Robertson-type Theorems for Countable Groups of Unitary Operators

0.16 MB·English

by  David R. Larson

ContemporaryMathematics
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Robertson-type Theorems for Countable Groups of Unitary

n
Operators
a
J

0 David R. Larson, Wai-Shing Tang, and Eric Weber
2

] Abstract. Let G be a countably infinite group of unitary operators on a
A complexseparableHilbertspaceH. LetX={x1,...,xr}andY ={y1,...,ys}
O be finite subsets of H, r<s, V0 =spanG(X),V1 =spanG(Y)and V0 ⊂V1.
We provethe followingresult: LetW0 beaclosedlinearsubspace of V1 such
h. that V0⊕W0 = V1 (i.e., V0+W0 = V1 and V0∩W0 = {0}). Suppose that
t G(X) andG(Y)areRieszbases forV0 andV1 respectively. Thenthereexists
a a subset Γ = {z1,...,zs−r} of W0 such that G(Γ) is a Riesz basis for W0 if
m
and only if g(W0) ⊆ W0 for every g in G. We first handle the case where
[ thegroupisabelianandthenuseacancellationtheorem ofDixmiertoadapt
thistothenon-abeliancase. Correspondingresultsfortheframecaseandthe

1 biorthogonal casearealsoobtained.
v
6
0
5
1. Introduction
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Wefirstconsiderwavelet-typeproblemsassociatedwithcountablyinfiniteabelian
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groupsofunitary operatorsona complex separable Hilbert space. Other results in
0
such and similar settings can be found in [4, 9, 10, 17]. We then adapt this to
/
h
non-abelian groups using a cancellation theorem of Dixmier.
t
a Section 2 is a revisit of Robertson’s theorems [12] in the setting of countably
m infinite abelian groups of unitary operators. Recently, Han, Larson, Papadakis
: and Stavropoulos [10] extended Robertson’s original results to this setting, with
v
proofsinvolvingthespectraltheoremforanabeliangroupofunitaryoperatorsand
i
X certainnon-trivialfactsonvonNeumannalgebras. Usingsomesimpleobservations
r on harmonic analysis, we show how Robertson’s original elementary proofs in [12]
a
still work in this new setting. In section 3, we consider the problem of existence
of oblique multiwavelets, and show that oblique multiwavelets exist under a very
naturalassumption. Theresultshereextendthoseof[14, 16]. Finallyinsection4,
we discuss the non-abelian case using a cancellation theorem of Dixmier.
Let us set up some notations and terminologies. Throughoutthis paper, let H
denote a complex separable Hilbert space. The inner product of two vectors x and
2000 Mathematics Subject Classification. 46C99, 47B99,46B15.
Keywordsandphrases. Biorthogonalsystems,Rieszbases,multiwavelets,unitaryoperators.
Thefirstandthirdauthors werepartiallysupportedbygrants fromtheNSF.
The second author’s research was supported in part by Academic Research Fund #R-146-
000-073-112,NationalUniversityofSingapore.
(cid:13)c2006 American Mathematical Society
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2 DAVIDR.LARSON,WAI-SHINGTANG,ANDERICWEBER
y in H is denoted by hx,yi. A countable indexed family {v } of vectors in H
n n∈J
is a Riesz basis for its closed linear span V = span{v } if there exist positive
n n∈J
constants A and B such that
(1.1) A |a |2 ≤k a v k2 ≤B |a |2, ∀{a }∈ℓ2(J).
n n n n n
X X X
A countable indexed family {vn}n∈J is a frame for its closed linear span V if there
exist positive constants A and B such that
(1.2) Akfk2 ≤ |hf,v i|2 ≤Bkfk2, ∀f ∈V.
n
X
ItiswellknownthataRieszbasisforaHilbertspaceisaframeforthesamespace.
Two families {v } and {v˜ } in H are biorthogonal if
n n
(1.3) hv ,v˜ i=δ ∀n, m.
n m n,m
If V and W are closed linear subspaces of H such that V ∩W = {0} and the
vector sum V =V +W is closed, then we write V =V ⊕W and call this a direct
1 1
