ArticlePDF Available

Abstract

We consider the C*-algebra generated by Toeplitz operators acting on the Bergman space over the upper half-plane whose symbols depend on the imaginary part of the argument only. Such algebra is known to be commutative, and is isometrically isomorphic to an algebra of bounded complex-valued functions on the positive half-line. In the paper we prove that the latter algebra consists of all bounded functions f that are very slowly oscillating in the sense that the composition of f with the exponential function is uniformly continuous or, in other words, $$\lim_{\frac{x}{y} \to 1} \left|f(x) - f(y)\right| = 0.
Vertical Toeplitz operators
on the upper half-plane
and very slowly oscillating functions
Crispin Herrera Ya˜nez, Egor A. Maximenko, Nikolai Vasilevski
This is a preprint of the article published in
Integral Equations and Operator Theory 77:2 (2013), 149–166.
The final publication is available at Springer via
http://dx.doi.org/10.1007/s00020-013-2081-1
Abstract
We consider the C-algebra generated by Toeplitz operators acting on the Bergman
space over the upper half-plane whose symbols depend on the imaginary part of the
argument only. Such algebra is known to be commutative, and is isometrically iso-
morphic to an algebra of bounded complex-valued functions on the positive half-line.
In the paper we prove that the latter algebra consists of all bounded functions fthat
are very slowly oscillating in the sense that the composition of fwith the exponential
function is uniformly continuous or, in other words,
lim
x
y1f(x)f(y)= 0.
MSC: Primary 47B35; Secondary 47B32, 32A36, 44A10, 44A15.
Key words: Bergman space, Toeplitz operators, invariant under horizontal shifts, Laplace
transform, very slowly oscillating functions.
This work was partially supported by CONACYT Project 102800 and by IPN-SIP Project
2013-0633, M´exico.
1
1 Introduction
The paper is devoted to the description of a certain class of Toeplitz operators acting on the
Bergman space over the upper half-plane and of the C-algebra generated by them.
Let Π = {z=x+iy C|y > 0}be the upper half-plane, and let =dxdy be
the standard Lebesgue plane measure on Π. Recall that the Bergman space A2(Π) is the
(closed) subspace of L2, dµ) which consists of all function analytic in Π. It is well known
that A2(Π) is a reproducing kernel Hilbert space whose (Bergman) reproducing kernel has
the form
KΠ,w(z) = 1
π(wz)2;
thus the Bergman (orthogonal) projection of L2, dµ) onto A2(Π) is given by
(P f )(w) = hf, KΠ,wi.
Given a function gL(Π), the Toeplitz operator Tg:A2(Π) → A2(Π) with generating
symbol gis defined by Tgf=P(gf).
One of the phenomena in the theory of Toeplitz operators on the Bergman space is
that (contrary to the Hardy space case) there exists a rich family of symbols that generate
commutative algebras of Toeplitz operators (see for details [13, 14]). There are three model
classes of such symbols: elliptic, which is realized by radial symbols, functions depending on
|z|, on the unit disk, parabolic, which is realized by symbols depending on y= Im(z) on the
upper half-plane, and hyperbolic, which is realized by homogeneous of order zero symbols on
the upper half-plane. All other classes of symbols, that generate commutative algebras of
Toeplitz operators, are obtained from the above three model classes by means of the M¨obius
transformations.
In each case of a commutative algebra of Toeplitz operators there is an (explicitly defined)
unitary operator Rthat reduces each Toeplitz operator Tafrom the algebra to a certain
(again explicitly given) multiplication operator by γa, being a function (in the parabolic
and hyperbolic cases), or a sequence (in the elliptic case). This “spectral” function (or
sequence) γa“carries” many substantial properties of corresponding Toeplitz operators, such
as boundedness, norm, compactness, spectrum, essential spectrum, etc.
A very important task to be done in this connection is to describe the properties of such
“spectral” functions and algebras generated by them, understanding thus in more detail the
properties of Toeplitz operators and algebras generated by them.
