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# Radial Toeplitz Operators on the Unit Ball and Slowly Oscillating Sequences

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## Abstract

In the paper we deal with Toeplitz operators acting on the Bergman space A2(Bn) of square integrable analytic functions on the unit ball Bn in Cn. A bounded linear operator acting on the space A2(Bn) is called radial if it commutes with unitary changes of variables. Zhou, Chen, and Dong [9] showed that every radial operator S is diagonal with respect to the standard orthonormal monomial basis (eα)α⊂Nn . Extending their result we prove that the corresponding eigenvalues depend only on the length of multiindex α, i.e. there exists a bounded sequence (λk)∞k=0 of complex numbers such that S eα = λ|α|eα. Toeplitz operator is known to be radial if and only if its generating symbol g is a radial function, i.e., there exists a function a, defined on [0,1], such that g(z) = a(|z|) for almost all z ∈ Bn. In this case Tgeα = γn,a(|α|)eα, where the eigenvalue sequence γn,a(k) ∞k =0 is given by γn,a(k) = 2(k+n) Z 1 0 a(r) r2k+2n-1 dr = (k+n) ∫ 10a(√r) rk+n-1 dr. Denote by Γn the set {γn,a : a 2 L∞([0,1])}. By a result of Súarez [8], the C-algebra generated by Γ1 coincides with the closure of Γ1 in ̀∞ and is equal to the closure of d1 in ̀∞, where d1 consists of all bounded sequences x = (xk)∞k=0 such that sup k≥0 (k+1) |xk+1 - xk| < +∞. We show that the C*-algebra generated by Γn does not actually depend on n, and coincides with the set of all bounded sequences (xk)∞k =0 that are slowly oscillating in the following sense: |xj - xk| tends to 0 uniformly as j+1/ k+1 → 1 or, in other words, the function x : {0,1,2, . . .} → C is uniformly continuous with respect to the distance ρ( j, k) = |ln( j + 1) - ln(k + 1)|. At the same time we give an example of a complexvalued function a L1([0,1], r dr) such that its eigenvalue sequence γn,a is bounded but is not slowly oscillating in the indicated sense. This, in particular, implies that a bounded Toeplitz operator having unbounded defining symbol does not necessarily belong to the C-algebra generated by Toeplitz operators with bounded defining symbols.
Radial Toeplitz operators on the unit ball
and slowly oscillating sequences
Sergei M. Grudsky
Departamento de Matem´aticas, CINVESTAV del I.P.N.,
Apartado Postal 14-740, 07000 M´exico, D.F., M´exico
Egor A. Maximenko
Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional,
C.P. 07730, M´exico, D.F., M´exico
Nikolai L. Vasilevski
Departamento de Matem´aticas, CINVESTAV del I.P.N.,
Apartado Postal 14-740, 07000 M´exico, D.F., M´exico
This is a draft version of a published research article:
Grudsky, Sergei M.; Maximenko, Egor A.; Vasilevski, Nikolai L.
Radial Toeplitz operators on the unit ball and slowly oscillating sequences.
Commun. Math. Anal. 14 (2013), no. 2, 77–94.
http://projecteuclid.org/euclid.cma/1356039033.
We are grateful to Roberto Mois´es Barrera Castel´an who found some errors in the
published article, namely, in the proof of Proposition 4.3. In this version we ﬁxed
the errors.
Abstract
In the paper we deal with Toeplitz operators acting on the Bergman space
A2(Bn) of square integrable analytic functions on the unit ball Bnin Cn. A
bounded linear operator acting on the space A2(Bn) is called radial if it com-
mutes with unitary changes of variables. Zhou, Chen, and Dong [9] showed that
every radial operator Sis diagonal with respect to the standard orthonormal
monomial basis (eα)αNn. Extending their result we prove that the correspond-
ing eigenvalues depend only on the length of multi-index α, i.e. there exists a
bounded sequence (λk)
k=0 of complex numbers such that Seα=λ|α|eα.
Toeplitz operator is known to be radial if and only if its generating symbol
gis a radial function, i.e., there exists a function a, deﬁned on [0,1], such that
g(z) = a(|z|) for almost all zBn. In this case Tgeα=γn,a (|α|)eα, where the
eigenvalue sequence γn,a(k)
k=0 is given by
γn,a(k) = 2(k+n)Z1
0
a(r)r2k+2n1dr = (k+n)Z1
0
a(r)rk+n1dr.
Denote by Γnthe set {γn,a :aL([0,1])}. By a result of Su´arez [8], the C-
algebra generated by Γ1coincides with the closure of Γ1in and is equal to
the closure of d1in , where d1consists of all bounded sequences x= (xk)
k=0
such that
sup
k0(k+ 1) |xk+1 xk|<+.
We show that the C-algebra generated by Γndoes not actually depend on n,
and coincides with the set of all bounded sequences (xk)
k=0 that are slowly
oscillating in the following sense: |xjxk|tends to 0 uniformly as j+1
k+1 1
or, in other words, the function x:{0,1,2, . . .} → Cis uniformly continuous
with respect to the distance ρ(j, k) = |ln(j+ 1) ln(k+ 1)|. At the same
time we give an example of a complex-valued function aL1([0,1], r dr ) such
that its eigenvalue sequence γn,a is bounded but is not slowly oscillating in the
indicated sense. This, in particular, implies that a bounded Toeplitz operator
having unbounded deﬁning symbol does not necessarily belong to the C-algebra
generated by Toeplitz operators with bounded deﬁning symbols.
AMS (MOS) subject classiﬁcation: Primary 47B35; Secondary 32A36, 44A60.
Keywords. Radial Toeplitz operator, Bergman space, unit ball, slowly oscillating
sequence.
2
1 Introduction and main results
Bergman space on the unit ball
We shall use some notation and well-known facts from Rudin [3] and Zhu [10].
Denote by ,·i the usual inner product in Cn:hz, wi=Pn
j=1 zjwj. Let |·|be
the Euclidean norm in Cninduced by this inner product, and let Bnbe the unit
ball in Cn. Denote by dv the Lebesgue measure on Cn=R2nnormalized so that
v(Bn) = 1, and denote by the surface measure on the unit sphere S2n1=Bn
normalized so that σ(S2n1) = 1. Let N={0,1,2, . . .}. Given a multi-index αNn
and a vector zCn, we understand the symbols |α|,α! and zαin the usual sense:
|α|=
n
X
j=1
αj, α! =
n
Y
j=1
αj!, zα=
n
Y
j=1
zαj
j.
Consider the Bergman space A2=A2(Bn, v) of all square integrable analytic func-
tions on Bn. Denote by (eα)αNnthe standard orthonormal monomial basis in A2:
eα(z) = r(n+|α|!)
n!α!zα.
The reproducing kernel Kzof the space A2at a point zBnsatisﬁes hf, Kzi=f(z)
for all f∈ A2, and is given by the following formula:
Kz(w) = X
αNn
eα(z)eα=1
(1 − hw, zi)n+1 .