sum. In this case, the map P :V −→V defined by
1 1
(1.4) P(v+w)=v, v ∈V, w ∈W,
is called the (oblique) projection of V on V along W. For the special case when
1
V and W are orthogonal,we shall write V =V ⊕⊥W and call this an orthogonal
1
direct sum. We write V⊥ for the orthogonalcomplement of V in H.
Let G be a discrete group. For every g in G, let χ denote the characteristic
g
function of {g}. Then {χ : g ∈ G} is an orthonormal basis for ℓ2(G). For each g
g
in G, define l :ℓ2(G)−→ℓ2(G) by (l a)(h)=a(g−1h), h∈G. Then l (χ )=χ
g g g h gh
for all g, h in G. The left regular representation λ of G is the homomorphism
λ:g 7→l .
g
2. Robertson’s Theorem for Countable Abelian Groups of Unitary
Operators
Let B(H) be the space of all bounded linear maps on H. A unitary system U
inB(H) is a setofunitary operatorsonH whichcontainsthe identity operatoron
H. AclosedlinearsubspaceM ofH isawanderingsubspaceforU ifU(M)⊥V(M)
for all U,V in U with U 6= V. It was first observed in [7] that Robertson’s results
onwanderingsubspacesfor the cyclicgroupgeneratedbya singleunitaryoperator
[12]haveinterestingconnectiontotheexistenceoforthonormalwaveletsassociated
withorthonormalmultiresolutions. Subsequently,itwasnotedin[8]thatanalogues
of Robertson’s results hold for the group generated by a finite set of commuting
unitary operators. Recently, Han, Larson, Papadakis and Stavropoulos extended
Robertson’sresultstothesetting ofacountableabeliangroupofunitaryoperators
[10, Theorem 4]. Their proof uses the spectral theorem for an abelian group of
unitary operators and certain non-trivial facts on von Neumann algebras. The
purpose of this section is to show, with the help of some simple observations, how
Robertson’s original elementary proof in [12] can easily be carried over, almost
verbatim, to this new setting.
Lemma 2.1. Let M and K be wandering subspaces of H for a countable group
G of unitary operators on H. If ⊕⊥g(M)⊆ ⊕⊥g(K), then dim(M)≤
g∈G g∈G
dim(K). P P
ROBERTSON-TYPE THEOREMS 3
Proof. WemodifytheargumentsduetoI.Halperin,givenin[6,p. 17],forthe
caseofasingleunitaryoperator. Theassertionistrivialifdim(K)=∞. Therefore
suppose that dim(K) = k < ∞. Let {x ,...,x } be an orthonormal basis for K.
1 k
Then {gx : g ∈ G,j = 1,...,k} is an orthonormal basis for K = ⊕⊥g(K).
j 1 g∈G
Let {yi}i∈I be an orthonormal basis for M, so {gyi : g ∈ G,i ∈ I}Pis orthonormal
in K . By Bessel’s inequality,
1
|hx ,gy i|2 ≤kx k2 =1, j =1,...,k.
j i j
XX
g∈G i∈I
Hence
dim(M) = ky k2
i
X
i∈I
k
= |hy ,gx i|2
i j
XXX
i∈I g∈Gj=1
k
= |hx ,g−1y i|2
j i
XXX
j=1g∈G i∈I
≤ k.
(cid:3)
Let us recallsome results on harmonic analysis (see, for example, [13]). Let G
be a locally compact abelian group, and Gˆ = {γ :G−→T continuous characters}
the dual group of G.
Fact 1: G is isomorphic and homeomorphic to Gˆˆ, the dual group of Gˆ, via the
canonical map g −→e , where e (γ)=γ(g), γ ∈Gˆ.
g g
Fact 2: The Fourier transform on L1(G) is the map ∧:L1(G)−→C (Gˆ) defined
0
by
fˆ(γ)= f(g)γ(g)dg, γ ∈Gˆ, f ∈L1(G).
Z
G
Identifying Gˆˆ with G as in Fact 1, the Fourier transform on L1(Gˆ) is the map
∧:L1(Gˆ)−→C (Gˆˆ)=C (G) defined by
0 0
fˆ(g)= f(γ)γ(g)dγ, g ∈G, f ∈L1(Gˆ).
Z
Gˆ
Fact 3: If G is compact abelian, then Gˆ forms an orthonormal basis for L2(G).
Correspondingly,ifGisdiscreteabelian,thenGˆˆ ={e :g ∈G}formsanorthonor-
g
mal basis for L2(Gˆ).
Fact 4: Suppose that G is a discrete abeliangroup(so Gˆ is compactand L2(Gˆ)⊂
L1(Gˆ)). Iff isinL2(Gˆ),thenfˆ(g)=hf, e i, g ∈G,andfˆisinℓ2(G). Moreover,
g
(i) the Fourier transform ∧:L2(Gˆ)−→ℓ2(G) is a unitary operator;
(ii) eˆ =χ for every g in G;
g g
(iii) for all f,h in L2(Gˆ), we have fh=fˆ∗hˆ, i.e.,
fh(g)= fˆc(m)hˆ(gm−1), g ∈G.
X
m∈G
