Toeplitz operators with quasi-radial quasi-homogeneous symbols and bundles of Lagrangian frames

0.21 MB·English

by  Raul Quiroga-Barranco

TOEPLITZ OPERATORS WITH QUASI-RADIAL
QUASI-HOMOGENEOUS SYMBOLS AND BUNDLES OF
LAGRANGIAN FRAMES
2
1
0
RAULQUIROGA-BARRANCOand ARMANDOSANCHEZ-NUNGARAY
2

n
Abstract. We prove that the quasi-homogenous symbols on the projective
a
J spacePn(C)yieldcommutativealgebrasofToeplitzoperatorsonallweighted
Bergman spaces, thus extending to this compact case known results for the
0 unitballBn. ThesealgebrasareBanachbutnotC∗. Weprovetheexistenceof
1 astronglinkbetweensuchsymbolsandalgebraswiththegeometryofPn(C).
ThelatterisalsoprovedforthecorrespondingsymbolsandalgebrasonBn.
]
A
O
.
h 1. Introduction
t
a
ThestudyofcommutativealgebrasofToeplitzoperatorshasshowntobeavery
m
interesting subject. Some previous results in this topic serve as backgroundto this
[
work. First,itwasshowntheexistenceofsymbolsdefininginterestingcommutative

1 C∗-algebras of Toeplitz operators on bounded symmetric domains (see [2], [7], [8]
v and [9]). Also, it was exhibited in [11] the existence of Banach algebras,which are
1 not C∗, of commutative Toeplitz operators on the unit ball Bn. And, in [6] we
6
constructed commutative C∗-algebras of Toeplitz operators on complex projective
1
2 spaces.
. A remarkable fact is that the currently known commutative C∗-algebras of
1
ToeplitzoperatorsonBnarenaturallyassociatedtoAbeliansubgroupsofthegroup
0
2 of biholomorphisms of Bn. In fact, their systematic description is best understood
1 withthe useofsuchgroupsofbiholomorphisms(see[8]and[9]). Furthermore,this
: provided the guiding light to construct commutative C∗-algebras of Toeplitz oper-
v
i ators in the complex projective space Pn(C): the currently known C∗-algebras for
X Pn(C) are naturally associated and described from the maximal tori of the group
r of isometric biholomorphisms of Pn(C) (see [6]).
a
TheknownBanach(notC∗)algebrasofcommutativeToeplitzoperatorsfirstin-
troducedin[11]forBn aregivenbythesocalledquasi-homogeneoussymbols. Such
symbols are defined in terms of radial and spherical coordinates of components in
Bn (see Section 3 below). However, their introduction lacked the stronger connec-
tionwiththe geometryofthe domainobservedforthecommutativeC∗-algebrasof
Toeplitz operators on Bn.
Given these lines of research,there are two natural problems to consider. First,
todeterminewhetherornotthereareanyinterestingBanachalgebras,thatarenot
1991 Mathematics Subject Classification. Primary47B35;Secondary 32A36,32M15,53C12.
Keywordsandphrases. Toeplitzoperators,commutativeBanachalgebras,Lagrangianframes,
complexprojectivespace.
Thefirstnamedauthor waspartiallysupportedbySNI-MexicoandbyaConacytgrant. The
secondnamedauthorwaspartiallysupportedbyaConacytpostdoctoral fellowship.
1
2 RAULQUIROGA-BARRANCOand ARMANDOSANCHEZ-NUNGARAY
C∗, ofcommutative Toeplitz operatorsonPn(C). And second,to find any possible
special links between the geometry of Bn and the Banach algebras, introduced in
[11], of commutative Toeplitz operators. The goal of this work is to solve these
problems.