The first essential step in this direction was done by Su´arez [9, 10], who proved, in
particular, that the set of Toeplitz operators with bounded radial symbols (the elliptic case)
is dense in the C-algebra generated by these operators, and that the l-closure of the set of
corresponding “spectral” sequences coincides with the l-closure of a certain set, which he
denotes by d1and which is commonly used in Tauberian theory. Then in [4] it was shown
that this closure coincides with the C-algebra of all slowly oscillating sequences introduced
2
by Schmidt [7, Definition 10], i.e., of all bounded sequences x= (xn)
n=0 such that
lim
m+1
n+1 1|xmxn|= 0,
which gives thus an isometric characterization of the elliptic case commutative algebra.
In this paper we study the commutative C-algebra VT (L) generated by Toeplitz opera-
tors of the model parabolic case, i.e., by Toeplitz operators with bounded symbols depending
on y= Im(z) (we call such symbols vertical). The main result of the paper states that the
set of Toeplitz operators with bounded vertical symbols is dense in the above C-algebra,
and that the algebra VT (L) itself is isometrically isomorphic to the (introduced in the
paper) C-algebra VSO(R+) of very slowly oscillating functions, the functions that are uni-
formly continuous with respect to the logarithmic metric ρ(x, y) = |ln(x)ln(y)|on R+or,
equivalently, the functions satisfying the condition
lim
x
y1f(x)f(y)= 0.
The paper is organized as follows. In Sections 2 and 3 we give various equivalent descriptions
of vertical operators (operators that are invariant under horizontal shifts) and of vertical
Toeplitz operators. In Sections 4 and 5 we introduce the algebra VSO(R+) and prove the
above stated main result on density. In Section 6 we give an example of a bounded Toeplitz
operator Tawith unbounded vertical symbol awhose “spectral” function γadoes not belong
to the algebra VSO(R+). This means that in spite of its boundedness Tadoes not belong
to the C-algebra generated by Toeplitz operators with bounded vertical symbols. In other
words, admitting bounded Toeplitz operators with unbounded symbol we enlarge the algebra
VT (L).
Note that the technique used in the paper for the parabolic case is more simple and
efficient than the general one of [9, 10]. Instead of the n-Berezin transform (a special kind of
an approximative unit introduced and used by Su´arez), we use another approximative unit
based on a certain Dirac sequence.
2 Vertical operators
Let L(A2(Π)) be the algebra of all linear bounded operators acting on the Bergman space
A2(Π). Given hR, let Hh∈ L(A2(Π)) be the horizontal translation operator defined by
Hhf(z) := f(zh).
We call an operator S∈ L(A2(Π)) vertical (or horizontal translation invariant) if it com-
mutes with all horizontal translation operators:
hR, HhS=SHh.
3
In this section we find a criterion for an operator from A2(Π) to be vertical. First we
recall some known facts on translation invariant operators on the real line.
Introduce the standard Fourier transform
(F f )(s) := 1
2πZR
eist f(t)dt,
being a unitary operator on L2(R).
For each hR, the translation operator τh:L2(R)L2(R) is defined by
τhf(s) := f(sh).
An operator Son L2(R) is called translation invariant if τhS=Sτh, for all hR. It is well
known (see, for example, [5, Theorem 2.5.10]) that an operator Son L2(R) is translation
invariant if and only if it is a convolution operator, i.e., if and only if there exists a function
σL(R) such that
S=F1MσF. (2.1)
We introduce as well the associated multiplication by a character operator MΘhf(s) :=
Θh(s)f(s), where Θh(s) := eish.
Note that τhand MΘhare related via the Fourier transform,
MΘhF=F τh.(2.2)
Lemma 2.1. Let M∈ L(L2(R)). The following conditions are equivalent:
(a) Mis invariant under multiplication by Θhfor all hR:
MMΘh=MΘhM.
(b) Mis the multiplication operator by a bounded measurable function:
σL(R) such that M=Mσ.
Proof. The part (b)(a) is trivial: MσMΘh=MσΘh=MΘhMσ. The implication (a)(b)
follows from the relation (2.2) and the result about the translation invariant operators cited
above.
Old proof. Assuming (a), by (2.2) we have
F1M F τh=F1MMΘhF=F1MΘhMF =τhF1MF,
which implies that F1MF commutes with translations. Then (2.1) implies
F1MF =F1MσF.
Since Fis unitary, (b) holds.
