Commutative $C^*$-algebras of Toeplitz operators on complex projective spaces

0.22 MB·English

by  Raul Quiroga-Barranco

COMMUTATIVE C∗-ALGEBRAS OF TOEPLITZ OPERATORS
ON COMPLEX PROJECTIVE SPACES
2
1 RAULQUIROGA-BARRANCOANDARMANDOSANCHEZ-NUNGARAY
0
2
Abstract. We prove the existence of commutative C∗-algebras of Toeplitz
n operatorsoneveryweightedBergmanspaceoverthecomplexprojectivespace
a Pn(C). The symbols that define our algebras are those that depend only
J
on the radial part of the homogeneous coordinates. The algebras presented

0 have anassociated pairofLagrangian foliationswithdistinguishedgeometric
1 propertiesandarecloselyrelatedtothegeometryofPn(C).

]
A
O
1. Introduction
.
h The existence of nontrivial commutative C∗-algebras of Toeplitz operators on
t
a bounded domains has shown to be a remarkably interesting phenomenon (see [2],
m [8], [9], [10], [13]). Large families of symbols defining such commutative algebras
[ on every weighted Bergman space have been proved to exist on the unit ball and
Reinhardt domains. Also, in [7] it was proved the existence of commutative C∗-
1
algebrasofToeplitzoperatorsforthesphere. Alloftheexamplesexhibitedinthese
v
7 references come equiped with a distinguished geometry and can be described in
4 terms of the isometry group of the domain.
1 InthisworkwestudythecomplexprojectivespacePn(C)anditshyperplaneline
2
bundle H. It is well known that these objects provide the usual setup to consider
.
1 quantization through the use of weighted Bergman spaces (see [11]). Note that,
0 due to the compactness of Pn(C), the weight of the Bergmanspaces has a discrete
2 set of values. Within this setup for Pn(C), we prove the existence of commutative
1
C∗-algebrasofToeplitzoperatorsonallweightedBergmanspaces. Asinthecaseof
:
v previousworks,ouralgebrascanbedescribedintermsofthesymmetriesofPn(C).
Xi To elaborate on this, recall that for every hyperplane P ⊂ Pn(C), the open set
r Pn(C)\P can be canonically identified with the complex plane Cn. With respect
a to such identification, we prove that the symbols that depend only on the radial
components of the coordinates of Cn define commutative C∗-algebras of Toeplitz
operators on every weighted Bergman space over Pn(C) (see Theorem 5.5); this
sort of symbols are called separately radial. Furthermore, we construct unitary
equivalences of the weighted Bergman spaces with suitable (finite dimensional)
Hilbert spaces such that the Toeplitz operators with separately radial symbols are
simultaneously turned into multiplication operators.
We alsoprovethat our commutativeC∗-algebrashavea distinguished geometry
thatdescribesthem. Recallthattheconnectedcomponentofthegroupofisometries
1991 Mathematics Subject Classification. Primary47B35;Secondary 32A36,32M15,53C12.
Keywordsandphrases. Toeplitzoperator,Bergmanspace,complexprojectivespace,commu-
tativeC∗-algebra,Lagrangianfoliation,Abeliangroup.
ThefirstnamedauthorwaspartiallysupportedbySNI-MexicoandbytheConacytgrantno.
82979. Thesecondnamedauthor waspartiallysupportedbyaConacyt postdoctoral fellowship.
1
2 R.QUIROGA-BARRANCOANDA.SANCHEZ-NUNGARAY
of Pn(C) is given by the group SU(n+1) of (n+1)×(n+1) unitary matrices
with determinant 1. It is proved that the separately radial symbols are those
invariantundertheactionofthediagonalmatricesofSU(n+1)(seeLemma6.1);in
particular,thisprovidesadescriptionoftheseparatelyradialsymbolsthatdepends
onPn(C)only,andnotonasetofcoordinates. Also,itallowsustoshowthatevery
commutativeC∗-algebraobtainedfromseparatelyradialsymbolshasanassociated
pair ofLagrangianfoliations with specialgeometric properties (see Theorem6.11).