The Berezin transform of a bounded linear operator S:A2→ A2is a function
BnCdeﬁned by
(B(S))(z) = hSKz, Kzi
kKzk2= (1 − |z|2)n+1 hSKz, Kzi.
It is well known that the Berezin transform Bis injective: if B(S) is identically zero,
then S= 0. A proof of this fact for the one-dimensional case is given by Stroethoﬀ
[7].
Given a function gL1(Bn), the Toeplitz operator Tgis deﬁned on a dense
subset of A2by
(Tg(f))(z) := ZBn
Kzgf dv.
If gL(Bn), then Tgis bounded and kTgk≤kgk.
Radial operators on the unit ball
Following Zhou, Chen and Dong [9] we recall the concept of a radial function on Bn
and of a radial operator acting on A2. The radialization of a measurable function
f:BnCis given by
f(Uz)dH (U),
3
where dH is the normalized Haar measure on the compact group Unconsisting of
the unitary matrices of order n.
where. For a continuous function fthis means that f(z) = f(|z|) for all zBn.
Given a unitary matrix U∈ Un, denote by ΨUthe corresponding “change of a
variable operator” acting on A2:
Uf)(z) := f(Uz).
Here Uis the conjugated transpose of U. Note that ΨUis a unitary operator on
the space A2, its inverse is ΨU, and the formula ΨU1U2= ΨU1ΨU2holds for all
U1, U2∈ Un.
Given a bounded linear operator S:A2→ A2, its radialization Rad(S) is deﬁned
by
ΨUSΨUdH(U),
where the integration is understood in the weak sense.
A bounded linear operator Sis called radial if SΨU= ΨUSfor all U∈ Unor,
Zhou, Chen, and Dong [9] proved that the Berezin transform “commutes with
the radialization” in the following sense: for every bounded linear operator Sacting
in A2
It follows that Sis radial if and only if B(S) is radial. In the one-dimensional case
(i.e., for n= 1) these facts were proved by Zorboska [11].
For each αNnwe denote by Pαthe orthogonal projection onto the one-
dimensional space generated by eα:
Pα(x) := hx, eαieα.
Given a bounded sequence λ= (λm)
m=0 of complex numbers, denote by Rλthe
following operator (radial operator with eigenvalue sequence λ):
Rλ:= X
αNn
λ|α|Pα,
where the convergence of the series is understood in the strong operator topology.
The Berezin transform of Rλwas computed in [1, 9]:
(B(Rλ))(z) = (1 − |z|2)n+1
X
m=0
2(m+n)!
m! (n1)! λm|z|2m.(1.1)
Theorem 1.1. Let Sbe a bounded linear radial operator in A2. Then there exists
a bounded complex sequence λsuch that S=Rλ.
4
Zhou, Chen, and Dong [9] proved one part of this theorem, namely, that Sis
diagonal with respect to the monomial basis. In Section 2 we prove the remaining
part: the eigenvalues of Sdepend only on the length of the multi-index.
Radial Toeplitz operators on the unit ball
Zhou, Chen, and Dong [9] proved that a Toeplitz operator Tgis radial if and only
if its generating symbol gis radial, i.e., if there exists a function adeﬁned on [0,1]
such that g(z) = a(|z|) for almost all zBn. Then Tgis diagonal with respect
to the orthonormal monomial basis, and the corresponding eigenvalues depend only
on the length of multi-indices. Denote the eigenvalue sequence of such operator by
γn,a:
Tgeα=γn,a(|α|)eα.
An explicit expression of the eigenvalues γn,a (m) in terms of awas found by Grudsky,
γn,a(m)=(m+n)Z1
0
a(r)rm+n1dr, (1.2)
or, changing a variable,
γn,a(k) = 2(m+n)Z1
0
a(r)r2m+2n1dr. (1.3)
Denote by Γn(L([0,1])), or Γnin short, the set of all these eigenvalue sequences,
which are generated by the radial Toeplitz operators with bounded generating func-
tions:
Γn:= Γn(L([0,1])) = γn,a :aL([0,1]).(1.4)
Deﬁne γ1,a and Γ1by (1.3) and (1.4) with n= 1:
γ1,a(k) = 2(k+ 1) Z1
0
a(r)r2k+1 dr, (1.5)
Γ1:= Γ1(L([0,1])) = γ1,a :aL([0,1]).(1.6)
Denote by d1(N) the set of all bounded sequences x= (xj)jNsatisfying the condi-
tion
sup
kN(k+ 1)(∆x)k<+,
where (∆x)k=xk+1 xk.
Then the C-algebra generated by radial Toeplitz operators with bounded gen-
erated symbols is isometrically isomorphic to the C-algebra generated by Γn.
Theorem 1.2 (Su´arez [8]).The C-algebra generated by Γ1coincides with the topo-
logical closure of Γ1in (N), being the topological closure of d1(N)in (N).
5
Slowly oscillating sequences
Denote by SO(N) the set of all bounded sequences that slowly oscillate in the sense
of Schmidt [5] (see also Landau [2] and Stanojevi´c and Stanojevi´c [6]):
SO(N) := nx: lim
j+1
k+1 1|xjxk|= 0o.
In other words, SO(N) consists of all bounded functions NCthat are uniformly
continuous with respect to the “logarithmic metric” ρ(j, k) := |ln(j+ 1) ln(k+ 1)|.
In Section 3 we give some properties and equivalent deﬁnitions of the C-algebra
SO(N).
In Section 4 we prove that the C-algebra generated by Γndoes not actually
depend on n. Applying Theorem 1.2 and some standard approximation techniques
(de la Vall´ee-Poussin means) we obtain the main result of the paper.
Theorem 1.3. For each nthe C-algebra generated by Γncoincides with the topo-
logical closure of Γnin and is equal to SO(N).
As shown by Grudsky, Karapetyants and Vasilevski [1], if aL1([0,1], r2n1dr)
and the sequence γn,a is bounded, then γn,a (m+ 1) γn,a(m)0. At the same
time, in this situation γn,a does not necessarily belong to SO(N). The next result is
proved in Section 5.
Theorem 1.4. There exists a function aL1([0,1], r dr)such that γn,a (N)\
SO(N).
That is, a bounded Toeplitz operator having unbounded deﬁning symbol does not
necessarily belong to the C-algebra generated by Toeplitz operators with bounded
deﬁning symbols.
2 Diagonalization of radial operators in the monomial
basis
Lemma 2.1 (Zhou, Chen, and Dong [9]).Let S:A2→ A2be a bounded radial
operator and αbe a multi-index. Then eαis an eigenfunction of S, i.e., hSeα, eβi= 0
for every multi-index βdiﬀerent from α.
Proof. For a reader convenience we give here a proof, slightly diﬀerent from [9].
Choose an index j∈ {1, . . . , n}such that αj6=βjand a complex number tsuch
that |t|= 1 and tαj6=tβj. For example, put
t=ewhere φ=π
|αjβj|.
Denote by Uthe diagonal matrix with (j, j)st entry equal to t1and all other
diagonal entries equal to 1:
U= diag(1,...,1, t1
|{z}
jst position
,1,...,1).