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4 DAVIDR.LARSON,WAI-SHINGTANG,ANDERICWEBER
(iv) for every g in G, we have ∧◦M = l ◦∧, where M :L2(Gˆ)−→L2(Gˆ) is
g g g
the multiplication operator defined by
(M f)(γ)=γ(g)f(γ), γ ∈Gˆ.
g
Using Fact 4(i)–(iii), the proofs of Theorem 1 and Theorem 2 in [12] can now
be carried over verbatim to our new setting, with Z and [0, 2π) there replaced by
G and G respectively. Hence we have
Thbeorem 2.2. ([10, Theorem 4]) Let G be a countably infinite abelian group
of unitary operators on H, let X and Y be wandering subspaces of H for G such
that
(a) ⊕⊥g(X)⊆ ⊕⊥g(Y),
g∈G g∈G
(b) dPim(Y)<∞. P
Then there exists a wandering subspace X′ of H for G such that
(i) g(X)⊥h(X′) for all g,h∈G,
(ii) ⊕⊥g(X)⊕⊥ ⊕⊥g(X′)= ⊕⊥g(Y),
g∈G g∈G g∈G
(iii) Pdim(X)+dim(X′)P=dim(Y). P
Related results for the setting of frames can be found in [9] and [17].
3. Oblique Multiwavelets and Biorthogonal Multiwavelets
Throughoutthis section, let G be a countably infinite abelian groupof unitary
operators on a complex separable Hilbert space H. Let X = {x ,...,x } and
1 r
Y ={y ,...,y }befinitesubsetsofH,r<s,V =spanG(X), V =spanG(Y)and
1 s 0 1
(3.1) V ⊂V .
0 1
Theorem3.1. LetW beaclosedlinearsubspaceofV suchthatV ⊕W =V .
0 1 0 0 1
Suppose that G(X) and G(Y) are Riesz bases for V and V respectively. Then there
0 1
exists a subset Γ = {z1,...,zs−r} of W0 such that G(Γ) is a Riesz basis for W0 if
and only if
(3.2) g(W )⊆W , g ∈G.
0 0
Such types of oblique multiwavelets were previously studied in [1, 2, 15, 16].
For the proof of Theorem 3.1, we need the following elementary result from [11,
Lemma 3.5].
Lemma 3.2. Let M,M′ and N be linear subspaces of a vector space X such
that
X =M ⊕N =M′⊕N.
Let P be the projection of X on M along N and let Q be the projection of X on M′
along N. Then P1 =P|M′ :M′ −→M and Q1 =Q|M :M −→M′ are invertible,
and P−1 =Q .
1 1
Proof of Theorem 3.1. The“onlyif”partisobvious. Conversely,supposethat(3.2)
holds. Let V =V ∩V⊥, the orthogonalcomplement of V in V . Then we have
1 0 0 1
V =V ⊕⊥V =V ⊕W .
1 0 0 0
By [11, Proposition 2.2] and Theorem 2.2, there exists Z = {w1,...,ws−r} ⊂ V
such that G(Z) is an orthonormal basis for V. Let P : V −→ V be the oblique
1 1
ROBERTSON-TYPE THEOREMS 5
projection of V on W along V . By Lemma 3.2, P| : V −→ W is invertible.
1 0 0 V 0
Hence {Pgw : g ∈ G, j = 1,...,s−r} is a Riesz basis for W . Since g(V ) = V
j 0 0 0
for everyg ∈G, and (3.2) holds, P commutes with g| for everyg ∈G. Therefore,
V1
G(Γ)isaRieszbasisforW0,wherezj =Pwj, j =1,...,s−r,andΓ={z1,...,zs−r}.
(cid:3)
A corresponding result for frames is also valid.
Theorem 3.3. Let W bea closed linear subspaceof V suchthat V ⊕W =V
0 1 0 0 1
(i.e., V +W =V and V ∩W ={0}). Suppose that G(X) and G(Y) are frames
0 0 1 0 0
for V and V respectively. Then there exists a subset Γ = {z ,...,z } of W such
0 1 1 p 0
that G(Γ) is a frame for W if and only if
0
(3.3) g(W )⊆W , g ∈G.
0 0
Note that here the number p is not necessarily equal to s−r, as in the corre-
sponding case for Riesz bases. We omit the proof of Theorem 3.3, which is similar
to that of Theorem 3.1. Instead of applying Theorem 2.2, we use the following
simple observation(see [9]): if {vn}n∈J is a frame for V1 and P⊥ is the orthogonal
projection of V1 onto V =V1∩V0⊥, then {P⊥(vn)}n∈J is a frame for V.
AsacorollaryoftheobliquecaseinTheorem3.1,weobtainthefollowingresult
for the biorthogonalcase, which is an extension of [14, Theorem 3.6].
Corollary 3.4. Let X = {x ,...,x }, X˜ = {x˜ ,...,x˜ }, Y = {y ,...,y } and
1 r 1 r 1 s
Y˜ = {y˜ ,...,y˜ } be finite subsets of H. Let G(X),G(X˜),G(Y) and G(Y˜) be Riesz
1 s
bases for their closed linear spans V ,V˜ ,V and V˜ respectively, G(X) biorthogonal
