalRadius ...... 106
5 NumericalRadiusInequalitiesofProductofOperators ............ 111
5.1 InequalitiesInvolvingAdjointOperators ....................... 111
5.2 InequalitiesforProductofTwoorMoreOperators .............. 116
6 NumericalRadiusofOperatorMatricesandApplications .......... 125
6.1 BasicInequalitiesof2×2OperatorMatrices .................. 126
6.2 Inequalitiesof2×2Off-DiagonalOperatorMatrices ............ 131
xi