6
Then Uis a unitary matrix, ΨUeα=tαjeα, and
tαjhSeα, eβi=hSΨUeα, eβi=hΨUSeα, eβi=hSeα,ΨUeβi=tβjhSeα, eβi.
Since tαj6=tβj, it follows that hSeα, eβi= 0.
Lemma 2.2 (Berezin transform of basic projections).Let αNnand zB. Then
B(Pα)(z) = (1 − |z|2)n+1 qα(z),
where qα:BCis the square of the absolute value of eα:
qα(z) = |eα(z)|2=(n+|α|)!
n!α!|zα|2.
Proof. We calculate PαKzfor an arbitrary zB:
PαKz=Pα
X
βNn
eβ(z)eβ
=eα(z)eα.
The reproducing property of Kzimplies that heα, Kzi=eα(z). Therefore
B(Pα)(z) = 1
Kz(z)hPαKz, Kzi= (1 − |z|2)n+1 heα(z)eα, Kzi
= (1 − |z|2)n+1 |eα(z)|2.
Lemma 2.3. For each mN, the function z7→ |z|2mis n
m+ntimes the arithmetic
mean of the functions qαwith |α|=m:
|z|2m=m!n!
(m+n)! X
|α|=m
qα(z) = n
m+n
m! (n1)!
(m+n1)! X
|α|=m
qα(z).
Proof. Apply the multinomial theorem and the deﬁnition of qα:
|z|2m=
n
X
j=1 |zj|2
m
=X
|α|=m
m!
α!
n
Y
j=1 |zj|2αj
=X
|α|=m
m!
α!|zα|2=m!n!
(m+n)! X
|α|=m
qα(z).
Lemma 2.4. Let αNn. Then for all zB,
n|z|2|α|.
7
Proof. Express the integration over Unthrough the integration over S2n1:
n+|α|!
n!α!|(Uz)α|2dH (U) = n+|α|!
n!α!|z|2|α|ZS2n1|ζα|2(ζ).
The value of the latter integral is well known (e.g., see [3, Proposition 1.4.9]):
ZS2n1|ζα|2(ζ) = (n1)! α!
(n1 + |α|)!.
of Pαis the arithmetic mean of all Pβwith |β|=|α|:
(n1 + |α|)! X
βNn
|β|=|α|
Pβ.(2.1)
Proof. We shall prove that both sides of (2.1) have the same Berezin transform, then
(2.1) will follow from the injectivity of the Berezin transform. We use the fact the
Berezin transform “commutes with the radialization” [9], and apply then Lemmas
2.2 and 2.4:
=n+|α|
n|z|2|α|(1 − |z|2)n+1.
On the other hand, by Lemmas 2.4 and 2.3,
(n1)! |α|!
(n1 + |α|)! X
|β|=|α|B(Pβ)(z) = (1 − |z|2)n+1 (n1)! |α|!
(n1 + |α|)! X
|β|=|α|
qβ(z)
=n+|α|
n|z|2|α|(1 − |z|2)n+1.
Lemma 2.6 (radialization of a diagonal operator).Let (cα)αNnbe a bounded family
of complex numbers. Consider the operator S:A2→ A2given by
S=X
αNn
cαPα.
Then
X
m=0
m! (n1)!
(m+n1)! X
|β|=m
cβ
X
|α|=m
Pα
.
Proof. Follows from Lemma 2.5 and the fact that the sum of a converging serie of
mutually orthogonal vectors does not depend on the order of summands.
8
Proof of Theorem 1.1. Let Sbe a bounded linear radial operator in A2. By Lemma
2.1,
S=X
αNn
cαPα.
Since Rad(S) = S, it follows from Lemma 2.6 that the coeﬃcients cαdepend only
on |α|. Deﬁning λmequal to cαfor some αwith |α|=m, we obtain
S=
X
m=0
λm
X
|α|=m
Pα
=Rλ.
3 Slowly oscillating sequences
Deﬁnition 3.1 (logarithmic metric on N).Deﬁne ρ:N×N[0,+) by
ρ(j, k) := ln(j+ 1) ln(k+ 1).
The function ρis a metric on Nbecause it is obtained from the usual metric
d:R×R[0,+), d(t, u) := |tu|, via the injective function NR,j7→ ln(j+1).
Deﬁnition 3.2 (modulus of continuity of a sequence with respect to the logarithmic
metric).Given a complex sequence x= (xj)jN, deﬁne ωρ,x : [0,+)[0,+] by
ωρ,x(δ) := sup|xjxk|:j, k N, ρ(j, k)δ.
Deﬁnition 3.3 (slowly oscillating sequences).Denote by SO(N) the set of the
bounded sequences that are uniformly continuous with respect to the logarithmic
metric:
SO(N) = λ(N): lim
δ0+ωρ,λ(δ) = 0.
Note that the class SO(N) plays an important role in Tauberian theory, see
Landau [2], Schmidt [5, §9], Stanojevi´c and Stanojevi´c [6].
For every sequence xthe function ωρ,x : [0,+)[0,+] is increasing (in the
non-strict sense). Therefore the condition limδ0+ωρ,x(δ) = 0 is equivalent to the
following one: for all ε > 0 there exists a δ > 0 such that ωρ,x(δ)< ε.
The same class SO(N) can be deﬁned using another special metric ρ1on N:
Deﬁnition 3.4. Deﬁne ρ1:N×N[0,+) by
ρ1(j, k) = |jk|
max(j+ 1, k + 1) = 1 min(j+ 1, k + 1)
max(j+ 1, k + 1).
Proposition 3.5. ρ1is a metric on N.
Proof. Clearly ρ1is non-negative, symmetric, and ρ1(j, k) = 0 only if j=k. We
have to prove that for all j, k, p N
ρ1(j, p) + ρ1(p, k)ρ1(j, k)0.(3.1)
9
Denote the left-hand side of (3.1) by Λ(j, k, p). Since Λ(j, k, p) is symmetric with
respect to jand k, assume without loss of generality that jk. If jpk, then
Λ(j, k, p) = 1j+ 1
p+ 1+1p+ 1
k+ 11j+ 1
k+ 1
=pj
p+ 1 pj
k+ 1 =(pj)(kp)
(k+ 1)(p+ 1) 0.
If jk < p, then
Λ(j, k, p) = (pk)(j+k+ 2)
(k+ 1)(p+ 1) 0.
If p<jk, then
Λ(j, k, p) = (jp)(j+k+ 2)
(j+ 1)(k+ 1) 0.
Proposition 3.6 (relations between ρand ρ1).
1. For all j, k N,
ρ1(j, k)ρ(j, k).(3.2)
2. For all j, k Nsatisfying ρ1(j, k)1
2,
ρ(j, k)2 ln(2)ρ1(j, k).(3.3)
Proof. Since the functions ρand ρ1are symmetric and vanish on the diagonal
(ρ(j, j) = ρ1(j, j ) = 0), consider only the case j < k. Denote k+1
j+1 1 by t, then
ρ(j, k) = ln(1 + t), ρ1(j, k) = 1 1
1 + t=t
1 + t.
