Compactness Criteria for Sets and Operators in the Setting of Continuous Frames

0.19 MB·English

by  M. Mantoiu

Compactness Criteria for Sets and Operators
in the Setting of Continuous Frames
3 M.Ma˘ntoiuand D.Parra ∗
1
0
2 Abstract

n To a generalized tight continuous frame in a Hilbert space H, indexed by a locally compact
a
space Σ endowedwith a Radonmeasure, oneassociates[9, 21] a coorbittheoryconvertingspaces
J
offunctionsonΣ inspacesofvectorscomparablewithH. Ifthecontinuousframeisprovidedby
2
theactionofasuitablefamilyofboundedoperatorsonafixedwindow,asymboliccalculusemerges

] [16], assigning operators in H to functions on Σ. We give some criteria of relative compactness
A for sets and for families of compact operators, involving tightness properties in terms of objects
F canonically associated to the frame. Particular attention is dedicated to a magnetic version of the
. pseudodifferentialcalculus.
h
t
a
m
[ Address

1
DepartamentodeMatema´ticas, UniversidaddeChile,
v
7 LasPalmeras3425, Casilla653,Santiago,Chile
4 E-mail: [e-mail]
2
E-mail: [e-mail]
0
.
1
0 Acknowledgements: Theauthorsaresupported byNu´cleoCientifico ICMP07-027-F”Mathemati-
3
calTheoryofQuantumandClassicalMagneticSystems”. ThefirstnamedauthorisgratefultotheErwin
1
: Schro¨dinger InstituteandtotheMittag-LefflerInstitute, wherepartofthisthispaperhasbeenwritten.
v
i
X
1 Introduction
r
a
The main goal of this article is to provide compactness criteria for bounded subsets Ω of some Banach
spacesY intermsofgeneralized continuous frames[7,9,21,16]. Itwillbeconvenient inthisIntroduc-
tiontorefer totheframeworkof[16],lessgeneral thanthat of[9,21],buthaving arichermathematical
structure.
In [16] the framework is built on a family {π(s)|s ∈ Σ} of bounded operators acting in a Hilbert
space H indexed bythe points ofalocally compact space Σendowed withaRadon measure µ. Under
∗2010MathematicsSubjectClassification:Primary46B50,42B36,Secundary47G30,46E30.
KeyWords:compactset,compactoperator,coorbitspace,pseudodifferentialoperator
1
certainconvenient axioms(thesquareintegrability condition (2.11)isbasic), oneintroduces andstudies
a map φπ from H × H into L2(Σ), and a symbolic calculus f 7→ Π(f) sending functions on Σ into
operators onH(inthespiritofapseudodifferential theory). Formallythedefinitions are
[φπ(u,v)](s) := hπ(s)u,vi, u,v ∈ H, s ∈ Σ (1.1)
and
Π(f):= dµ(s)f(s)π(s)∗, f ∈ L2(Σ), (1.2)
ZΣ
but with suitable interpretations (using Gelfand triples for instance) they can be pushed to much more
generalsituations.
If Σ is a locally compact group and π : Σ → B(H) is a (maybe projective) unitary, strongly con-
tinuous (maybeirreducible) representation, theframework isstandard [7]. Thefunction φπ iscalled the
representation coefficient mapandΠistheintegrated formofthisrepresentation.
However, in many physically or/and mathematically motivated situations Σ is not a group. Even
when it is, π is not a projective representation; the operators π(s)π(t) might not be connected to π(st)
(when the later exists) in some simple way. The need of a formalism covering non-group-like situa-
tions motivated the approach in [16], to which we refer for more technical details, for constructions
of involutive algebras and of coorbit spaces of vectors and symbols and for relevant examples. Actu-
ally coorbit spaces of vectors have been previously defined in [9, 21] starting with a continuous frame
W := {w(s)|s ∈ Σ} ⊂ H, following the fundamental approach of [7]. These references, besides a
deepinvestigation, alsocontain manyexamples andmotivational issues towhichwesendtheinterested
reader. In such a generality, however, the symbolic calculus Π and the connected developments of [16]
arenotavailable.
