Di u = ∂ u/∂ xi ;if α = (α1 , ... , αn ) with αi non-negativeintegers, Dα u =
where |α |= α1 + ... + αn .
Ω :anopensetin Rn ; Ω isa domain ifitisalsoconnected.
∂Ω :boundaryof Ω ; Ω :closureof Ω ; Ω c = Rn \Ω .
Ω ⊂⊂ Ω : Ω isacompactsubsetof Ω .
dist (x, ∂Ω):distancefrom x to Ω c .
N:positiveintegers; N0 = N ∪{0}; Z:allintegers.
f (t ) g (t ) as t → a:thereexistpositiveconstants c1 , c2 suchthat c1 f (t )/g (t ) c2 for |t a| ( = 0) smallenough,if a ∈ R;andforlargeenough ±t if a =±∞
T G :restrictionoftheoperator(orfunction) T toset G
f + = max (f ,0), f =− min(f ,0).
A ⊂ B forsets A, B allowsfor A = B.
Embedding:aboundedlinearinjectivemapofaBanachspace X toanothersuchspace Y .
lp (1 p ∞):complexsequencespacewithnorm (ξj ) p = ( |ξj |p )1/p when1 p < ∞ and (ξj ) ∞ = sup j |ξj | when p =∞.
Allvectorspacesthatwillbementionedwillbeassumedtobeoverthecomplexfield,unless otherwisestated.ThenormonanormedvectorspaceXwillusuallybedenotedby • X , orby • ifnoambiguityispossible.
GivenanyBanachspaces X and Y ,thevectorspaceofallboundedlinearmapsfrom X to Y willbedenotedby B (X , Y ),orby B (X ) if X = Y ;withthenorm • definedby T = sup{ Tx : x 1}, B (X , Y ) isaBanachspace.Itisnaturaltotrytodistinguish membersof B (X , Y ) thathaveparticularlygoodproperties.Compactlinearmapscome intothiscategory,sincetheyhavepropertiesreminiscentoflinearmapsactingbetween finite-dimensionalspaces.
Definition1.1. LetXandYbeBanachspacesandletT : X → Ybelinear.ThemapTissaid tobe compact if,andonlyif,foranyboundedsubsetBofX,theclosure T (B) ofT (B) is compact.
Evidently T iscompactif,andonlyif,givenanyboundedsequence (xn ) in X ,thesequence (Txn ) containsaconvergentsubsequence.Notealsothatif T iscompact,itiscontinuous, sinceotherwisetherewouldbeasequence (xn ) in X suchthat xn = 1forall n ∈ N,and Txn →∞ as n →∞,whichisimpossible.
Examples.
(i) IfT ∈ B (X , Y ) andthedimensionoftherange R (T ) := T (X ) ofT, dim R (T ),is finite,thenTmustbecompact,sinceifBisaboundedsubsetofXthen T (B) isclosed andbounded,andhencecompact.
(ii) Noteveryboundedlinearmapiscompact:takeX = Y = l2 ,foreachn ∈ N lete(n) be theelementofl2 withmthcoordinate δmn (equalto1ifm = n,and0otherwise),and observethattheidentitymapofl2 toitselfiscontinuousbutnotcompact,becausethe sequence (e(n) ) hasnoconvergentsubsequence.
foralls ∈ JandforallxintheBanachspaceC(J ) ofallcontinuouscomplex-valued functionsonJ(withnorm • givenby x = max {|x(s)| : s ∈ J }).ThenKisalinear mapofC(J ) intoitself,andinfact,Kiscompact.Toseethis,setM = max {|k(s, t )| : s,t ∈ J };then Kx M (b a) x forallx ∈ C(J ),andsoifBisaboundedsubset ofC(J ) thenK (B) isbounded.Moreover,foralls1 ,s2 ∈ Jandanyx ∈ C(J ), |(Kx)(
Thus,givenany ε > 0,thereexistsa δ> 0 suchthat |(Kx)(s1 ) (Kx)(s2 )| < ε if x 1 and |s1 s2 | <δ(s1 , s2 ∈ J ).Hence {Kx : x ∈ C(J ), x 1} isequicontinuousandbounded,andthusrelativelycompact,bytheArzela–AscoliTheorem(cf. Yosida[265],p.85).
Denoteby K (X , Y ) thefamilyofallcompactlinearmapsfrom X to Y ,andput K (X ) = K (X , X ).Thefollowingpropositionisawell-knownconsequenceofthe definitionofcompactness:
Proposition1.2. LetX,Y,ZbeBanachspaces.Then K (X , Y ) isaclosedlinearsubspaceof B (X , Y );ifT1 ∈ B (X , Y ) andT2 ∈ B (Y , Z) thenT2 T1 iscompactifeitherT1 orT2 is compact.