Deﬁne f: (0,+)(0,+) by
f(t) := ln(1 + t)
11
1+t
.
Then
f0(t) = tln(1 + t)
t2>0,
and thus fis strictly increasing on (0,+). Since limt0+f(t) = 1 and f(1) =
2 ln(2), we see that f(t)>1 for all t > 0 and f(t)2 ln(2) for all t(0,1].
Substituting tby k+1
j+1 1 we obtain (3.2) and (3.3).
Corollary 3.7. The set SO(N)can be deﬁned using the metric ρ1instead of ρ:
SO(N) = nλ(N): lim
δ0+sup
ρ1(j,k)δ|λjλk|= 0o.
Let us mention some simple properties of SO(N).
10
Proposition 3.8. SO(N)is a closed subalgebra of the C-algebra (N).
Proof. It is a general fact that the set of the uniformly continuous functions on some
metric space Mis a closed subalgebra of the C-algebra of the bounded continuous
functions on M. In our case M= (N, ρ). Since
ωρ,f+gωρ,f +ωρ,g, ωρ,λf =|λ|ωρ,f ,
ωρ,fg ωρ,f kgk+ωρ,gkfk, ωρ,f=ωρ,f ,
the set SO(N) is closed with respect to the algebraic operations. The topological
closeness of SO(N) in (N) follows from the inequality
ωρ,f (δ)2kfgk+ωρ,g(δ).
Proposition 3.9 (comparison of SO(N) to c(N)).The set of the converging se-
quences c(N)is a proper subset of SO(N).
Proof. 1. Denote by N:= N∪ {∞} the one-point compactiﬁcation (Alexandroﬀ
compactiﬁcation) of N. The topology on Ncan be induced by the metric
dN(j, k) :=
j
j+ 1 k
k+ 1.
If σc(N), then σis uniformly continuous with respect to the metric dN, but dNis
less or equal than ρ:
dN(j, k) = |jk|
(j+ 1)(k+ 1) |jk|
max(j+ 1, k + 1) =ρ1(j, k)ρ(j, k).
2. The sequence x= (xj)jNwith xj= cos(ln(j+ 1)) does not converge but belongs
to SO(N) since
xjxk=cos(ln(j+ 1)) cos(ln(k+ 1))ln(j+ 1) ln(k+ 1)=ρ(j, k).
We deﬁne now the left and right shifts of a sequence. Given a complex sequence
x= (xj)jN, deﬁne the sequences τL(x) and τR(x) as follows:
τL(x) := (x1, x2, x3, . . .), τR(x) := (0, x0, x1, . . .).
More formally,
τL(x)j:= xj+1;τR(x)j:= (0, j = 0;
xj1, j ∈ {1,2,3, . . .}.
Note that τL(τR(x)) = xfor every sequence x.
Both τLand τRare bounded linear operators on (N). In the following two
propositions we show that SO(N) is an invariant subspace of each one of these
operators.
11
Proposition 3.10. For every xSO(N),τL(x)SO(N).
Proof. The image of τL(x) is a subset of the image of x, therefore kτL(x)k≤kxk.
If δ > 0, j, k N,j < k and ρ(j, k)δ, then
ρ(j+ 1, k + 1) = ln k+ 2
j+ 2 = ln k+ 1
j+ 1 + ln 1 + 1
k+ 1ln 1 + 1
j+ 1
<ln k+ 1
j+ 1 =ρ(j, k)δ.
It follows that ωρ,τL(x)(δ)ωρ,x(δ) and lim
δ0+ωρ,τL(x)(δ) = 0.
Proposition 3.11. For every xSO(N),τR(x)SO(N).
Proof. The sequences xand τR(x) have the same image up to one element zero:
{τR(x)j:jN}={xj:jN}∪{0}.
Therefore kτR(x)k=kxk.
2. Let δ0,1
3,j, k N,j < k and ρ(j, k)δ. Then j1, k2, and
ρ1(j1, k 1) = kj
k=k+ 1
k·(k+ 1) (j+ 1)
k+ 1 3
2ρ1(j, k).
Applying Proposition 3.6 we see that
ρ1(j1, k 1) 3
2ρ(j, k) = 3
2δ1
2
and
ρ(j1, k 1) 2 ln(2)ρ1(j1, k 1) 2 ln(2) 3
2δ= 3 ln(2)δ.
Thus for every δ0,1
3,
ωρ,τR(x)(δ)ωρ,x(3 ln(2)δ).
Therefore lim
δ0+ωρ,τR(x)(δ) = 0.
4 Γnis a dense subset of SO(N)
First we prove that Γnis contained in SO(N).
Proposition 4.1. Let aL([0,1]). Then γn,a SO(N). More precisely,
kγn,ak≤ kak,(4.1)
and for all j, k N,
γn,a(j)γn,a(k)2kakρ(j, k).(4.2)
12
Proof. The inequality (4.1) follows directly from (1.3):
|γn,a(j)| ≤ 2(n+j)Z1
0
r2n+2j1kakdr =kak.
The proof of (4.2) is based on an idea communicated to us by K. M. Esmeral Garc´ıa.
Since both sides of (4.2) are symmetric with respect to the indices jand k, without
loss of generality we consider the case j < k. First factorize a(r) and bound it by
kak:
γn,a(j)γn,a(k)=Z1
0(n+j)r2n+2j1(n+k)r2n+2k1a(r)dr(4.3)
≤ kakZ1
0(n+j)r2n+2j1(n+k)r2n+2k1dr. (4.4)
Denote by r0the unique solution of the equation (n+j)r2n+2j1(n+k)r2n+2k1= 0
on the interval (0,1):
r0=n+j
n+k1
2(kj).
The function r7→ (n+j)r2n+2j1(n+k)r2n+2k1takes positive values on the
interval (0, r0) and negative values on the interval (r0,1). Dividing the integral (4.4)
on two parts by the point r0, we obtain:
γn,a(j)γn,a(k)2kakr2n+2j
0r2n+2k
0= 2kakr2n+2j
0ρ1(j, k).
Since r0<1 and ρ1(j, k)ρ(j, k), the inequality (4.2) follows.
Deﬁnition 4.2. Denote by d1(N) the set of the bounded sequences xsuch that
sup
jN(j+ 1)|xj+1 xj|<+.
Proposition 4.3. d1(N)is a proper subset of SO(N).
Proof. 1. Let xd1(N) and
M= sup
jN(j+ 1)|xj+1 xj|.
Then for all j, k Nwith j < k we have
|xkxj| ≤
k1
X
q=j|xq+1 xq| ≤ M
k1
X
q=j
1
q+ 1 2M
k1
X
q=j
1
q+ 2
= 2M
k1
X
q=j
ρ1(q, q + 1) 2M
k1
X
q=j
ρ(q, q + 1) = 2(j, k ).
Therefore d1(N) is contained in SO(N).
13
2. Consider the sequence
xj:= sin πblog2(j+ 2)c
plog2(j+ 2) .