Such properties come from the group A(n) of isometries of Pn(C) defined by the
diagonal matrices in SU(n + 1). It turns out that A(n) is a maximal Abelian
subgroup of the group of isometries of Pn(C) (see Section 6). We prove that, up
to an isometry, A(n) is the only such maximal Abelian subgroup. This behavior is
in contrastwith the existence of n+2 maximal Abelian subgroups of the group of
isometries of Bn, which provides n+2 nonequivalent commutative C∗-algebras of
Toeplitzoperators(see[8]and[9]). ThisiswhyfortheprojectivespacePn(C)there
exist, up to an isometry, only one commutative C∗-algebra of Toeplitz operators
that has an associated pair of foliations as described in Theorem 6.11.
As for the contents of this work, in Section 2 we present some differential geo-
metric preliminaries onPn(C). Section3 recallsthe basic materialonquantization
on compact K¨ahler manifolds. Section 4 recollects the known results specific to
thequantizationforcomplexprojectivespacesincludingweightedBergmanspaces.
Sections 5 and 6 containour main results and proofs onthe existence ofcommuta-
tive C∗-algebrasandtheir geometricdescription; ourtechniques are closelyrelated
to those found in [8] and [9].
2. Geometric preliminaries on Pn(C)
Recall that Pn(C) is the complex n-dimensional manifold that consists of the
elements [w] = Cw\{0}, where w ∈ Cn+1\{0}. For every j = 0,...,n, we have
an open set
U ={[w]∈Pn(C):w 6=0}
j j
and a biholomorphism ϕ :U →Cn given by
j j
1
ϕ ([w])= (w ,...,w ,...,w )=(z ,...,z ),
j 0 j n 1 n
w
j
where the numbers zk are known as tbhe homogeneous coordinates with respect
to the map ϕ . The collection of all such maps yields the holomorphic atlas of
j
Pn(C) and provides (equivalent) realizations of Cn as an open dense conull subset
of Pn(C).
We refer to Example 6.3 of [5] for the details on the following construction of
the Fubini-Study metric on Pn(C).
For every j =0,...,n consider the function f :U →C given by
j j
n n
w w
k k
(2.1) f ([w])= =1+ z z ,
j k k
w w
j j
k=0 k=1
X X
for the above homogeneous coordinates z with respect to ϕ . Then, it is easily
k j
seen that
∂∂logf =∂∂logf
j k
COMMUTATIVE TOEPLITZ OPERATORS ON Pn(C) 3
on U ∩U for every j,k = 0,...,n. In particular, there is a well defined closed
j k
(1,1)-form ω on Pn(C) given by
(2.2) ω =i∂∂logf ,
j
on U . This yields the canonical K¨ahler structure on Pn(C) for which it is the
j
Hermitiansymmetricspacewithconstantpositiveholomorphicsectionalcurvature.
The corresponding Riemannian metric is known as the Fubini-Study metric. We
refer to [5] for these and the rest of the remarks in this section on the geometry of
Pn(C) induced by ω.
With respect to the chartϕ , we have the following induced K¨ahler form on Cn
0
(1+|z|2) n dz ∧dz − n z z dz ∧dz
ω =(ϕ−1)∗(ω)=i k=1 k k k,l=1 k l k l.
0 0 (1+|z|2)2
P P
The volume element on Pn(C) with respect to the Fubini-Study metric is defined
by
1
Ω= ωn.
(2π)n
The following result is a consequence of the definition of the volume element of a
Riemannian metric and its properties for Hermitian symmetric spaces (see [3]).
Lemma 2.1. The volume element on Cn induced by the Fubini-Study metric of
Pn(C) is given by
1 1 dV(z)
Ω= ωn =
(2π)n 0 πn(1+|z |2+···+|z |2)n+1
1 n
where dV(z)=dx ∧dy ∧···∧dx ∧dy is the Lebesgue measure on Cn.
1 1 n n
We also recall that the Hopf fibration of Pn(C) is given by
π :S2n+1 →Pn(C)
w 7→[w],
where S2n+1 ⊂Cn+1 is the unit sphere centered at the origin.