0 0 1 1
to G(X˜), G(Y) biorthogonal to G(Y˜), and
V ⊂V , V˜ ⊂V˜ .
0 1 0 1
Let W =V ∩V˜⊥ and W˜ =V˜ ∩V⊥. If r <s, then
0 1 0 0 1 0
(i) there exists a subset Γ:={z1,...,zs−r} of W0 such that G(Γ) is a Riesz basis
for W and G(X ∪Γ) is a Riesz basis for V , and
0 1
(ii) there exists a subset Γ:={z˜1,...,z˜s−r} of W˜0 such that G(Γ) is a Riesz basis
for W˜ and G(X˜ ∪Γ) is a Riesz basis for V˜ , and G(Γ) is biorthogonal to
0 e 1 e
G(Γ).
e e
Proof. Foreveryg ∈G, wehaveg(V )=V andg(V˜ )=V˜ forj =0,1. As g
j j j j
is unitary, using the definitions of W and W˜ , g(W )=W and g(W˜ )=W˜ too.
0 0 0 0 0 0
SinceV1 =V0⊕W0,byTheorem3.1,thereexistsasubsetΓ:={z1,...,zs−r}ofW0
such that G(Γ) is a Riesz basis for W . By [14, Proposition 3.1], W˜ ⊕W⊥ = H.
0 0 0
Hence by [16, Lemma 3.1], there exists a subset Γ := {z˜1,...,z˜s−r} of W˜0 such
that G(Γ) is a Riesz basis for W˜ and G(Γ) is biorthogonalto G(Γ). The remaining
0 e
assertions follow easily from [14, Theoerm 2.1]. (cid:3)
e e
4. Non-Abelian Groups
We consider now the possibility that G is not abelian. The proofs of the major
statements all follow from the following fundamental result:
Theorem 4.1 (Cancellation Theorem). Let G be a locally compact group, and
let ρ be a unitary representation of G whose commutant is a finite von Neumann
6 DAVIDR.LARSON,WAI-SHINGTANG,ANDERICWEBER
algebra. Suppose ρ is equivalent to σ ⊕σ as well as σ ⊕σ . Then σ is equivalent
1 2 1 3 2
to σ .
3
TheproofofthistheoremisanapplicationofaresultduetoDixmier[5,III.2.3
Proposition6]. WewillusetheCancellationTheoreminthefollowingspecificcase,
altered slightly from [3].
Proposition4.2. SupposeG isadiscretegroup, andρis arepresentation ofG
which is equivalent to a finite multiple of the left regular representation of G. Then
thecommutant of ρ is a finitevon Neumann algebra, whence thesubrepresentations
of ρ satisfy the cancellation property. That is to say, if ρ is equivalent to σ ⊕σ
1 2
as well as σ ⊕σ , then σ is equivalent to σ .
1 3 2 3
Theorem 4.3. The statements in Theorems 2.2, 3.1, 3.3, and Corollary 3.4
still hold when the assumption that G is abelian is removed.
Proof. WeproveTheorem2.2,withouttheassumptionthatG isabelian. The
other statements follow analogously.
Let K = ⊕⊥g(Y) and let ρ denote the action of G on K. Since Y is a
g∈G
complete wandPering subspace of K for G and is finite dimensional, we have that ρ
is equivalent to a finite multiple of the left regular representation λ of G, whence
wecanapplythe cancellationtheorem. LetK = ⊕⊥g(X);letσ denotethe
1 g∈G 1
action of G on K , and let σ denote the action oPf G on K ∩K⊥, the orthogonal
1 2 1
complement of K in K.
1
For every positive integer N, let λ denote the N multiple of the left regular
N
representationλ of G. We have the following equivalences:
λ ⊕λ ≃λ ≃ρ≃σ ⊕σ
dim(X) dim(Y)−dim(X) dim(Y) 1 2
as well as
σ ≃λ ,
1 dim(X)
since X is a complete wandering subspace for σ on K .
1 1
Therefore we have that
σ ⊕λ ≃σ ⊕σ .
1 dim(Y)−dim(X) 1 2
By the cancellation theorem, we have that
σ ≃λ ,
2 dim(Y)−dim(X)
from which it follows that K ∩ K⊥ contains a complete wandering subspace of
1
dimension dim(Y)−dim(X); call this subspace X′. The items (i) (ii) and (iii)
from Theorem 2.2 now follow. (cid:3)
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Department of Mathematics, Texas A&M University, College Station, TX 77843,
USA
E-mail address: [email protected]
DepartmentofMathematics,NationalUniversity of Singapore,2ScienceDrive2,
117543,Republicof Singapore
E-mail address: [email protected]
Department of Mathematics, Iowa State University, 396 Carver Hall, Ames, IA
50011,USA
E-mail address: [email protected]