Thispropositionimpliesthat K (X ) isaclosedtwo-sidedidealintheBanachalgebra B (X )
Ithasalreadybeennotedthatif T ∈ B (X , Y ) isof finiterank,thatis,dim R (T )< ∞, then T ∈ K (X , Y ).Inparticular, B (X , Y ) = K (X , Y ) ifeither X or Y isfinite-dimensional. Thefollowingresultcomplementsthis,andthrowsnewlightonExample(ii)above.
Proof. Let yn ∈ Xn \Xn 1 .Thefunction y → y yn ispositiveandcontinuouson Xn 1 andapproachesinfinityas y →∞;henceithasaminimum,at zn ∈ Xn 1 ,say,and 0 < zn yn x + zn yn forany x ∈ Xn 1 .Thepoint xn := (yn zn )/ yn zn thenhasalltherequired properties. ✷
Proof. Let x ∈ X \M .Since M isclosed, d := dist (x, M )> 0.Thus,givenany θ ∈ (0,1), thereexists mθ ∈ M suchthat x mθ d/θ .Theelement xθ := (x mθ )/ x mθ hasallthepropertiesneeded. ✷
Compactnessofalinearmapispreservedbythetakingoftheadjoint.Beforethis resultisgiveninaformalway,someremarksaboutnotationaredesirable.Givenany normedvectorspace X ,bythe adjointspaceX ∗ of X ismeantthesetofall conjugate linear continuousfunctionalson X ;thatis, f ∈ X ∗ if,andonlyif, f : X → C iscontinuousand f (α x + β y) = α f (x) + β f (y) forall α , β ∈ C andall x, y ∈ X .Ourchoiceof conjugatelinear functionals,ratherthanthemorecommon linear functionals,isdictatedsolelybythe conveniencethatwillresultlateroninthebook.Withtheusualdefinitionsofadditionand multiplicationbyscalars,theadjointspace X ∗ becomesaBanachspacewhengiventhenorm • definedby
f = sup{|(f , x)| : x = 1},
where (f , x) isthevalueof f at x.(Thisisoftendenotedby f (x) asabove.Strictlyspeaking, weshouldwrite (f , x)X ,butthesubscriptwillbeomittedifnoambiguityispossible.The sameomissionwillbemadeforinnerproductsinaHilbertspace.)Givenany T ∈ B (X , Y ),
theadjointof T isthemap T ∗ ∈ B (Y ∗ , X ∗ ) definedby (T ∗ g , x) = (g , Tx) forall g ∈ Y ∗ and x ∈ X ;notethat (α S + β T )∗ = α S∗ + β T ∗ forall α , β ∈ C andall S, T ∈ B (X , Y ).These conventionsaboutadjointswillapplyevenwhentheunderlyingspacesareHilbertspaces, andtherewillthereforebenoneoftheusualawkwardnessaboutthedistinctionbetween Banach-spaceandHilbert-spaceadjointsofamapthathastobemadewhenlinear,rather thanconjugate-linear,functionalsareused.Ofcourse,manyoftheresultstobegivenbelow wouldalsoholdhad X ∗ beendefinedtobethespaceofallcontinuouslinearfunctionals on X .NotethattheRieszRepresentationTheorem(cf.Taylor[231],Theorem4.81-C) enablesanyHilbertspace H tobeidentifiedwith H ∗ ;andthatinviewofthis,givenany T ∈ B (H1 , H2 )(H1 and H2 beingHilbertspaces), T ∗ ∈ B (H2 , H1 ).If H1 = H2 = H thenboth T and T ∗ belongto B (H ):themap T issaidtobe self-adjoint if T = T ∗
Theorem1.6. LetXandYbeBanachspacesandletT ∈ B (X , Y ).ThenT ∈ K (X , Y ) if,and onlyif,T ∗ ∈ K (Y ∗ , X ∗ ).
Thewell-knownproofmaybefoundinYosida[265],p.282. Thenotionoftheadjointofamap T willalsobeneededwhen T isunbounded.Thuslet D (T ) bealinearsubspaceof X thatis dense in X (i.e. D (T ) = X ) andlet T : D (T ) → Y be linear.Let D (T ∗ ) ={g ∈ Y ∗ :thereexists f ∈ X ∗ suchthat (f , x) = (g , Tx) forall x ∈ D (T )}; theadjoint T ∗ of T isthemap T ∗ : D (T ∗ ) → X ∗ definedby T ∗ g = f ,i.e. (T ∗ g , x) = (g , Tx) forall x ∈ D (T ) andall g ∈ D (T ∗ ).Notethatitisessentialthat D (T ) bedensein X for T ∗ tobewell-defined.Amoredetaileddiscussionofself-adjointmapsinaHilbertspacewill begivenin§3.4.