On one hand, we define quasi-homogeneous symbols in the complex projective
spacePn(C),andshowthattheseprovideBanachalgebrasofcommutativeToeplitz
operatorson every weightedBergmanspace ofPn(C). The results that exhibit the
commuting Toeplitz operatorsare obtainedin Section 4, where Theorem 4.5 is the
mainresult. Onthe otherhand,wealsoprovethatthe Banachalgebrasdefinedby
quasi-homogeneoussymbolsturnouttohaveastrongconnectionwiththegeometry
of the supporting space, for both Bn and Pn(C); the main results in this case are
presentedinSection5. Inparticular,weprovethatthequasi-homogeneoussymbols,
for both Bn and Pn(C), can be associated to an Abelian group of holomorphic
isometriesofthe correspondingspace(see Theorem5.1). Suchgroupis a subgroup
of a maximal torus in the corresponding isometry group.
We further prove that the groups associated to quasi-homogeneous symbols af-
ford pairs of foliations with distinguished Lagrangian and Riemannian geometry
known as Lagrangianframes (see Section 5 below and [7], [8] and [9]). This recov-
ers the behavior observed for the C∗-algebras of commutative Toeplitz operators
constructed in [8], [9] and [6], for which such Lagrangian frames appear as well.
Nevertheless, it is important to note a key difference between the C∗ case and
the Banach case. For the C∗-algebras of Toeplitz operators on Bn and Pn(C), as
constructed in [8], [9] and [6], the Lagrangian frames are obtained for the whole
space, i.e. they come from Lagrangian submanifolds of the whole space, either Bn
or Pn(C). But for the Banach algebras defined by quasi-homogeneous symbols the
Lagrangianframes are obtained on the fibers of a suitable fibration over either Bn
or Pn(C). The existence of such fibration and the Lagrangian frames on its fibers
are obtained in Theorem 5.4. It is also proved in Theorem 5.6 that a full maximal
torus of isometries continues to play an importante role, since the complement to
the group that defines the fiberwise Lagrangian frames acts by automorphisms of
such frames.
TheauthorswanttoacknowledgetheinputofPedroLuisdelAngel,fromCimat,
provided through several conversations with him.
2. Preliminaries on the Geometry and Analysis of Pn(C)
In this section we establish our notation concerning the n-dimensional complex
projective space Pn(C). For further details we refer to the bibliography.
There is a natural realization of Cn as an open conull dense subset given by
Cn →Pn(C), z 7→[1,z],
which defines a biholomorphism onto its image. Note that the points of Pn(C) are
denoted by [w], the complex line through w ∈ Cn+1 \{0}. We will refer to this
embedding as the canonical embedding of Cn into Pn(C).
Let us denote with ω the canonical K¨ahler structure on Pn(C) that defines the
Fubini-Study metric, whose volume is then given by Ω = (ω/2π)n. These induce
QUASI-RADIAL QUASI-HOMOGENEOUS SYMBOLS AND LAGRANGIAN FRAMES 3
on Cn the following K¨ahler form and volume element, respectively
(1+|z|2) n dz ∧dz − n z z dz ∧dz
ω =i k=1 k k k,l=1 k l k l,
0 (1+|z|2)2
P P
1 dV(z)
Ω = ,
0 πn(1+|z |2+···+|z |2)n+1
1 n
where dV(z) denotes the Lebesgue measure on Cn.
LetH denotethedualbundleofthetautologicallinebundleofPn(C). Werecall
that H carries a canonical Hermitian metric h obtained from the (flat) Hermitian
metric of Cn+1. Then, it is also wellknownthat the curvature Θ of (H,h) satisfies
the identity
Θ=−iω,
which amounts to say that (H,h) is a quantum line bundle over Pn(C).
We will denote with Γ(Pn(C),Hm) and Γ (Pn(C),Hm) the smooth and holo-
hol
morphic sections of Hm, respectively. Note that Hm denotes the m-th tensorial
power of H. Clearly, both of these spaces lie inside L (Pn(C),Hm).