For every jand kwith k > j,
|xkxj| ≤ πblog2(k+ 2)c
plog2(k+ 2) πblog2(j+ 2)c
plog2(j+ 2)
πlog2(k+ 2)
plog2(k+ 2) π(log2(j+ 2) 1)
plog2(j+ 2)
=πplog2(k+ 2) plog2(j+ 2)+π
plog2(j+ 2)
=πlog2k+2
j+2
plog2(k+ 2) + plog2(j+ 2) +π
plog2(j+ 2).
Thus xSO(N). On the other hand, if j= 2k23, then
|xj+1 xj|=|xj|=
sin π(k21)
plog2(2k21)!
=
sin π(k21)
plog2(2k21)!
.
Appying the inequality |sin(t)| ≥ 2|t|
π, which holds for all twith |t| ≤ π
2, we obtain:
|xj+1 xj| ≥ 2 k(k21)
plog2(2k21)!2kpk211
k=1
plog2(j+ 3).
Therefore x /d1(N).
Lemma 4.4. Let x(N)and δ(0,1). Denote by ythe sequence of the de la
Vall´ee-Poussin means of x:
yj=1
1 + bc
j+bc
X
k=j
xk.(4.5)
Then yd1(N)and
kyxkωρ,x(δ).(4.6)
Proof. Note that for all jN, the sum in the right-hand side of (4.5) contains
1 + bcterms. Therefore
|yj| ≤ 1
1 + bc
j+bc
X
k=jkxk=kxk.
14
For jN, let us estimate the diﬀerence |yj+1 yj|:
|yj+1 yj|=
1
1 + b(j+ 1)δc
j+b(j+1)δc
X
k=j
xk1
1 + bc
j+bc
X
k=j
xk
b(j+ 1)δc − bc
(1 + b(j+ 1)δ)(1 + bc)
j+b(j+1)δc
X
k=j|xk|+|xj+b(j+1)δc|
1 + b(j+ 1)δc
kxk(bc+ 1)
(j+ 1)δ(1 + bc)+kxk
(j+ 1)δ
=kxk
(j+ 1)δ.
Thus yd1(N). Let us prove (4.6). If jkj+bjδc, then
ρ(j, k) = ln k+ 1
j+ 1 ln k
jln(1 + δ)δ.
Therefore
|yjxj| ≤ 1
1 + bc
j+bc
X
k=j|xkxj| ≤ ωρ,x(δ).
Proposition 4.5. d1(N)is a dense subset of SO(N).
Proof. Let ε > 0. Using the fact that ωρ,x(δ)0 as δ0, choose a δ > 0 such
that ωρ,x(δ)< ε. Deﬁne yby (4.5). Then yd1(N) and kxyk< ε by Lemma
4.4.
Theorem 1.3 follows from Proposition 4.5 and Theorem 1.2:
Proposition 4.6. Γ1is a dense subset of SO(N).
Proof. Proposition 4.1 implies that Γ1is contained in SO(N). Let xSO(N) and
ε > 0. Applying Proposition 4.5 ﬁnd a sequence yd1(N) such that
kyxk<ε
2.
Using Theorem 1.2 we ﬁnd a function aL([0,1]) such that kγ1,a yk<ε
2.
Then
kγ1,a xk≤ kγ1,a yk+kyxk< ε.
Lemma 4.7. Let aL([0,1]). Then γn,a =τn1
L(γ1,a).
Proof. Follows directly from the deﬁnitions of γn,a and γ1,a, see (1.3) and (1.5).
Proposition 4.8. Γnis a dense subset of SO(N).
15
Proof. By Proposition 4.1, Γnis a subset of SO(N).
Let xSO(N) and ε > 0. Denote τn1
R(x) by y. By Proposition 3.11, ySO(N).
Using Proposition 4.6 ﬁnd a function aL([0,1]) such that kyγ1,ak< ε. Then
apply Lemma 4.7:
kxγn,ak=kτn1
L(y)τn1
L(γ1,a)k=kτn1
L(yγ1,a)k≤ kyγ1,ak< ε.
We ﬁnish this section with an important observation. The results stated up to
this moment do not take into account the multiplicities of the eigenvalues. In this
connection we recall that for each bounded radial operator Rλon A2(Bn) with the
eigenvalue sequence λ(N), the equality
Rλeα=λpeα
holds for all multi-indices αNnsatisfying |α|=p, and there are n+p1
n1such
multi-indices.
As was mentioned, for each natural number nthe C-algebra generated by
Toeplitz operators on A2(Bn) with bounded radial symbols is isomorphic and iso-
metric to the C-algebra of multiplication operators Rλon 2(N) whose eigenvalue
sequences belong to SO(N), and thus its Cstructure does not depend on n. At the
same time these algebras, when nis varied, are quite diﬀerent if we count multiplic-
ities of eigenvalues, that is when the operators forming the algebra are considered
by their action on the basis elements of the corresponding Hilbert space A2(Bn).
Let us consider in more detail sequences of eigenvalues with multiplicities. For-
mula for the rising sum of binomial coeﬃcients states that
p
X
m=0 n+m1
n1=n+p1
n.
Now, for every jNthere exists a unique pin Nsuch that
n+p1
nj < n+p
n.
Denote this pby πn(j), and say that the index jis located on the p-st “level”.
Given a sequence λ, deﬁne Φn(λ) as the sequence obtained from λby
repeating each λpaccording to its multiplicity. That is,
Φn(λ) :=
(n+p1
n)elements
z }| {
λ0
|{z}
(n1
n1)
times
, λ1
|{z}
(n
n1)
times
, λ2
|{z}
(n+1
n1)
times
, λ3
|{z}
(n+2
n1)
times
, . . . , λp
|{z}
(n+p1
n1)
times
, . . ..
Since the isometric homomorphism Φnof (N) is injective, the C-algebra gener-
ated by the set {Φn(γn,a) : aL[0,1]}coincides with Φn(SO(N)), that is, with
the C-algebra obtained from SO(N) by applying the mapping Φn.
16
Note that for all p, q with p<qthe following estimates hold:
ln q+ 1
p+ 1 ln n+q
nln n+p
nnln q+ 1
p+ 1,
which implies that Φn(SO(N)) coincides with the C-algebra SOrep,n(N), a subal-
gebra SO(N), which consists of all sequences having the same elements on each
“level”:
SOrep,n(N) := nµSO(N) : if πn(j) = πn(k),then µj=µko.
That is, the described above eigenvalue repetitions do not change in essence a slowly
oscillating behavior of sequences.
5 Example
In this section we construct a bounded sequence λ= (λj)jNsuch that λ=γn,a
for a certain function aL1([0,1], r dr) but λ /SO(N). This implies that the
corresponding radial Toeplitz operator is bounded, but it does not belong to the
C-algebra generated by radial Toeplitz operators with bounded symbols.
Proposition 5.1. Deﬁne f:{zC:<(z)0} → Cby
f(z) := 1
z+nexp i
3πln2(z+n),(5.1)
where ln is the principal value of the natural logarithm (with imaginary part in
(π, π]). Then there exists a unique function AL1(R+,eudu)such that fis the
Laplace transform of A:
f(z) = Z+
0
A(u)ezu dz.