We denote with SU(n+1) the Lie group of (n+1)×(n+1) unitary matrices
with determinant 1. In other words, for A a complex (n+1)×(n+1) matrix
with det(A) = 1, we have A ∈ SU(n+1) if and only if A∗A = I . For Z
n+1 n+1
the group of (n + 1)-th roots of unity in C, we also consider the quotient Lie
group PSU(n+1,C)=SU(n+1,C)/Z I , and we denote with λ the natural
n+1 n+1
quotientmapofSU(n+1)ontoPSU(n+1). Itis wellknownthatthese Lie groups
areconnectedand compact. The followingresult is a consequence ofExample 10.5
of [5] and Section 4 in Chapter VIII of [3].
Proposition2.2. Thenaturalaction ofSU(n+1)onS2n+1 inducesaholomorphic
action of PSU(n+1) on Pn(C) that satisfies the following properties:
• λ(A)[w]=[Aw] for every w ∈S2n+1 and A∈SU(n+1).
• The induced PSU(n+1)-action on Pn(C) realizes the connected component
Iso (Pn(C))ofthegroupofisometriesofPn(C)fortheFubini-Studymetric.
0
Hence, we have an isomorphism of Lie groups Iso (Pn(C))≃PSU(n+1).
0
4 R.QUIROGA-BARRANCOANDA.SANCHEZ-NUNGARAY
3. Quantum line bundles and quantization on compact Ka¨hler
manifolds
Let M be a K¨ahler manifold with K¨ahler form ω. We will now recall the notion
of a quantum line bundle over M which allows to consider the Berezin-Toeplitz
quantization for suitable sections over M. We refer to [1] and [11] for further
details.
Suppose that L → M is a holomorphic line bundle. For any such line bundle,
wewilldenotewithΓ(U,L)thespaceofsmoothsectionsofLoveranopensetU of
M. A Hermitian metric h on L is a smooth choice of Hermitian inner products on
the fibers of L. For such metric, the pair (L,h) is called a Hermitian line bundle.
A connection D on L is given by assignments
D :Γ(U,L)→Γ1(U,L),
whereU isanopensubsetofM andΓk(U,L)denotesthespaceofL-valuedk-forms
over U; also, D must be complex linear and satisfy the following property
D(fζ)=df ⊗ζ+fDζ,
foreveryf ∈C∞(U)andζ ∈Γ(U,L). WecanextendDasaderivationonL-valued
forms so that it defines maps D : Γk(U,L) → Γk+1(U,L). Then, the curvature of
D is defined as D2 : Γ(U,L) → Γ2(U,L). It is well known that D2 is linear with
respect to the multiplication by smooth functions, which implies that D2 defines a
L⊗L∗-valued 2-form. The latter is called the curvature of D.
Also,theconnectionD issaidtobecompatiblewithhifthefollowingconditions
are satisfied:
• For every ζ ,ζ ∈Γ(U,L) and X a smooth vector field over U we have
1 2
X(h(ζ ,ζ ))=h(D ζ ,ζ )+h(ζ ,D ζ ).
1 2 X 1 2 1 X 2
• For every holomorphic section ζ ∈ Γ(U,L) and X a smooth vector field
over U of type (0,1) we have
D ζ =0.
X
The followingresultestablishes the existence and uniqueness ofcompatible con-
nections. It also provides an expression for the curvature form. We refer to [1] for
its proof.
Proposition 3.1. For a Hermitian line bundle (L,h) over a K¨ahler manifold M,
there exist a unique connection D that is compatible with h. The curvature can be
considered as a complex-valued (1,1)-form Θ. Moreover, if ζ is a local holomorphic
section of L on the open set U, then on this open set the curvature is given by
Θ=∂∂log(h(ζ,ζ)).
The connection from Proposition 3.1 is called the Hermitian connection of the
Hermitian line bundle.
A Hermitian line bundle (L,h) over a K¨ahler manifold M is called a quantum
line bundle if it satisfies the condition
Θ=−iω,
where Θ is the curvature of L and ω is the K¨ahler form of M. By Proposition3.1,
this is equivalent to requiring
ω =i∂∂log(h(ζ,ζ)),
COMMUTATIVE TOEPLITZ OPERATORS ON Pn(C) 5
for every open set U ⊂M and every local holomorphic section of L defined on U.
For any Hermitian line bundle (L,h) and m∈Z , we denote Lm =L⊗···⊗L
+
(m times), which is itself a Hermitian line bundle with the induced metric. We
will denote the latter by h(m). Furthermore, it is easy to see that the Hermitian
connection of Lm is the induced connection of D to the tensor product. We will
denote such connection by D(m).