2
Foreverym∈Z andwithrespecttothecanonicalembeddingofCnintoPn(C),
+
we define the weigthed measure on Pn(C) with weight m by
(n+m)! Ω(z)
dν (z)=
m m! (1+|z |2+···+|z |2)m
1 n
(n+m)! dV(z)
= .
πnm! (1+|z |2+···+|z |2)n+m+1
1 n
Asimplycomputationshowsthatdν isaprobabilitymeasureforallm∈Z . For
m +
simplicity, we will use the same symbol dν to denote the weighted measures for
m
bothPn(C)andCn. Itisalsostraightforwardtoshowthatthecanonicalembedding
of Cn into Pn(C) induces a canonical isometry
Φ:L (Cn,ν )→L (Pn(C),Hm)
2 m 2
with respect to which we will identify these spaces in the rest of this work. Also,
we will denote with h·,·i the inner product of this Hilbert spaces.
m
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
These Bergmanspaces are finite-dimensional and are described by the following
well known result.
Proposition 2.1. For every m∈Z , the Bergman space A2 (Pn(C)) satisfies the
+ m
following properties.
(i) A2 (Pn(C)) can be identified with the space P(m)(Cn+1) of homogeneous
m
polynomials of degree m over Cn+1.
(ii) For Φ : L (Cn,ν ) → L (Pn(C),Hm) the canonical isometry described
2 m 2
above, we have Φ(A2 (Pn(C)))=P (Cn), the space of polynomials on Cn
m m
of degree at most m.
In what follows, we will use this realization of the Bergman spaces without
further notice.
4 RAULQUIROGA-BARRANCOand ARMANDOSANCHEZ-NUNGARAY
Recall the following notation for multi-indices α,β ∈Nn and z ∈Cn
|α|=α +···+α ,
1 n
α!=α !...α !, α∈Nn,
1 n
zα =zα1...zαn,
δ =δ ...δ .
α,β α1,β1 αn,βn
The Bergman space A2 (Pn(C)) has a basis consisting of the polynomials zα =
m
zα1...zαn where α∈Nn and |α|≤m. Hence we will consider the set
1 n
J (m)={α∈Nn :|α|≤m}.
n
More precisely, an easy computation shows that the set
1
m! 2
(2.1) zα :α∈J (m)
n
α!(m−|α|)!
((cid:18) (cid:19) )
is an orthonormal basis of A2 (Pn(C)).
m
For ψ ∈ L (Pn(C),Hm), and considering the identification Φ, we define the
2
Bergman projection by
(n+m)! ψ(w)K(z,w)dV(w)
B (ψ)(z)=
m πnm! ZCn (1+w1w1+···+wnwn)n+m+1
where
K(z,w)=(1+z w ···+z w )m.
1 1 n n
This operator satisfies the well known reproducing property.
Proposition 2.2. If ψ ∈ L (Pn(C),Hm), then B (ψ) belongs to the weighted
2 m
Bergman space A2 (Pn(C)). Also, B (ψ)=ψ if ψ ∈A2 (Pn(C)).
m m m
Using this, we define the Toeplitz operator T on A2 (Pn(C)) with bounded
a m
symbol a∈L (Pn(C)) by
∞
T (ϕ)=B (aϕ),
a m
for every ϕ∈A2 (Pn(C)).
m
LetusdenotebySn theunitsphereinCn. Inthiswork,wewillusethefollowing
identity on the n-sphere Sn
(2.2) ξαξβdS(ξ)=δ = 2πnα!
α,β
ZSn (n−1+|α|)!
where dS be the corresponding surface measure on Sn (see [12]).
3. Toeplitz operators with quasi-homogeneous symbols
Quasi-homogeneous symbols were introduced in [11] on the unit ball Bn in Cn.