Proof. For every zCwith <(z)0 we can write ln(z+n) as ln |z+n|+iarg(z+n)
with π
2<arg(z+n)<π
2. Then
|f(z)|=1
|z+n|exp i
3πln |z+n|+iarg(z+n)2
=1
|z+n|exp 2 arg(z+n)
3πln |z+n|
=1
|z+n|1+ 2 arg(z+n)
3π
.
Since |z+n| ≥ 1 and 1
3<2 arg(z+n)
3π<1
3,
|f(z)| ≤ 1
|z+n|2/3.
17
Therefore for every x > 0,
ZR|f(x+iy)|2dy ZR
dy
((x+n)2+y2)2/3<ZR
dy
(1 + y2)2/3<+,
and fbelongs to the Hardy class H2on the half-plane {zC:<(z)>0}. By
Paley–Wiener theorem (see, for example, Rudin [4, Theorem 19.2]), there exists a
function AL2(0,+) such that for all x > 0
f(x) = Z+
0
A(u)eux du.
The uniqueness of Afollows from the injective property of the Laplace transform.
Applying H¨older’s inequality we easily see that AL1(R+,eudu):
Z+
0|A(u)|eudu ≤ kAk2Z+
0
e2udu1/2
=kAk2
2.
Proposition 5.2. The sequence λ= (λj)jN, where
λj:= exp i
3πln2(j+n),(5.2)
belongs to (N)\SO(N). Moreover there exists a function aL1([0,1], r dr)such
that λ=γn,a.
Proof. Since |λj|= 1 for all jN, the sequence λis bounded. Let Abe the function
from Proposition 5.1. Deﬁne a: [0,1] Cby
a(r) = A(2 ln r).
Then
Z1
0|a(r)|r dr =1
2Z1
0|a(t)|dt =1
2Z1
0|A(ln(t))|dt
=1
2Z+
0|A(u)|eudu < +,
and
γn,a(j)=(j+n)Z1
0
a(r)rj+n1dr = (j+n)Z1
0
A(ln r)rj+n1dr
= (j+n)Z+
0
A(u) e(j+n)udu = (j+n)f(j+n) = λj.
Let us prove that λ /SO(N). For every j, k Nwe have
|λjλk|=exp i
3πln2(j+n)ln2(k+n)1.
18
Replace jby the following function of k:
j(k) := k+$k+n ln1/2(k+n)%. Then j(k)k k+n=1 ln1/2(k+n)+O1 k+n and ln(j(k) + n) = ln(k+n) + ln 1 + j(k)k k+n = ln(k+n) + 1 ln1/2(k+n)1 2 ln(k+n)+O 1 ln3/2(k+n)!. Denote ln2(j(k) + n)ln2(k+n) by Lkand consider the asymptotic behavior of Lk as k→ ∞: Lk:= ln2(j(k) + n)ln2(k+n) = 1 + 2 ln1/2(k+n) + O1 ln(k+n). Since Lkincreases slowly for large k, for every K > 0 there exists an integer kK such that Lk+ 1 is close enough to an integer multiple of 6π2, say to 62: Lk+ 1 62. For such k, |λj(k)λk|=exp i 3π(Lk+ 1 62)exp i 3π1 exp i 3π16= 0. It means that |λj(k)λk|does not converge to 0 as kgoes to inﬁnity. On the other hand, ρ(j(k), k) = ln j(k)+1 k+ 1 (k+n) (k+ 1) ln1/2(k+n)0. It follows that λ /SO(N). Acknowledgments This research was partially supported by the projects CONACyT 102800, IPN-SIP 20120730, and by PROMEP via “Proyecto de Redes”. 19 References [1] Sergei M. Grudsky, Alexei N. Karapetyants, and Nikolai L. Vasilevski, Toeplitz operators on the unit ball in Cnwith radial symbols. J. Operator Theory,49 (2003), pp 325-346. [2] Edmund Landau, ¨ Uber die Bedeutung einiger neuen Grentzwerts¨atze der Herren Hardy und Axer. Prace Mat. Fiz.,21 (1910), pp 97-177. [3] Walter Rudin, Function theory in the unit bal l in Cn, Springer-Verlag, Berlin and New York, 1980. [4] Walter Rudin, Real and complex analysis, McGraw-Hill, New York, 1987. [5] Robert Schmidt, ¨ Uber divergente Folgen and lineare Mittelbildungen. Math. Z.,22 (1925), pp 89-152. [6] ˇ Caslav V. Stanojevi´c and Vera B. Stanojevi´c, Tauberian retrieval theory. Publications de l’Institut Mathematique,71 (2002), pp 105-111. [7] Karel Stroethoﬀ, The Berezin transform and operators on spaces of analytic functions. Linear Operators, Banach Center Publications,38 (1997), pp 361-380. [8] Daniel Su´arez, The eigenvalues of limits of radial Toeplitz operators. Bull. London Math. Soc.,40 (2008), pp 631-641. [9] Ze-Hua Zhou, Wei-Li Chen, and Xing-Tang Dong, The Berezin transform and radial op- erators on the Bergman space of the unit ball. Complex Analysis and Operator Theory, doi:10.1007/s11785-011-0145-2 (2011), pp 1-17. [10] Kehe Zhu, Spaces of Holomorphic Functions on the Unit Ball , Springer, 2004. [11] Nina Zorboska, The Berezin transform and radial operators. Proceedings of the American Mathematical Society,131 (2003), pp 793-800. 20 ... Several authors have studied intensively this sort of operators. Among those references studying either radial or separately radial operators one can find [1,4,6,11,13,14,16,17,22,27]. An interesting feature of this sort of operators is that they commute with each other. ... ... This is one of the reasons for the importance of proving the existence of commutative C *algebras generated by Toeplitz operators. Another one, is the fact that they provide concrete examples of commutative algebras (see [4,14]). ... ... As found in previous works (see, for example, [4,6,14]) it is useful to consider the subspaces of sequences in ∞ spaces associated to diagonalizable Toeplitz operators. For this reason, we state the following immediate consequence of Theorem 5.3. ... Article Full-text available For a partition $$\varvec{k} = (k_1, \dots , k_m)$$ of n consider the group $$\mathrm {U}(\varvec{k}) = \mathrm {U}(k_1) \times \dots \times \mathrm {U}(k_m)$$ block diagonally embedded in $$\mathrm {U}(n)$$ and the center $$\mathbb {T}^m$$ of $$\mathrm {U}(\varvec{k})$$. We study the Toeplitz operators with $$\mathbb {T}^m$$-invariant symbols acting on the weighted Bergman spaces on the unit ball $$\mathbb {B}^n$$. We introduce the $$(\varvec{k},j)$$-quasi-radial quasi-homogeneous symbols as those that are invariant under the group $$\mathrm {U}(\varvec{k},j,\mathbb {T})$$ obtained from $$\mathrm {U}(\varvec{k})$$ by replacing the factor $$\mathrm {U}(k_j)$$ with its center $$\mathbb {T}$$. These symbols are used to build commutative Banach non-$$C^*$$ algebras generated by Toeplitz operators. These algebras generalize those from the literature and show that they can be built using groups. We describe the action of such Toeplitz operators on monomials through explicit integral formulas involving the symbols. We prove that every Toeplitz operator with $$\mathbb {T}^m$$-invariant symbol has an associated Toeplitz operator with $$\mathrm {U}(\varvec{k})$$-invariant symbol in terms of which we can describe some properties. ... Several authors have studied intensively this sort of operators. Among those references studying either radial or separately radial operators one can find [1], [4], [6], [11], [13], [14], [16], [17], [22], [27]. An interesting feature of this sort of operators is that they commute with each other. ... ... This is one of the reasons for the importance of proving the existence of commutative C * -algebras generated by Toeplitz operators. Another one, is the fact that they provide concrete examples of commutative algebras (see [4] and [14]). ... ... As found in previous works (see, for example, [6], [4] and [14]) it is useful to consider the subspaces of sequences in ℓ ∞ spaces associated to diagonalizable Toeplitz operators. For this reason, we state the following immediate consequence of Theorem 5.3. ... Preprint Full-text available For a partition$\boldsymbol{k} = (k_1, \dots, k_m)$of$n$consider the group$\mathrm{U}(\boldsymbol{k}) = \mathrm{U}(k_1) \times \dots \times \mathrm{U}(k_m)$block diagonally embedded in$\mathrm{U}(n)$and the center$\mathbb{T}^m$of$\mathrm{U}(\boldsymbol{k})$. We study the Toeplitz operators with$\mathbb{T}^m$-invariant symbols acting on the weighted Bergman spaces on the unit ball$\mathbb{B}^n$. We introduce the$(\boldsymbol{k},j)$-quasi-radial quasi-homogeneous symbols as those that are invariant under the