We recall that the volume form of the K¨ahler manifold (M,ω) is given by Ω =
ωn/(2π)n, where n=dimC(M). Note that for a quantum line bundle, this volume
can be considered as coming from the geometry of either M or L.
Inthe restofthissectionweassumethatM iscompact,andsothatΩhasfinite
total volume. On each space Γ(M,Lm) we define the Hermitian inner product
hζ,ξi= h(m)(ζ,ξ)Ω,
ZM
whereζ,ξ ∈Γ(M,Lm). TheL -completionofthelatterHermitianspaceisdenoted
2
by L (M,Lm). Since M is compact, the space of global holomorphic sections of
2
Lm, denoted by Γ (M,Lm), is finite dimensional and so closed in L (M,Lm). In
hol 2
particular, we have an orthogonalprojection
Π :L (M,Lm)→Γ (M,Lm),
m 2 hol
for every positive integer m.
4. Quantization and Bergman spaces on Pn(C)
In this section we recollect some known properties and facts of the projective
spacePn(C)thatprovideitsquantizationandtheBergmanspacesonit. Forfurther
details on the less elementary facts we refer to [1] and [11].
Recall that the tautological or universal line bundle of Pn(C) is given by
T ={([w],z)∈Pn(C)×Cn+1 :z ∈Cw},
and assigns to every point in Pn(C) the line in Cn+1 that such point represents. It
is well known that T is a holomorphic line bundle. Furthermore, T has a natural
Hermitian metric h inherited from the usual Hermitian inner product on Cn+1.
0
Let us denote by H = T∗ the dual line bundle with the corresponding induced
metric h dual to the metric h on T. The line bundle H is called the hyperplane
0
line bundle. The following well known result provides a quantization for Pn(C) as
described in Section 3.
Proposition 4.1. The Hermitian line bundle (H,h) is a quantum line bundle over
Pn(C).
Foreverym∈Z andwithrespecttothe coordinatesgivenbyϕ ,theweighted
+ 0
measure on Pn(C) with weight m is given by
(n+m)! Ω(z)
dν (z)= ,
m m! (1+|z |2+···+|z |2)m
1 n
which has the following explicit expression
(n+m)! dV(z)
dν (z)=
m πnm! (1+|z |2+···+|z |2)n+m+1
1 n
where,asbefore,dV(z)=dx ∧dy ∧···∧dx ∧dy istheLebesguemeasureonCn.
1 1 n n
Forthe sakeofsimplicity, we willuse the samesymbol dν to denote the weighted
m
6 R.QUIROGA-BARRANCOANDA.SANCHEZ-NUNGARAY
measures for both Pn(C) and Cn. Note that that dν is a probability measure.
m
Following the remarks of Section 3, the Hilbert space L (Pn(C),Hm) denotes the
2
L -completionofΓ(Pn(C),Hm)withrespecttotheinnerproductdefinedusingthe
2
measure dν .
m
The line bundle H can be trivialized over each subset U ⊂ Pn(C) (U as in
j j
Section 2) so that the corresponding set of transition functions for Hm are given
as follows
gm :U ∩U →C∗
kj j k
wm
[w]7→ k .
wm
j
In particular,there is a trivialization of Hm over U . Then, every section ζ of Hm
0
restricted to U can be considered as a map ζ| : U → C. Since ϕ−1 : Cn → U
0 U0 0 0 0
defines a biholomorphism, the composition ζ = ζ| ◦ϕ−1 maps Cn → C. Then,
U0 0
we have the following well known result.
b
Proposition 4.2. The map given by
Φ :L (Pn(C),Hm)→L (Cn,ν )
0 2 2 m
ζ 7→ζ,
is an isometry of Hilbert spaces.
b
The weighted Bergman space on Pn(C) with weight m∈Z is defined by
+
A2 (Pn(C))={ζ ∈L (Pn(C),Hm):ζ is holomorphic}
m 2
=Γ (Pn(C),Hm).
hol
As remarked before, since Pn(C) is compact, the space A2 (Pn(C)) is finite di-
m
mensional. Furthermore,thenextwellknownresultprovidesanexplicitdescription
of these Bergman spaces.