To get the situation treated in [16] one sets essentially w(s) := π(s)∗w for some fixed nomalised
vector(window)ofH. Itisfruitful toconsider thepartial function u 7→ φπ(u) := φπ(u,w)forfixedw
w
andclearlythiscanbegeneralized toanisometry
H ∋u 7→ φ (u) := hu,w(·)i ∈L2(Σ) (1.3)
W
forcontinuousframeswhicharenotdefinedbyfamiliesofoperators. Thenecessarynotionsfrom[9,16,
21]arebrieflyreviewedinsection2.
Let us come back to compactness issues. Let us fix an infinite dimensional Banach space Y and a
boundedsubsetΩofY. WeassumethatY issomehowdefinedinthesetting(H,Σ,π,φπ,Π). Tipically
it will be one of the coorbit spaces of vectors constructed in terms of the frame; the Hilbert space H
itself is a particular but important example. To be relatively compact Ω needs extra properties beyond
boundedness, and it is natural to search for such properties in terms of the maps π,φπ or Π. The
followingdefinition(inspired by[6])willbeconvenient.
Definition1.1. AssumethattheBanachspaceY isendowedwithastructureofBanachleftmoduleover
anormedalgebraA,meaningthataleftmodulestructureA×Y ∋(a,y) 7→ a·y ∈ Y isgivenandthe
relation ka·yk ≤kak kyk issatisfied forevery a ∈ Aandy ∈ Y. LetA0 ⊂ A;wesay thatthe
Y A Y
boundedset Γ ⊂ Y isA0-tightifforeveryǫ > 0thereexistsa ∈ A0 withsup ka·y−yk ≤ ǫ.
y∈Γ Y
2
In various situations, depending on the meaning of · and k · k , tightness could have a specific
Y
intrepretation(equicontinuity, uniformconcentration, etc). NotethatY isnaturallyaBanachleftmodule
over B(Y), the Banach algebra of all bounded lineal operators in Y, so very often we choose A0 ⊂
B(Y).
MostofourresultswillinvolvecharacterizationofrelativecompactnessofΩintermsofitstightness
with respect to a (finite) family of Banach module structures {A × Y 7→ Y} and corresponding
j j∈J
subsets {A0 ⊂ A } . Anoccuring generalization isusingforcharacterization tightness oftheimage
j j j∈J
Ω′ := ψ(Ω)ofΩintoanotherBanachleftmodule.
Forillustration, let us reproduce here aslightly simplified version of Theorem 4.1. Weask the map
π∗(·) := π(·)∗ : Σ → B(H) to be strongly continuous, to satisfy π∗(s ) = 1 for some s ∈ Σ and to
1 1
verify condition (2.11). Note that C (Σ) is contained in the C∗-algebra C (Σ) of complex continuous
c 0
functions onΣvanishing atinfinity,whichactsonL2(Σ)bypointwisemultiplication.
Theorem1.2. AboundedsubsetΩof Hisrelativelycompactifandonlyifanyoneofthenextequivalent
conditions holds:
1. Forsome(every) w ∈Hthefamilyφπ(Ω)isC (Σ)-tightinL2(Σ).
w c
2. ThesetΩisΠ[C (Σ)]-tight;hereweusetheBanachmoduleB(H)×H → H.
c
3. Onehas lim sup kπ∗(s)u−π∗(s )uk= 0 foreverys ∈ Σ.
0 0
s→s0u∈Ω
Two possible generalizations can be taken into account: (a) replace W := {π(s)∗w | s ∈ Σ}
by a general continuous frame and (b) replace H by a coorbit space. Both these generalizations are
considered in section 3, but only involving the characterization 1 of relative compactness of Ω in terms
oftightness ofthesetφ (Ω). Oneobtaines anextension ofthemainresult of[5],whichrequired Σto
W
be a locally compact group and w(s) = π(s)∗w for some irreducible integrable unitary representation
π : Σ → B(H). Although substantially moregeneral, ourresult allowsalmostthesameproofasin[5];
weinclude thisproofforconvenience andbecause sometechnical detailsaredifferent.