group$\mathrm{U}(\boldsymbol{k},j,\mathbb{T})$obtained from$\mathrm{U}(\boldsymbol{k})$by replacing the factor$\mathrm{U}(k_j)$with its center$\mathbb{T}$. These symbols are used to build commutative Banach non-$C^*$algebras generated by Toeplitz operators. These algebras generalize those from the literature and show that they can be built using groups. We describe the action of such Toeplitz operators on monomials through explicit integral formulas involving the symbols. We prove that every Toeplitz operator with$\mathbb{T}^m$-invariant symbol has an associated Toeplitz operator with$\mathrm{U}(\boldsymbol{k})$-invariant symbol in terms of which we can describe some properties. ... This may happen (see e.g. [8]) if and only if the sequence ϕ = {ϕ(k)} k∈Z+ belongs to SO(Z + ), the set of all bounded sequences which slowly oscillate in the sense of Schmidt [12] SO ... ... [8]) if and only if ϕ belongs to VSO(R + ), the set of all bounded functions which satisfy the condition VSO(R + ) = ϕ ∈ L ∞ (R + ) : lim ... Article Full-text available For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols. ... This may happen (see e.g. [8]) if and only if the sequence ϕ = {ϕ(k)} k∈Z + belongs to SO(Z + ), the set of all bounded sequences which slowly oscillate in the sense of Schmidt [12] SO(Z + ) = ς ς ς ∈ ℓ ∞ : lim j+1 k+1 →1 |ς j − ς k | = 0 . ... ... This happens (see e.g. [8]) if and only if ϕ belongs to VSO(R + ), the set of all bounded functions which satisfy the condition ... Preprint Full-text available For three standard models of commutative algebras generated by Toeplitz operators in the weighted analytic Bergman space on the unit disk, we find their representations as the algebras of bounded functions of certain unbounded self-adjoint operators. We discuss main properties of these representation and, especially, describe relations between properties of the spectral function of Toeplitz operators in the spectral representation and properties of the symbols. ... Our treatment of this example is close to [37,Chapters 4,6] and [14], where L 2 (D, μ 2 ) is decomposed into L 2 (R/(2πZ)) ⊗ L 2 ([0, 1), r dr), and the Fourier transform over R/(2πZ) is applied to the equation defining H 1 . The C*algebra VT for this example was described in [16] using Suárez [34]. ... Article Full-text available Let G be a locally compact abelian group with a Haar measure, and Y be a measure space. Suppose that H is a reproducing kernel Hilbert space of functions on G×Y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\times Y$$\end{document}, such that H is naturally embedded into L2(G×Y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2(G\times Y)$$\end{document} and is invariant under the translations associated with the elements of G. Under some additional technical assumptions, we study the W*-algebra V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {V}}$$\end{document} of translation-invariant bounded linear operators acting on H. First, we decompose V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {V}}$$\end{document} into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces H^ξ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\widehat{H}}_\xi$$\end{document}, ξ∈G^\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\xi \in {\widehat{G}}$$\end{document}, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {V}}$$\end{document}. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to V\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {V}}$$\end{document}, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors. ... We recall that a ∈ L ∞ (D) is called radial if and only if a(z) = a(|z|), for every z ∈ D. Besides the references mentioned above, we can also refer to [5,13,15,17,22,25,26,30,34,35] for examples of research on Toeplitz operators with radial symbols and some generalizations to the unit ball B n . From these references we would like to highlight [22] where it was discovered the importance of Toeplitz operators with radial symbols. ... Article Full-text available Let Dn×nI\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {D}}^{{\mathrm {I}}}_{n \times n}$$\end{document} be the Cartan domain of type I which consists of the complex n×n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \times n$$\end{document} matrices Z that satisfy Z∗Z<In\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Z^*Z < I_n$$\end{document}. For a symbol a∈L∞(Dn×nI)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a \in L^\infty ({\mathrm {D}}^{{\mathrm {I}}}_{n \times n})$$\end{document} we consider three radial-like type conditions: 1) left (right) U(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {U}}(n)$$\end{document}-invariant symbols, which can be defined by the condition a(Z)=a((Z∗Z)12)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(Z) = a\big ((Z^*Z)^\frac{1}{2}\big )$$\end{document} (a(Z)=a((ZZ∗)12)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(Z) = a\big ((ZZ^*)^\frac{1}{2}\big )$$\end{document}, respectively), and 2) U(n)×U(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathrm {U}}(n) \times {\mathrm {U}}(n)$$\end{document}-invariant symbols, which are defined by the condition a(A-1ZB)=a(Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$a(A^{-1}ZB) = a(Z)$$\end{document} for every A,B∈U(n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$A, B \in {\mathrm {U}}(n)$$\end{document}. We prove that, for n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}, these yield different sets of symbols. If a satisfies 1), either left or right, and b satisfies 2), then we prove that the corresponding Toeplitz operators Ta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_a$$\end{document} and Tb\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_b$$\end{document} commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}, symbols a whose corresponding Toeplitz operators Ta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_a$$\end{document} are non-normal. We use these facts to prove the existence, for n≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n \ge 2$$\end{document}, of commutative Banach non-C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document} algebras generated by Toeplitz operators. ... In the special case of the unit ball in C n , it was shown that Toeplitz operators with radial symbols acting on the Bergman space generate a commutative C * -algebra as in [21] and the corresponding C * -algebra is isometrically isomorphic to the space of bounded sequences that are uniformly continuous with respect to the logarithmic metric [22]. Toeplitz operators with radial symbols on the Fock space F(C) generate a commutative C * -algebra [23] which is isometrically isomorphic to the C * -algebra C b,u (N 0 , ρ 1 ) of bounded sequences that are uniformly continuous with respect to the square-root metric [14]. ... Article Full-text available The Fock space F(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}({\mathbb {C}}^n)$$\end{document} is the space of holomorphic functions on Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^n$$\end{document} that