Proposition 4.3. For every m∈Z , the Bergman space A2 (Pn(C)) satisfies the
+ m
following properties.
(1) With respect to the homogeneous coordinates of Pn(C), the Bergman space
A2 (Pn(C)) can be identified with the space P(m)(Cn+1) of homogeneous
m
polynomials of degree m over Cn+1.
(2) For Φ the isometry from Proposition 4.2, we have
0
Φ (A2 (Pn(C)))=P (Cn),
0 m m
where P (Cn) denotes the space of polynomials on Cn of degree ≤m.
m
Propositions 4.2 and 4.3 allow us to reduce our computations on the spaces
L (Pn(C),Hm)andA2 (Pn(C))tothespacesL (Cn,ν )andP (Cn),respectively.
2 m 2 m m
In what follows and when needed, we will use such reductions by applying the
corresponding identifications without further mention.
The Bergman space A2 (Pn(C)) identified with P (Cn) has the natural mono-
m m
mialbasiswhichwewilluse forourcomputations. Suchmonomialsaredenotedby
zp =zp1...zpn where p=(p ,...,p ) and |p|=p +···+p ≤m. We denote the
1 n 1 n 1 n
COMMUTATIVE TOEPLITZ OPERATORS ON Pn(C) 7
enumerating set by J (m)={p∈Zn :|p|≤m}. A direct computation shows that
n +
(n+m)! zpzpdV(z)
hzp,zpi =
m πnm! ZCn (1+|z1|2+···+|zn|2)n+m+1
(n+m)! tp1···tpndt ···dr
= 1 n 1 n
m! ZRn+ (1+t1+···+tn)n+m+1
p!(m−|p|)!
=
m!
for every p,q ∈ J (M), where p! = p !···p ! and p ∈ J (m). Also, it is easy to
n 1 n n
check that hzp,zqi =0 for all p,q ∈J (m) such that p6=q. In particular, the set
m n
1
m! 2
(4.1) zp :p∈J (m)
n
p!(m−|p|)!
((cid:18) (cid:19) )
is an orthonormal basis of A2 (Pn(C)).
m
The following result provides the classical description of the weighted Bergman
projectionsofL (Pn(C),Hm)ontoA2 (Pn(C)). Inparticular,theseprojectionsare
2 m
precisely the orthogonal maps Π from Section 3 for Pn(C).
m
Proposition 4.4. Let B : L (Pn(C),Hm) → L (Pn(C),Hm) be the operator
m 2 2
given by the expression
(n+m)! ψ(w)K(z,w)dV(w)
B (ψ)(z)= ,
m πnm! ZCn (1+|w1|2+···+|wn|2)n+m+1
whereK(z,w)=(1+z w ···+z w )m. Then,B satisfiesthefollowingproperties
1 1 n n m
(1) If ψ ∈L (Pn(C),Hm), then B (ψ)∈A2 (Pn(C)).
2 m m
(2) B (ψ)=ψ for every ψ ∈A2 (Pn(C)).
m m
Inparticular, B istheorthogonalprojection L (Pn(C),Hm)→A2 (Pn(C)). Also,
m 2 m
K(z,w) is the Bergman kernel for L (Pn(C),Hm).
2
5. Toeplitz operators with separately radial symbols
We introduce a decompositionfor the projectivespacePn(C) which is similarin
spirittothequasi-ellipticdecompositionusedforthen-dimensionalunitballin[8].
Consider the polar coordinates z = t r where t ∈ T and r ∈ R , for every
j j j j j +
j =1,...,n. This yields, for all points z ∈Cn, an identification
z =(z ,...,z )=(t r ,...,t r )=(t,r),
1 n 1 1 n n
where t = (t ,...,t ) ∈ Tn, r = (r ,...,r ) ∈ Rn. In particular, we have Cn =
1 n 1 n +
Tn×Rn, with the corresponding volume form
+
n n
dt
j
dV(z)= r dr .
j j
it
j
j=1 j=1
Y Y
Hence, for the measure ν on Cn introduced in Section 4, we obtain the decompo-
m
sition
L (Cn,ν )=L (Tn)⊗L (Rn,µ ),
2 m 2 2 + m
where
n
dt
L (Tn)= L T, j
2 2
2πit
j=1 (cid:18) j(cid:19)
O
8 R.QUIROGA-BARRANCOANDA.SANCHEZ-NUNGARAY
and the measure dµ of L (Rn,µ ) is given by
m 2 + m
n
(n+m)!