However,we willalsoconsiderquasi-homogeneoussymbolsforthe projectivespace
Pn(C) as functions defined on Cn. The basic definitions of the quasi-radial and
quasi-homogeneous symbols are very similar for both Bn and Cn, so we will recall
them here together. This will also be useful latter on for our common geomet-
ric treatment of quasi-homogeneous symbols on Bn and Pn(C). To simplify our
notation, from now on Un will denote one of either Bn or Cn, corresponding to
whether we are dealing with Bn or Pn(C), respectively. For the latter case, we are
considering Cn canonically embedded into Pn(C) as described in Section 2.
QUASI-RADIAL QUASI-HOMOGENEOUS SYMBOLS AND LAGRANGIAN FRAMES 5
Let k = (k ,...,k) ∈ Zl be a multi-index so that |k| = n. We will call such
1 l +
multi-index k a partition of n. For the sake of definiteness, we will assume that
k ≤ ··· ≤ k . This partition provides a decomposition of the coordinates z ∈ Un
1 l
as z =(z ,...,z ) where
(1) (l)
z =(z ,...,z ),
(j) k1+···+kj−1+1 k1+···+kj
foreveryj =1,...,l,andtheempty sumis0byconvention. Forz ∈Un,wedefine
r =|z | and
j (j)
z
(j)
ξ =
(j) r
j
if z 6= 0. Besides the quasi-radii (r ,...,r ), this provides a set of coordinates
(j) 1 l
(ξ(1),...,ξ(l))∈Sk1 ×···×Skl.
For a partitionk of n, a k-quasi-radialsymbol a onUn is a function a:Un →C
which depends only on the coordinates (r ,...,r) introduced above. In other
1 l
words, a(z) = a(r ,...,r ). The set of all k-quasi-radial functions is denoted by
1 l
R . The family of sets R , while k varies over the partitions of n, is a partially
k k
ordered by inclusion. The minimal element among these sets is the set R of
(n)
radial functions and the maximal element is the set R of separately radial
(1,...,1)
functions.
Also,fork apartitionofn,ak-quasi-homogeneoussymbolϕonUn isafunction
ϕ:Un →C of the form
ϕ(z)=a(r ,...,r )ξpξq
1 l
where a is a k-quasi-radialsymbol and p,q ∈Nn satisfy
p·q =p q +···+p q =0.
1 1 n n
Recall that every k the family R is contained in R . Hence, the Toeplitz
k (1,...,1)
operators T for symbols a ∈ R can be simultaneously diagonalized with respect
a k
to the monomial basis in the corresponding Bergman space (see [8] and [6]). Fur-
thermore, for the case Un = Cn the following result provides the multiplication
operator so obtained. Note that this result is similar to Lemma 3.1 from [11].
Lemma 3.1. Consider the case Un = Cn, and let k be a partition of n. For any
k-quasi-radial bounded measurable symbol a(r ,...,r ), we have
1 l
T zα =γ (α)zα,
a a,k,m
for every α∈J (m), where
n
γ (α)=γ (|α |,...,|α |)
a,k,m a,k,m (1) (l)
2l(n+m)!
(3.1) =
l
(m−|α|)! (k −1+|α |)
j=1 j (j)
l
Q
× a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
Proof. Let α∈Zn with |α|≤m. Then, we have
+
hT zα,zαi =hazα,zαi
a m m
(n+m)! a(r ,...,r )zαzαdV(z)
1 l
= .
πnm! ZCn (1+|z1|2+···+|zn|2)n+m+1
6 RAULQUIROGA-BARRANCOand ARMANDOSANCHEZ-NUNGARAY
Making the substitution z(j) = rjξ(j), where rj ∈ [0,∞) and ξ(j) ∈ Skj, for j =
1,...,l, we obtain
l
hazα,zαi = (n+m)! a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
m πnm! Rn 1 l j j
Z + jY=1
l
× ξα(j)ξα(j)dS(ξ )
(j) (j) (j)
j=1ZSkj
Y
2mα!(n+m)!