are square-integrable with respect to the Gaussian measure on Cn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {C}}^n$$\end{document}. This space plays an important role in several subfields of analysis and representation theory. In particular, it has for a long time been a model to study Toeplitz operators. Esmeral and Maximenko showed in 2016 that radial Toeplitz operators on F(C)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}({\mathbb {C}})$$\end{document} generate a commutative C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra which is isometrically isomorphic to the C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra Cb,u(N0,ρ1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{b,u}({\mathbb {N}}_0,\rho _1)$$\end{document}. In this article, we extend the result to k-quasi-radial symbols acting on the Fock space F(Cn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {F}}({\mathbb {C}}^n)$$\end{document}. We calculate the spectra of the said Toeplitz operators and show that the set of all eigenvalue functions is dense in the C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra Cb,u(N0k,ρk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{b,u}({\mathbb {N}}_0^k, \rho _k)$$\end{document} of bounded functions on N0k\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {N}}_0^k$$\end{document} which are uniformly continuous with respect to the square-root metric. In fact, the C∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^*$$\end{document}-algebra generated by Toeplitz operators with quasi-radial symbols is Cb,u(N0k,ρk)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C_{b,u}({\mathbb {N}}_0^k, \rho _k)$$\end{document}. ... We recall that a ∈ L ∞ (D) is called radial if and only if a(z) = a(|z|), for every z ∈ D. Besides the references mentioned above, we can also refer to [5,12,14,16,22,25,26,30,34,35] for examples of research on Toeplitz operators with radial symbols and some generalizations to the unit ball B n . From these references we would like to highlight [22] where it was discovered the importance of Toeplitz operators with radial symbols. ... Preprint Let$\mathrm{D}^\mathrm{I}_{n \times n}$be the Cartan domain of type I which consists of the complex$n \times n$matrices$Z$that satisfy$Z^*Z < I_n$. For a symbol$a \in L^\infty(\mathrm{D}^\mathrm{I}_{n \times n})$we consider three radial-like type conditions: 1) left (right)$\mathrm{U}(n)$-invariant symbols, which can be defined by the condition$a(Z) = a\big((Z^*Z)^\frac{1}{2}\big)$($a(Z) = a\big((ZZ^*)^\frac{1}{2}\big)$, respectively), and 2)$\mathrm{U}(n) \times \mathrm{U}(n)$-invariant symbols, which are defined by the condition$a(A^{-1}ZB) = a(Z)$for every$A, B \in \mathrm{U}(n)$. We prove that, for$n \geq 2$, these yield different sets of symbols. If$a$satisfies 1), either left or right, and$b$satisfies 2), then we prove that the corresponding Toeplitz operators$T_a$and$T_b$commute on every weighted Bergman space. Furthermore, among those satisfying condition 1), either left or right, there exist, for$n \geq 2$, symbols$a$whose corresponding Toeplitz operators$T_a$are non-normal. We use these facts to prove the existence, for$n \geq 2$, of commutative Banach non-$C^*$algebras generated by Toeplitz operators. ... Our treatment of this example is close to [37,Chapters 4,6] and [14], where L 2 (D, µ 2 ) is decomposed into L 2 (R/(2πZ))⊗L 2 ([0, 1), r dr), and the Fourier transform over R/(2πZ) is applied to the equation defining H 1 . The C*-algebra VT for this example was described in [16] using Suárez [34]. ... Preprint Full-text available Let$G$be a locally compact abelian group with a Haar measure, and$Y$be a measure space. Suppose that$H$is a reproducing kernel Hilbert space of functions on$G\times Y$, such that$H$is naturally embedded into$L^2(G\times Y)$and is invariant under the translations associated with the elements of$G$. Under some additional technical assumptions, we study the W*-algebra$\mathcal{V}$of translation-invariant bounded linear operators acting on$H$. First, we decompose$\mathcal{V}$into the direct integral of the W*-algebras of bounded operators acting on the reproducing kernel Hilbert spaces$\widehat{H}_\xi$,$\xi\in\widehat{G}$, generated by the Fourier transform of the reproducing kernel. Second, we give a constructive criterion for the commutativity of$\mathcal{V}$. Third, in the commutative case, we construct a unitary operator that simultaneously diagonalizes all operators belonging to$\mathcal{V}\$, i.e., converts them into some multiplication operators. Our scheme generalizes many examples previously studied by Nikolai Vasilevski and other authors.
... In the special case of the unit ball in C n , it was shown that Toeplitz operators with radial symbols acting on the Bergman space generate a commutative C * -algebra as in [21] and the corresponding C * -algebra is isometrically isomorphic to the space of bounded sequences that are uniformly continuous with respect to the logarithmic metric [22]. Toeplitz operators with radial symbols on the Fock space F(C) generate a commutative C * -algebra [23] which is isometrically isomorphic to the C * -algebra C b,u (N 0 , ρ 1 ) of bounded sequences that are uniformly continuous with respect to the square-root metric [14]. ...
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Reproducing kernel Hilbert spaces of square integrable holomorphic functions on complex domains play an important role in several subfields of analysis and representation theory. A well known example is the Fock space F(C^n) of holomorphic functions on C^n that are square-integrable with respect to the Gaussian measure. Esmeral and Maximenko showed in 2016 that radial Toeplitz operators on F(C) generates a commutative C^*-algebra which is isometrically isomorphic to C^*-algebra C_{b,u}(N_0). In this article we extend this result to k-quasi-radial symbols acting on the Fock space F(C^n). We calculate the spectra of the said Toeplitz operators and show that the set of all eigenvalue functions is dense in the C^*-algebra C_{b,u}(N_0^k) of bounded functions on N_0^k which are uniformly continuous with respect to the square-root metric. In fact the C^*-algebra generated by Toeplitz operators with quasi-radial symbols is C_{b,u}(N_0^k).
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The paper is devoted to the study of Toeplitz operators with radial symbols on the weighted Bergman spaces on the unit ball in C n . Ad-mitting "badly" behaved unbounded symbols we get new qualitative features. In particular, contrary to known results, a Toeplitz operator with the same (unbounded) symbol now can be bounded in one weighted Bergman space and unbounded in another, compact in one weighted Bergman space and bounded but not compact in another, compact in one weighted Bergman space and unbounded in another. In our case of radial symbols, the Wick (or covariant) symbol of a Toeplitz operator gives complete information about the operator, providing its spectral decomposition.
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