In fact the characterizations 2 and 3, suitably modified, would also be available in coorbit spaces.
Howeverthis wouldneed manypreparations from thepaper [16](submitted for publication) and would
involvesomeimplicitassumptionsrequiringalotofexemplifications. Therefore,atleastforthemoment,
we decided to include compactness characterization in terms of π and Π only for the important case of
Hilbertspaces.
We are also interested in families of compact operators. Two Banach spaces X and Y being given,
the problem of deciding when a set K of compact operators : X → Y is relatively compact in the
operatornormtopologyisalreadyaclassicalone;formoredetailsandmotivationscf. [1,10,18,19,22]
andreferences therein. Clearly,compactness resultsforsubsetsofY (asthosegiveninsections 3and4)
arecrucial,butextrarefinamentsareneeded: ForK tobearelativelycompactsetofcompactoperators,
it is necessary but not sufficient that {Sx|kxk ≤ 1,S ∈ K } be relatively compact in Y; this even
X
happens inHilbert spaces. Wediscuss thisproblem insection 5;ofcourse, ifK := {S}isasingleton,
onegetseasilycriteriafortheoperator S tobecompact.
3
InafinalSectionwetreatwhatwethinktobeanimportantexample,themagneticWeylcalculus[16,
14],whichdescribes thequantization ofaparticlemovinginRn undertheactionofavariablemagnetic
field B (a closed 2-form on Rn). It is a physically motivated extension of the usual pseudodifferential
theory in Weyl form, which can be recovered for B = 0. One reason for including this here is that
it definitely stays outside the realm of projective group representations and the results on compactness
existingintheliteraturedonotapply. Butitisarathersimpleparticularcaseoftheformalismdevelopped
in [16] and the compactness criteria of the present paper work very well. We decided to present only
theHilbertspace theory, having inviewcertain applications tothespectral theory ofmagnetic quantum
Hamiltonians that will hopefully addressed in the future. A second reason to treat the magnetic Weyl
calculus here is that it presents extra mathematical structure which has important physical implications
and which also enlarges the realm of compacness criteria. If the magnetic field is zero, part of our
Theorem6.2reproducestheclassicalRiesz-KolmogorovTheorem(cf. [5,6,11]forusefuldiscussions).
Extensions of this classical result can be found in [11] and especially in [6]; since these references use
essentiallythegroup-theoreticframework,theycannotbeappliedtooursection6. Itwouldbeinteresting
to generalize the double module formalism of [6] to cover at least the magnetic Weyl calculus and its
generalization tonilpotent Liegroups[20,2,3].
2 Coorbit spaces and quantization rules associated to continuous frames
Westartwithsomenotations andconventions:
We denote by H the conjugate of the (complex separable) Hilbert space H; it coincides with H as
an additive group but it is endowed with the scalar multiplication α · u := αu and the scalar product
hu,vi′ := hu,vi. Ifu,v ∈ Htherankoneoperator λ ≡ h·,vihu|isgivenbyλ (w) := hw,viu.
u,v u,v
Let Σ be a Hausdorff locally compact and σ-compact space endowed with a fixed Radon measure
µ. ByC(Σ)onedenotes thespaceofallcontinuous functions onΣ,containing theC∗-algebra BC(Σ)
composed of bounded continuous functions. The closure in BC(Σ) of the space C (Σ) of continuous
c
compactlysupportedcomplexfunctionsonΣistheC∗-algebraC (Σ)ofcontinuousfunctionsvanishing
0
atinfinity. TheLebesguespaceL2(Σ;µ) ≡ L2(Σ)willalsobeused, withscalarproduct hu,vi =:
L2(Σ)
hu,vi .
(Σ)
For Banach spaces X,Y we set B(X,Y) for the space of linear continuous operators from X to Y
andusetheabbreviationB(X) := B(X,X). TheparticularcaseX′ := B(X,C)referstothetopological
dual ofX . ByK(X,Y) wedenote thecompact operators from X toY. IfH isaHilbert space, B (H)
2
is the two-sided ∗-ideal of all Hilbert-Schmidt operators in B(H); it is a Hilbert space with the scalar
producthS,TiB2(H) := Tr(ST∗).