=
m! l (k −1+|α |)
j=1 j (j)
l
Q
× a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
and the result follows from (2.2) (cid:3)
We now find the action of the Toeplitz operators with quasi-homogeneous sym-
bols on the canonical monomial basis. Note again that the following result corre-
sponds to Lemma 3.3 from [11].
Lemma 3.2. Consider the case Un = Cn, and let aξpξq = a(r ,...,r)ξpξq be
1 l
a k-quasi-homogeneous symbol for k a partition of n. Then, the Toeplitz operator
Taξpξq acts on monomials zα with α∈Zn+ and |α|≤m as follows
γ˜ (α)zα+p−q for α+p−q ∈J (m)
Taξpξqzα =(0a,k,p,q,m for α+p−q 6∈Jnn(m)
where
2l(α+p)!(n+m)!
γ˜ (α)=
a,k,p,q,m l
(α+p−q)!(m−|α+p−q|)! (k −1+|α +p |)
j=1 j (j) (j)
l
Q
(3.2) × a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)+p(j)−q(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
Proof. Let α,β ∈Zn satisfy |α|,|β|≤m. Then, we have
+
hTaξpξqzα,zβim =haξpξqzα,zβim
(n+m)! a(r ,...,r )ξpξqzαzβdV(z)
1 l
= .
πnm! ZCn (1+|z1|2+···+|zn|2)n+m+1
QUASI-RADIAL QUASI-HOMOGENEOUS SYMBOLS AND LAGRANGIAN FRAMES 7
Applying the change of the variables z = r ξ , where r ∈ [0,∞) and ξ ∈
(j) j (j) j (j)
Skj, for j =1,...,l, this yields
haξpξqzα,zβi = (n+m)! a(r ,...,r )(1+r2)−(n+m+1)
m πnm! Rn 1 l
Z +
l
× r|α(j)|+|β(j)|+2kj−1dr
j j
j=1
Y
l
× ξα(j)+p(j)ξβ(j)+q(j)dS(ξ )
(j) (j) (j)
j=1ZSkj
Y
2l(α+p)!(n+m)!
(3.3) =δ
α+p,β+q l
m! (k −1+|α +p |)
j=1 j (j) (j)
× a(r ,...Q,r )(1+r2)−(n+m+1)
1 l
Rn
Z +
l
× r2|α(j)+p(j)−q(j)|+2kj−1dr
j j
j=1
Y
Observethat this expressionis nonzeroif andonly if β =α+p−q, whicha priori
belongs to J (m). We conclude the result from the orthonormality of the basis
n
defined in (2.1). (cid:3)
4. Commutativity results for quasi-homogeneous symbols on Pn(C)
In the rest of this section we will restrict ourselves to the case Un = Cn. The
resultsinthissectionshowthatthecommutingidentitiesprovedin[11]fortheunit
ball Bn have corresponding ones for the complex projective space Pn(C).
Theorem 4.1. Let k ∈ Zl be a partition of n and p,q ∈ Nn a pair of orthogonal
+
multi-indices. If a and a are non identically zero k-quasi-radial bounded symbols,
1 2
then the Toeplitz operators Ta1 and Ta2ξpξq commute on each weighted Bergman
space A2 (Pn(C)) if and only if |p |=|q | for each j =1,...,l.
m (j) (j)
Proof. Let α ∈ J (m) be given. First note that if α+p−q 6∈ J (m), then the
n n
Lemmas3.1and3.2implythatbothTa1Ta2ξpξqzα andTa2ξpξqTa1zα vanish. Hence,
we can assume that α+p−q ∈ J (m). Applying again Lemmas 3.1 and 3.2 we
n
8 RAULQUIROGA-BARRANCOand ARMANDOSANCHEZ-NUNGARAY
obtain
2m(α+p)!(n+m)!
Ta1Ta2ξpξqzα = (α+p−q)!(m−|α+p−q|)! l (k −1+|α +p |)
j=1 j (j) (j)
l
Q
× a (r ,...,r )(1+r2)−(n+m+1) r2|α(j)+p(j)−q(j)|+2kj−1dr
2 1 l j j
Rn
Z + jY=1
2l(n+m)!