dµ = (1+r2+···+r2)−n−m−1 r dr .
m m! 1 n j j
j=1
Y
We note that the Bergman space A2 (Pn(C)) is given, in the local coordinates
m
from ϕ , as the (closed) subspace of L (Cn,ν ) which consists of all functions
0 2 m
satisfying the equations
∂ϕ 1 ∂ ∂
= +i ϕ=0, j =1,...,n,
∂z 2 ∂x ∂y
j (cid:18) j j(cid:19)
or, in polar coordinates,
∂ϕ t ∂ t ∂
j j
= − ϕ=0, j =1,...,n.
∂z 2 ∂r r ∂t
j (cid:18) j j j(cid:19)
Now consider the discrete Fourier transform F:L (T)→l =l (Z) defined by
2 2 2
dt
F:f 7→ c = f(f)t−j ,
j
2πit
(cid:26) ZS1 (cid:27)j∈Z
so that, in particular, the operator F is unitary with inverse given by
F−1 =F∗ :{cj}j∈Z 7→f = cjtj
j∈Z
X
Now,letus considerthe operatoru:l ⊗L ((0,1),rdr)→l ⊗L ((0,1),rdr) given
2 2 2 2
by the composition
t ∂ t ∂
u=(F⊗I) − (F−1⊗I).
2 ∂r r∂t
(cid:18) (cid:19)
Then, it is easy to check (see Subsection 4.1 of [12]) that u acts by
1 ∂ j
{cj(r)}j∈Z 7→ − cj(r)
2 ∂r r
(cid:26) (cid:18) (cid:19) (cid:27)j∈Z
Introduce the unitary operator
U =F ⊗I :L (Tn)⊗L (Rn,ν )→l (Zn)⊗L (Rn,ν )
(n) 2 2 + m 2 2 + m
whereF =F⊗···⊗F(ntimes). Then,theimageA2 =U(A2 (Pn(C)))underU
(n) m m
of the Bergman space is the closed subspace of l (Zn)⊗L (Rn,ν ) which consist
2 2 + m
of all sequences {cp(r)}p∈Zn, r =(r1,...,rn)∈Rn+, satisfying the equations
1 ∂ p
j
− c (r)=0,
p
2 ∂r r
(cid:18) j j(cid:19)
for all p ∈ Z and j = 1,...,n. The general solution of this system of equations
j
has the form
c (r)=αmc rp,
p p p
for all p∈Zn, where c ∈C, rp =rp1···rpn and α =α is given by
p 1 n p (|p1|,...,|pn|)
−1
(n+m)! tp1···tpnt dt ···t d 2
αm = 1 n 1 1 n n
p m! ZRn+ (1+t1+···+tn)n+m+1!
1
m! 2
=
p!(m−|p|)!
(cid:18) (cid:19)
COMMUTATIVE TOEPLITZ OPERATORS ON Pn(C) 9
We have that every function c (r) = αmc rp has to be in L (Rn,ν ), and this
p p p 2 + m
integrability condition implies c = 0 for every p ∈ Zn \J (m). Hence, A2 ⊂
p n m
l (Zn)⊗L (Rn,ν ) coincides with the space of all sequences that satisfy
2 2 + m
1
m! 2 c rp p∈J (m)
(5.1) cp(r)= p!(m−|p|)! p n ,
 (cid:16) 0 (cid:17) p∈Zn\Jn(m)
and for all such sequences we have
k{cp(r)}p∈Jnkl2(Zn)⊗L2(Rn+,νm) =k{cp}p∈Jnkl2(Zn).
In particular, we have a natural isometric identification A2 = l (J (m)). These
m 2 n
constructions allow us to consider the isometric embedding
R :l (J (m))→l (Zn)⊗L (Rn,ν )
0 2 n 2 2 + m
defined by
1
m! 2 c rp p∈J (m)
R0 :{cp}p∈Jn(m) 7→cp(r)= (cid:16)p!(m−|p0|)!(cid:17) p p∈Zn/nJn(m) .