Werecallnowtheconcept oftightcontinuous frameandtheconstruction ofcoorbit spaces, slightly
modifying the approach of [9, 21]. Let us fix a family W := {w(s) | s ∈ Σ} ⊂ H that is a tight
continuous frame; the constant of the frame is assumed to be 1 by normalizing the measure µ. This
meansthatthemaps 7→ w(s)isassumedweaklycontinuous andforeveryu,v ∈ Honehas
hu,vi = dµ(s)hu,w(s)ihw(s),vi. (2.1)
ZΣ
4
ClearlyW istotalinHanddefinesanisometricoperator
φ :H → L2(Σ), [φ (u)](s) := hu,w(s)i (2.2)
W W
withadjointφ† : L2(Σ) → Hgiven(inweaksense)by
W
φ† (f)= dµ(s)f(s)w(s). (2.3)
W
ZΣ
The(Gramian)kernelassociated totheframeisthefunctionp :Σ×Σ → Cgivenby
W
p (s,t):= hw(t),w(s)i = [φ (w(t))](s) = [φ (w(s))](t), (2.4)
W W W
defining a self-adjoint integral operator P = Int(p ) in L2(Σ). One checks easily that P =
W W W
φ φ† isthefinalprojection oftheisometryφ ,soP L2(Σ) isaclosedsubspace ofL2(Σ). Since
W W W W
φ† φ = 1,onehastheinversion formula
W W (cid:2) (cid:3)
u= dµ(t)[φ (u)](t)w(t), (2.5)
W
ZΣ
leadingtothereproducing formulaφ (u) = P [φ (u)],i.e.
W W W
[φ (u)](s) = dµ(t)hw(t),w(s)i[φ (u)](t). (2.6)
W W
ZΣ
Thus P (Σ) := P L2(Σ) is a reproducing space with reproducing kernel p ; it is composed of
W W W
continuous functions onΣ.
(cid:2) (cid:3)
To extend the setting above beyond the L2-theory, one can supply an extra space of “test vectors”,
denoted by G, assumed to be a Fre´chet space continuously and densely embedded in H. Applying the
Riesz isomorphism we are led to a Gelfand triple (G,H,G′). The index σ refers to the fact that on
σ
the topological dual G′ we consider usually the weak-∗ topology. In certain circumstances one takes G
to be a Banach space and sometimes it can even be fabricated from the frame W and from some extra
ingredients, asinRemark2.1below. Butveryoften(thinkoftheSchwartzspace)theauxiliarspaceG is
onlyFre´chet.
We shall suppose that the family W is contained and total in G and that Σ ∋ s 7→ w(s) ∈ G is a
weakly continuous function. Then we extend φ to G′ by [φ (u)](s) := hu,w(s)i, where the r.h.s.
W W
denotes now the number obtained byapplying u ∈ G′ to w(s) ∈ G anddepends continuously on s. By
thetotalityofthefamilyW inG,thisextensionisinjective. Inaddition,Φ : G′ → C(Σ)iscontinuous
W
ifoneconsider onG′ theweak-∗ topology andonC(Σ)thetopology ofpointwise convergence.
As in [7, 9, 21] and many other references treating coorbit spaces, one uses φ (·) to pull back
W
subspacesoffunctionsonΣ. Solet(M,k·k )beanormedspaceoffunctionsonΣ(moreassumptions
M
onMwillbeimposedwhennecessary) andset
coW(M) ≡ co(M) := {u ∈ G′|φW(u) ∈ M}, kukco(M):=kφW(u)kM . (2.7)
5
Recalling the totality of the family W in G, one gets a normed space co(M),k·kco(M) and φW :
co(M) → M is an isometry. Without extra assumptions, even when M is a Banach space, co(M)
(cid:0) (cid:1)
might not be complete, so we define co(M) to be the completion. The canonical (isometric) extension
of φ to a mapping : co(M) → M will also be denoted by φ . If the norm topology of co(M)
W W
happenstobestronger thantheweak-∗etopology onG′,thencanonically co(M) ֒→ Gσ′ .