×
l
(m−|α+p−q|)! (k −1+|α +p −q |)
j=1 j (j) (j) (j)
l
Q
× a (r ,...,r )(1+r2)−(n+m+1) r2|α(j)+p(j)−q(j)|+2kj−1dr
1 1 l j j
Rn
Z + jY=1
×zα+p−q.
And similarly, we have
2l(n+m)!
Ta2ξpξqTa1zα = (m−|α|)! l (k −1+|α |)
j=1 j (j)
l
Q
× a (r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 1 l j j
Rn
Z + jY=1
2l(α+p)!(n+m)!
×
l
(α+p−q)!(m−|α+p−q|)! (k −1+|α +p |)
j=1 j (j) (j)
l
Q
× a (r ,...,r )(1+r2)−(n+m+1) r2|α(j)+p(j)−q(j)|+2kj−1dr
2 1 l j j
Rn
Z + jY=1
×zα+p−q
F|qrom| wwhheircehjw=e 1co,.n.c.l,uld.e that Ta1Ta2ξpξqzα = Ta2ξpξqTa1zα if and only if |p(j)| (cid:3)=
(j)
If we assume that |p |=|q | for all j =1,...,l, then the equations (3.1) and
(j) (j)
(3.2) combine together to yield the following identity.
2l(α+p)!(n+m)!
γ˜ (α)=
a,k,p,q,m l
(α+p−q)!(m−|α+p−q|)! (k −1+|α +p |)
j=1 j (j) (j)
l
Q
× a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
l
(α+p)! (k −1+|α |)
= j=1 j (j) γ (α)
(α+p−q)! Qlj=1(kj −1+|α(j)+p(j)|) a,k,m
l (Qα +p )!(k −1+|α |)
(j) (j) j (j)
(4.1) = γ (α)
a,k,m
(α +p −q )!(k −1+|α +p |)
j=1(cid:18) (j) (j) (j) j (j) (j) (cid:19)
Y
Asaconsequenceofthepreviouscomputations,wealsoobtainthefollowingvery
special property of Toeplitz operators with quasi-homogeneous symbols.
QUASI-RADIAL QUASI-HOMOGENEOUS SYMBOLS AND LAGRANGIAN FRAMES 9
Corollary 4.2. Let k ∈ Zl be a partition of n and p,q ∈ Nn a pair of orthogo-
+
nal multi-indices such that |p | = |q | for all j = 1,...,l. Then for each non
(j) (j)
identically zero k-quasi-radial function a, we have TaTξpξq =TξpξqTa =Taξpξq.
Consider k = (k ,...,k ) and a pair of multi-indices p,q such that p ⊥ q and
1 l
|p |=|q | for j =1,...l. we define
(j) (j)
p˜ =(0,...,0,p ,0,...,0), q˜ =(0,...,0,q ,0,...,0)
(j) (j) (j) (j)
where the only possibly non zero part is placed in the j-th position. In particular,
we have p = p˜ +...+p˜ and q = q˜ +...+q˜ . Now let T = T for
(1) (l) (1) (l) j ξp˜(j)ξp˜(j)
every j = 1,...,l. As a consequence of the previous computations we obtain the
following result.
Corollary 4.3. The Toeplitz operators T = T , for j = 1,...l mutually
j ξp˜(j)ξp˜(j)
commute and
m
Tj =Tξpξq
j=1
Y
Next,weobtainanecessaryandsufficientconditionfortwogivenquasi-homogeneous
symbols to determine Toeplitz operators that commute with each other.