For which the adjoint operator R0∗ :l2(Zn)⊗L2(Rn+,νm)→l2(Jn(m)) is given by
1
m! 2
R0∗ :{fp(r)}p∈Zn 7→((cid:18)p!(m−|p|)!(cid:19) ZRn+rpfp(r)dνm(r))p∈Jn(m).
It is easily seen that
R∗R =I : l (J (m))→l (J (m))
0 0 2 n 2 n
R R∗ =P : l (Zn)⊗L (Rn,ν )→l (J (m))
0 0 1 2 2 + m 2 n
whereP istheortogonalprojectionofl (Zn)⊗L (Rn,ν )ontol (J (M))=A2 .
1 2 2 + m 2 n m
Summarizing the above we have the next result.
Theorem5.1. Theoperator R=R∗U maps L (Pn(C),ν )ontoA2 =l (J (m)),
0 2 m m 2 n
and its restriction
R|A2m(Pn(C)) :A2m(Pn(C))→l2(Jn(m))
is an isometric isomorphism. The adjoint operator
R∗ =U∗R :l (J (m))→A2 (Pn(C))⊂L (Pn(C),ν )
0 2 n m 2 m
is an isometric isomorphism of l (J (m)) = A2 onto the subspace A2 (Pn(C)).
2 n m m
Furthermore
RR∗ =I :l (J (m))→l (J (m))
2 n 2 n
R∗R=B :L (Pn(C),ν )→A2 (Pn(C))
m 2 m m
where B is the Bergman projection of L (Pn(C),ν ) onto A2 (Pn(C))
m 2 m m
10 R.QUIROGA-BARRANCOANDA.SANCHEZ-NUNGARAY
We note that an explicit calculation yields
1
m! 2
R∗ =U∗R :{c } 7→U∗ c rp
0 p p∈Jn(m) ((cid:18)p!(m−|p|)!(cid:19) p )p∈Jn(m)
 1 
m! 2
= c (rt)p
p
p!(m−|p|)!
p∈XJn(m)(cid:18) (cid:19)
1
m! 2
= c zp,
p
p!(m−|p|)!
p∈XJn(m)(cid:18) (cid:19)
which implies the following result.
Corollary 5.2. With the notation of Theorem 5.1, the isometric isomorphism
R∗ :l (J (m))→A2 (Pn(C))⊂L (Pn(C),ν ) is given by
2 n m 2 m
1
m! 2
(5.2) R∗ :{c } 7→ c zp.
p p∈Jn(m) p!(m−|p|)! p
p∈XJn(m)(cid:18) (cid:19)
A similar direct computation yields the following result.
Corollary 5.3. With the notation of Theorem 5.1, the isometric isomorphism
R|A2m(Pn(C)) :A2m(Pn(C))→l2(Jn(m)) is given by
1
m! 2
(5.3) R:ψ 7→ ψ(z)zpdν (z) .
m
((cid:18)p!(m−|p|)!(cid:19) ZCn )p∈Jn(m)
We now introduce a special family of symbols on Cn.
Definition 5.4. We will call a function a(z), z ∈ Cn separately radial if a(z) =
a(r), i.e. a depends only on the radial components of z =(t,r).
The separately radial symbols give rise to a C∗-algebra of Toeplitz operators
that can be turned simultaneously into multiplication operators.
Theorem 5.5. Let a be a bounded measure separately radial function. Then the
ToeplitzoperatorT actingonA2 (Pn(C))isunitaryequivalenttothemultiplication
a m
operator γ I = RT R∗ acting on l (J (m)), where R and R∗ are given by (5.3)
a,m a 2 n
and (5.2) respectively. The sequence γ = {γ (p)} is explicitely given
a,m a,m p∈Jn(m)
by
2nm! a(r ,...,r )r2p1+1···r2pn+1dr ···dr
(5.4) γ (p)= 1 n 1 n 1 n.
a,m p!(m−|p|)!ZRn+ (1+r12+···+rn2)n+m+1
Proof. Using the previous results, the operator T is unitary equivalent to the
a
operator
RT R∗ =RB aB R∗ =R(R∗R)a(R∗R)R∗
a m m
=(RR∗)RaR∗(RR∗)=RaR∗
=R∗UaU∗R
0 0
=R∗(F ⊗I)a(F−1 ⊗I)R
0 (n) (n) 0
=R∗aR .
0 0