e
Remark 2.1. Following the approach of [7, 9, 21], we indicate now a possible choice for G adapted to
e
a given frame W in H. Let us consider a continuous admissible weight α : Σ× Σ → [1,∞) which
is bounded along the diagonal (α(s,s) ≤ C < ∞ for all s ∈ Σ), symmetric (α(s,t) = α(t,s) for all
s,t ∈ Σ)andsatisfiesα(s,t) ≤ α(s,r)α(r,t) forallr,s,t ∈ Σ. Itiseasytoseethat
Aα := {K : Σ×Σ → Cmeasurable|kKkAα< ∞} (2.8)
isaBanach∗-algebraofkernelswiththenorm
kKkA := max esssup dµ(t)|(αK)(s,t)|, esssup dµ(s)|(αK)(s,t)| . (2.9)
α
(cid:26) s∈Σ ZΣ t∈Σ ZΣ (cid:27)
Pickingsome(inessential)pointr ∈ Σonedefinestheweighta ≡ a :Σ → [1,∞)by a(s):= α(s,r).
r
We require that the kernel p given by (2.4) be an element of A ; Then it follows that P defines a
W α W
bounded operator in the weighted Lebesgue space L1(Σ). Then set G ≡ G := {v ∈ H | φ (v) ∈
a a,W W
L1(Σ)}withtheobviousnorm
a
kvk :=kφ (v)k = dµ(s)a(s)|[φ (v)](s)|. (2.10)
Ga,W W L1a(Σ) W
ZΣ
ThespaceG isaBanachspacecontinuously anddenselyembeddedinH. Inthisframework,coorbit
a,W
spaces were defined and thoroughly investigated in [9, 21]; if M is a Banach space then co(M) is
automatically complete. The dependence of these coorbit spaces on the frame W is also studied in
[9,21];wearegoingtoassumethattheframeW isfixed.
Following [16], we reconsider a particular case of the formalism described above. This particular
case has extra structure allowing to develop a symbolic calculus and to define and study corresponding
coorbit spaces of functions or ”distributions” on Σ; we shall only indicate the facts that are useful for
thepresent paper.
Letπ : Σ→ B(H)beamapsuchthatforeveryu,v ∈ Honehas
dµ(s)|hπ(s)u,vi|2 =kuk2kvk2 . (2.11)
ZΣ
Wesetπ(s)u =: π (s)andπ(s)∗u ≡ π∗(s)u =: π∗(s)foreverys ∈ Σandu ∈ H,getting familiesof
u u
functions {π : Σ → H | u ∈ H}and {π∗ : Σ → H | u ∈ H}. Onealso requires π∗ tobecontinuous
u u u
foreveryu.
ThemapΦπ : H⊗H → L2(Σ)uniquely definedby
[Φπ(u⊗v)](s) ≡ [φπ(u,v)](s) := hπ(s)u,vi
b
6
isisometric,by(2.11). Althoughthiswasnotnotneededin[16],wealsorequireΦπ tobesurjective. For
every normalized vector w ∈ H the map φπ : H → L2(Σ) given by φπ(u) := φπ(u,w) is isometric.
w w
Fixingw,itisclearthatweareintheaboveframeworkwiththetightcontinuous framedefinedby
W ≡ W(π,w) = {w(s) := π(s)∗w|s ∈ Σ}. (2.12)
Using existing notations one can write φ = φπ and w(·) = π∗(·). After introducing aFre´chet space
W w w
G continuously embedded inH,onecandefinecoorbit spacescoπ(M) := {u ∈ G′|φπ(u) ∈ M}asit
w w
wasdoneabove. Butwearenotgoingtoneedthem.