Theorem 4.4. Let k ∈ Zl be a partition of n and p,q ∈ Nn a pair of orthog-
+
onal multi-indices. Consider a(r ,...,r )ξpξq and b(r ,...,r )ξuξv two k-quasi-
1 l 1 l
homogeneous symbols on Pn(C) where a(r ,...,r ) and b(r ,...,r ) are k-quasi-
1 l 1 l
radial measurable and bounded symbols. Assume that |p | = |q | and |u | =
(j) (j) (j)
|v(j)| for all j = 1,...,l. Then, the Toeplitz operators Taξpξq and Tbξuξv commute
on each weighted Bergman space A2 (Pn(C)) if and only if for each j =1,...,l one
m
of the following conditions hold
(i) p =q =0
j j
(ii) u =v =0
j j
(iii) p =u =0
j j
(iv) q =v =0
j j
Proof. First,weobservethatforourhypotheses,thequantitiesTbξuξvTaξpξqzα and
TaξpξqTbξuξvzα are always simultaneously zero or non zero. Hence, we compute
TaξpξqTbξuξvzα and TbξuξvTaξpξqzα, for α ∈ Jn(m), assuming that both are non
zero.
By (4.1), we have the following expression
2l(α+p)!(n+m)!
TbξuξvTaξpξqzα = (α+p−q)!(m−|α|)! l (k −1+|α +p |)
j=1 j (j) (j)
l
Q
× a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
2l(α+p−q+u)!(n+m)!
×
l
(α+p−q+u−v)!(m−|α|)! (k −1+|α +u |)
j=1 j (j) (j)
l
Q
× b(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
×zα+p−q+u−v.
10 RAULQUIROGA-BARRANCOand ARMANDOSANCHEZ-NUNGARAY
Similarly, we also have
2l(α+u)!(n+m)!
TaξpξqTbξuξvzα = l
(α+u−v)!(m−|α|)! (k −1+|α +u |)
j=1 j (j) (j)
l
Q
× b(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
2l(α+u−v+p)!(n+m)!
×
(α+p−q+u−v)!(m−|α|)! l (k −1+|α +p |)
j=1 j (j) (j)
l
Q
× a(r ,...,r )(1+r2)−(n+m+1) r2|α(j)|+2kj−1dr
1 l j j
Rn
Z + jY=1
×zα+p−q+u−v
Therefore, we conclude that TbξuξvTaξpξqzα =TaξpξqTbξuξvzα if and only if
(α +u −v +p )!(α +u )! (α +p −q +u )!(α +p )
j j j j j j j j j j j j
=
(α +u −v )! (α +p −q )!
j j j j j j
for j = 1,...,l. And this equality holds if and only if for each j = 1,...,l one of
the following conditions is fulfilled
(i) p =q =0,
j j
(ii) u =v =0,
j j
(iii) p =u =0,
j j
(iv) q =v =0.
j j
(cid:3)
Finally,wepresentoneofourmainresults: theconstructionofaBanachalgebra
of Toeplitz operators on Pn(C). Our construction is parallel to the one presented
in [11].
As above, continue to consider k ∈ Zl a partition of n, and now let h ∈ Zl be
+ +
such that 1 ≤ h ≤ k −1, for all j = 1,...,l. Let ϕ be a k-quasi-homogeneous
j j
symbol on Cn of the form
ϕ(z)=a(r ,...,r )ξpξq,
1 l
where p,q ∈ Nn satisfy p·q = 0. Consider the decompositions of p and q given as
follows
p=(p ,...,p ), q =(q ,...,q )
(1) (l) (1) (l)
p =(p ,...,p ), q =(q ,...,q ),
(j) j,1 j,kj (j) j,1 j,kj
for j =1,...,l. With respect to this decomposition, we will now assume that
p =0, q =0,
j,r j,s
for r > h and s ≤ h , where j = 1,...,l. As before, we also assume that |p | =
j j (j)
|q | for all j, which now becomes
(j)
p +···+p =q +···+q .
j,1 j,hj j,hj+1 j,kj
Let us denote with R (h) the space of all symbols obtained through this construc-
k
tion. As a consequence of Theorem 4.4.