Next,theresultsoftheFredholm–Riesz–Schaudertheoryofcompactlinearmapswill begiven;thistheoryextendsinamostdirectwaythetheoryoflinearmapsinfinitedimensionalspaces.Thecompletepicturefollowsfromaseriesofauxiliaryresults,anumber ofwhichareofinterestintheirownright.Throughoutthediscussion X willstandfora non-trivial(thatis ={0})Banachspace, I willbetheidentitymapfrom X to X ,and,given any T ∈ B (X ) andany λ ∈ C,weshallwrite Tλ for T λI .Thenotionsofthe resolvent set andthe spectrum ofalinearmapwillalsobeneeded;thesewillbeexplainedinterms ofalinearmap S fromalinearsubspace D (S) of X to X .Theresolventset, ρ(S),of S isdefinedtobe {λ ∈ C : (S λI ) 1 existsandbelongsto B (X )}; C\ρ(S) iscalledthe spectrumof S andiswrittenas σ(S).Threedisjointsubsetsof σ(S) aredistinguished: the pointspectrum σp (S) := {λ ∈ σ(S) : S λI isnotinjective},the continuousspectrum σc (S) := {λ ∈ σ(S) : S λI isinjective,(S λI ) D (S) isdensein X butnotequalto X },and the residualspectrum σr (S) := {λ ∈ σ(S) : S λI isinjective, (S λI )D (S) isnotdensein X }.Theelementsof σp (S) are,ofcourse, eigenvalues of S.Ingeneralthethreesubsetsofthe spectrumgivenabovedonotexhaust σ(S);itisconceivablethat (S λI ) 1 couldexist,have domain X andyetbeunbounded.However,thispathologycannotoccurinanimportant case,namely,when S is closed:recallthatalinearmap T fromalinearsubspace D (T ) of X toanormedvectorspace Y issaidtobeclosedifits graph G (T ) := {{x, Tx} : x ∈ D (T )} isaclosedsubsetoftheproductspace X × Y ,when X × Y isgiventhenorm • definedby
{x, y} = ( x 2 + y 2 ) 1 2
Itiseasytoseethat T isclosedif,andonlyif,givenanysequence (xn ) in D (T ) suchthat xn → x and Txn → y,itfollowsthat x ∈ D (T ) and Tx = y.Formoredetailsonthesetopics, seeKato[134]III-5,6.Finally,wewrite N (T ) forthe kernel of T ,i.e. {x ∈ D (T ) : Tx = 0}, andsetnul T = dim N (T );nul T isthe nullity of T .
Theorem1.7. LetT ∈ K (X ) andsupposethat λ ∈ C\{0}.Then
(i) if R (Tλ ) = XthenTλ isinjectiveandT 1 λ ∈ B (X );
(ii) R (Tλ ) isclosed;
(iii) ifTλ isinjective,T 1 λ ∈ B (X );
(iv) either λ ∈ ρ(T ),inwhichcase R (Tλ ) = X,or λ ∈ σp (T ),inwhichcase R (Tλ ) isaproperclosedlinearsubspaceofX;
(v) nulTλ < ∞
Definition1.8. LetT ∈ B (X ) andsupposethat λ ∈ σp (T ).Thegeometricmultiplicityof λ isdefinedtobenulTλ .
Theorem1.9. LetT ∈ K (X ).Then σp (T ) isatmostcountableandhasnoaccumulationpointexceptpossibly0.Eachpointof σ(T )\{0} isaneigenvalueoffinitegeometric multiplicity.
Itisworthremarkingthatifdim X =∞ then0 ∈ σ(T ) forall T ∈ K (X ).Tosee this,supposethat0 ∈ ρ(T ) forsome T ∈ K (X );then T 1 ∈ B (X ) andso I = TT 1 iscompact.Theorem1.3nowgivesthedesiredcontradiction. Atthisstagetheadjoint T ∗ of T makesanimportantentrance.If T ∈ K (X ),ithas alreadybeenobservedthat T ∗ ∈ K (X ∗ );anditiseasytoseethat σ(T ) := σ(T ∗ ) ={λ : λ ∈ σ(T ∗ )},compactnessnotbeingneededforthislatterresult.Itfollowsthatif λ ∈ C\{0}, then λ ∈ σp (T ) if,andonlyif, λ ∈ σp (T ∗ ).
Theorem1.10. LetT ∈ K (X ) andlet λ ∈ C\{0}.Then nul Tλ = nul T ∗ λ .