To define the symbolic calculus Π, sending functions on Σ into bounded linear operators on H,
we make use of the rank one operators Λ(u ⊗ v) ≡ λ := h·,viu indexed by u,v ∈ H. This
u,v
defines both a map λ : H×H → F(H) with values in the ideal of finite-rank operators and a unitary
map Λ : H⊗H → B (H) from the Hilbert tensor product to the Hilbert space of all Hilbert-Schmidt
2
operators on H. Consequently Π := Λ◦(Φπ)−1 : L2(Σ) → B (H) will also be unitary; its action is
2
uniquely defibnedbyΠ[(φ(u,v)] = h·,viu. Alsorecall[16,Prop. 2.3]theformulavalidinweaksense
Π(f) = dµ(s)f(s)π∗(s). (2.13)
ZΣ
3 Compactness in coorbit spaces associated to continuous frames
Let us fix a tight continuous frame W := {w(s) | s ∈ Σ} contained and total in a Fre´chet space G
that is continuously embedded in the Hilbert space H. It is assumed that s 7→ hu,w(s)i is continuous
for every u ∈ G′. For any normed space M of functions on Σ we have defined the coorbit space
co (M) ≡ co(M)withcompletion co(M),whichwillbesupposed continuously embedded inG′ .
W σ
Oneconsiders abounded subset Ω ofco(M) and investigate when this subset is relatively compact
in terms of the canonical mapping φWe ≡ φ. We are guided by [5, Th. 4], but some preparations are
neededduetoourgeneralsetting. ThenexteabstractLemmawillbeapplied toY = co(M) ֒→ Gσ′ .
Lemma 3.1. Let S(G) a family of seminorms defining the topology of G. Assume that Y is a normed
e
spacecontinuously embeddedinG′ andletΩ ⊂ Y bebounded.
σ
1. Foreveryp ∈S(G)thereexistsapositive constantD suchthat
p
|hu,vi| ≤ D kuk p(v), ∀v ∈ G, u ∈Y.
p Y
2. Seen as a subset of G′, the set Ω is equicontinuous and (consequently) relatively compact in the
weak-∗ topology.
Proof. 1isstandard; actually thecondition isequivalent toY ֒→ G′ .
σ
2. Abaseofneighborhoods oftheorigininG is
U(p;δ) := {v ∈ G|p(v) < δ} |p ∈ S(G),δ > 0 .
(cid:8) (cid:9)
7
Assumethatkuk ≤M foreveryu∈ Ω. Letǫ > 0andp ∈ S(G). Using1,foreveryv ∈U p; ǫ
Y MDp
andeveryu∈ Ωonegets (cid:16) (cid:17)
|hu,vi| ≤ D kuk p(v) ≤ D Mp(v) ≤ ǫ,
p Y p
and this is equicontinuity. The statement concerning relative compactness follows from the Bourbaki-
AlaogluTheorem[15].
Let us denote by K(Σ) the family of characteristic functions of all compact subsets in Σ. It can
be seen as a subset of the normed algebra L∞(Σ) formed of L∞ functions on Σ which are essentially
c
compactlysupported.
We assume that M is a solid Banach space of functions with absolutely continuous norm (cf. [4];
see also [5]). We recall that such a space contains all the characteristic functions of sets M ⊂ Σ with
µ(M) < ∞and givenf,g : Σ → Ctwoµ-measurable functions, if|f(s)| ≤ |g(s)|almost everywhere
and g ∈ M then f ∈ M and kf k ≤kg k . It follows that M is a Banach L∞(Σ)-module. In
M M c
addition, for allf,g ∈ Mthefollowing dominated convergence theorem holds: whenever f : Σ → C
n
are measurable, |f | ≤ |g| and f → f µ-a.e. then kf −f k → 0. Any such space is reflexive [4,
n n n M
Ch. 1,Prop. 3.6&Th. 4.1].
Theorem3.2. LetusassumethatMisasolidBanachspaceoffunctions onΣwithabsolutely continu-
ousnorm. Thenthebounded subset Ωof co(M)isrelatively compactifandonlyifφ(Ω)isK(Σ)-tight
inM.
e
Proof. Westartwiththeonlyifpart. Byrelativecompactness ofΩ,foranyǫ > 0thereisafinitesubset
F suchthat
ǫ
min ku−vkco(M)≤ , ∀u∈ Ω.
v∈F e 2
Recalling that Σ has been assumed σ-compact, there isan increasing family {L |m ∈ N}ofcompact
m
m→∞
subsets of Σ with ∪ L = Σ. Since pointwisely |χ φ(v)| ≤ |φ(v)| and χ φ(v) −−−−→ φ(v),
m m Lm Lm
thereisacompactsetL ⊂ ΣwithcomplementLc suchthat
ǫ
max kχLcφ(v)kM≤ .
v∈F 2
Then,foreveryu∈ Ω,usingtheinformation aboveandthefactthatφ: co(M) → Misisometric,
kχLcφ(u)kM≤ min(kχLcφw(u−v)kM + kχLcφ(v)kM)
v∈F e
ǫ
≤ min kφ(u−v)k +
M
v∈F 2
ǫ
= min ku−vkco(M) + ≤ ǫ.
v∈F e 2
Wenow prove the converse. Knowingthat φ(Ω) isK(Σ)-tight inM, one needs toshow that every
sequence (un)n∈N ⊂ Ω has a convergent subsequence. By Lemma3.1 the bounded set Ω ⊂ co(M) is
relativelycompactinGσ′ ,so(un)n∈N hasa∗-weaklyconvergent subsequence uj → u∞ ∈ G′:
e
hu ,vi → hu ,vi forany v ∈ G. (3.1)
j ∞
8
Puttingv := w(s)in(3.1),wegetforeverys ∈ Σ
hu ,w(s)i = [φ(u )](s) → [φ(u )](s)= hu ,w(s)i.
j j ∞ ∞
Thereforethesequence(φ(u )) ispointwiseCauchy. Weshallconvertthisinthenormconvergence
j j∈N
kφ(u )−φ(u )k → 0 when j,k → ∞. (3.2)
j k M
Then the proof would be finished since φ : co(M) → M is isometric: (uj)j∈N will be Cauchy in
co(M),thusconvergent (tou ofcourse).
∞
By tightness, pick a compact subset L ⊂ Σesuch that kχLcφ(u)kM≤ ǫ for every u ∈ Ω; then we
geet
kχLcφ(uj −uk)kM≤ 2ǫ, ∀j,k ∈ N. (3.3)
Sinceco(M)iscontinuouslyembeddedinG′ ,foranyseminormp ∈S(G)thereexistpositiveconstants
σ
D ,D′ suchthatforeverys ∈ Σ
p p
e
sup|huj −uk,w(s)i| ≤ Dpsupkuj −ukkco(M) p[w(s)] ≤ Dp′ p[w(s)].
j,k j,k e
By our assumption on W and by the Uniform Boundedness Principle the family {w(s) | s ∈ L} is
boundedinG,soweget
|[φ(u −u )](s)| ≤ D′C , ∀j,k ∈ N, s ∈ L.
j k p p,L
AnyhowweobtainbytheDominatedConvergence Theorem
kχ φ(u −u )k → 0 when j,k → ∞. (3.4)
L j k M
Putting(3.4)and(3.3)together onegets(3.2)andthustheresult.
Remark3.3. LetS beanbounded operator fromthe Banachspace X toco(M). ThenS isacompact
operatorifandonlyifforeveryǫ>0thereexistsacompactsetL ⊂ Σsuchthat
e
kχLc◦φW ◦SkB(X,M)≤ ǫ. (3.5)
This follows easily applying Theorem 3.2 to the set Ω := S X and using the explicit form of the
[1]
operatornorm. HereX denotestheclosedunitballintheBanachspaceX .
[1] (cid:0) (cid:1)
4 Compactness in Hilbert spaces
To have an ampler setting, we turn now to the particular case described in the last part of Section 2.
Thus a family {π(s) | s ∈ Σ} of bounded operators in the Hilbert space H is given. We recall that
s 7→ π(s)∗ ∈ B(H) is strongly continuous and that (2.11) is verified for every u,v ∈ H. Then φπ :
w
H → L2(Σ) defined by [φπ(u)](s) := hπ(s)u,wi is well-defined and isometric for every normalized
w
vectorwoftheHilbertspaceH.
9
Theorem4.1. LetΩbeabounded subsetof H. Considerthefollowingassertions:
1. Ωisrelativelycompact.
2. Forevery w ∈Hthefamilyφπ(Ω)isK(Σ)-tightinL2(Σ).
w
3. Thereexists w ∈ Hsuchthatthefamilyφπ (Ω)isK(Σ)-tight inL2(Σ).
0 w0
4. Foreachǫ > 0thereexistsf ∈ C (Σ)with supkΠ(f)u−uk≤ ǫ(i.e. ΩisΠ[C (Σ)]-tight).
c c
u∈Ω
5. Onehas lim sup kπ(s)∗u−π(s )∗uk= 0 foreverys ∈ Σ.
0 0
s→s0u∈Ω
6. Foreveryǫ > 0ands ∈ Σthereexistsg ∈C (Σ)suchthat supkΠ(g)u−π(s )∗uk≤ ǫ.
0 c 0
u∈Ω
Then 1,2,3 and 4 are equivalent, they imply 5, which in its turn implies 6. Thus, if we assume that
π(s )∗ = 1 forsomes ∈ Σ,thenallthesixassertions areequivalent.
1 1
Proof. The equivalence of the points 1,2 and 3 follows from Theorem 3.2, since in this case H =
co L2(Σ) and M := L2(Σ) is indeed a solid Banach space of functions with absolutely continuous
norm.
(cid:2) (cid:3)
1 ⇒ 4. Let Ω ⊂ H be relatively compact and, for some ǫ > 0, let F be a finite subset such that for
each u ∈ Ω there exists v ∈ F with ku−v k≤ ǫ/4. Thesubspace F generated by F willbe finite-
u u
dimensionalandthusthecorrespondingprojectionP willbeafinite-rankoperatorsatisfyingPv = vfor
everyv ∈ F . Thenforeveryu ∈ Ω
kPu−uk ≤kPu−Pv k+ kPv −v k + kv −uk≤ 2 ku−v k≤ ǫ/2. (4.1)
u u u u u
Notice that {Π(f) | f ∈ C (Σ)} is a dense set of compact operators. To see this, use the fact that
c
Π : L2(Σ) → B (H) is an isometric isomorphism and that C (Σ) is dense in L2(Σ); the topology of
2 c
B (H)isstronger thanthatofB(H),whileK(Σ)istheclosureofB (H)intheoperator norm. Letnow
2 2
M := supu∈Ωkuk; by density there is some f ∈ Cc(Σ) with kP −Π(f)kB(H)≤ ǫ/2M . From this
andfrom(4.1)theconclusion followsimmediately.
4 ⇒ 1. To prove the converse, for ǫ > 0 choose f ∈ C (Σ) such that sup kΠ(f)u−uk≤ ǫ/2.
c u∈Ω
SinceΠ(f)isacompactoperatorandΩisbounded,therangeΠ(f)Ωisrelativelycompact,sothereisa
finiteset Gsuch that foreach u ∈ Ω there isan element vu ∈ Gwith kΠ(f)u−vuk≤ ǫ/2. Thenfor
u∈ Ωonehas
ku−vuk≤ku−Π(f)uk + kΠ(f)u−vuk≤ ǫ/2+ǫ/2 = ǫ,
sothesetΩistotally bounded.
4 ⇒ 5. Setting S⊥ := 1−S, we compute for s ∈ Σ, u ∈ Ω, f ∈ C (Σ) and s belonging to a
0 c
neighborhood V ofs :
0
kπ(s)∗u−π(s )∗uk ≤k[π(s)∗−π(s )∗]Π(f)uk + k[π(s)∗ −π(s )∗]Π(f)⊥uk
0 0 0
≤ sup kukk[π(s)∗ −π(s0)∗]Π(f)kB(H) +2sup kπ(t)∗kB(H) sup kΠ(f)⊥uk .
u∈Ω t∈V u∈Ω
10