Conversely, if (b) holds, then MσMΘh=MσΘh=MΘhMσ.
4
Let Θ+
hdenote the restriction of Θhto R+. The following lemma states that an operator
on L2(R+) commutes with MΘ+
hif and only if it is a multiplication operator.
Lemma 2.2. Let M∈ L(L2(R+)). The following conditions are equivalent:
(a) Mis invariant under multiplication by Θ+
hfor all hR:
MMΘ+
h=MΘ+
hM.
(b) Mis the multiplication operator by a bounded function:
σL(R+) such that M=Mσ.
Proof. To prove that (a) implies (b), define the restriction operator
P:L2(R)L2(R+), g 7→ g|R+,
and the zero extension operator
J:L2(R+)L2(R), Jf(x) := (f(x) if x > 0,
0 if x0.
For every hRthe following equalities hold:
JMΘ+
h=MΘhJ, P MΘh=MΘ+
hP.
If (a) holds, then the operator JM P is invariant under multiplication by Θh, for all hR:
J MP MΘh=J MMΘ+
hP=JMΘ+
hMP =MΘhJMP,
and by Lemma 2.2 there exists a function σ1L(R) such that JM P =Mσ1. Set σ=σ1|R+.
Then for all fL2(R+) and all xR+,
(Mσf)(x) = σ(x)f(x) = σ1(x)(Jf)(x) = (Mσ1Jf )(x)
= (J MP J f )(x)=(JMf )(x)=(Mf)(x),
and (b) holds. The implication (b)(a) follows directly, as in the previous lemma.
The Berezin transform [1, 2] of an operator S∈ L(A2(Π)) is the function Π Cdefined
by
B(S)(w) := hSKΠ,w, KΠ,wi
hKΠ,w, KΠ,w i.
5
Following [12, Section 2] (see also [14, Section 3.1]), we introduce the isometric isomorphism
R:A2(Π) L2(R+),
()(x) := x
πZΠ
φ(w) eiwx (w).
The operator Ris unitary, and its inverse R:L2(R+)→ A2(Π) is given by
(Rf)(z) = 1
πZR+pξf (ξ) eizξ .
The next theorem gives a criterion for an operator to be vertical, and is an analogue of the
Zorboska result [15] for radial operators.
Theorem 2.3. Let S∈ L(A2(Π)). The following conditions are equivalent:
(a) Sis invariant under horizontal shifts:
hRSHh=HhS.
(b) RSRis invariant under multiplication by Θ+
hfor all hR:
hRRSRMΘ+
h=MΘ+
hRSR.
(c) There exists a function σL(R+)such that
S=RMσR.
(d) The Berezin transform of Sis a vertical function, i.e., depends on Im(w)only.
Proof. (a)(b). Follows from the formulas RMΘ+
h=HhRand RHh=MΘ+
hR.
(b)(c). Follows from Lemma 2.2.
(c)(d). Using the residue theorem we get
(RKΠ,w)(x) = ix
πeiRe(w)xeIm(w)x.
Therefore
B(S)(w) = hMσRKΠ,w , RKΠ,w i
hKΠ,w, KΠ,w i= (2 Im(w))2Z+
0
(x) e2 Im(w)xdx,
and B(S)(w) depends only on Im(w).
(d)(a). Compute the Berezin transform of HhSHhusing the formula HhKΠ,w =
KΠ,w+h:
B(HhSHh)(w) = hSHhKΠ,w, HhKΠ,wi
kKΠ,wk2=hSKΠ,w+h, KΠ,w+hi
kKΠ,w+hk2
=B(S)(w+h) = B(S)(w).
Since the Berezin transform is injective [8], HhSHh=S.
Corollary 2.4. The set of all vertical operators on L(A2(Π)) is a commutative C-algebra
which is isometrically isomorphic to L(R+).
6
3 Vertical Toeplitz operators
In this section we establish necessary and sufficient conditions for a Toeplitz operator to be
vertical.
Lemma 3.1. Let gL(Π). Then Tgis zero if and only if g= 0 almost everywhere.
Proof. The corresponding result for Toeplitz operators on the Bergman space on the unit
disk is well known, see, for example, [14, Theorem 2.8.2]. To extend it to the upper half-plane
case, we introduce the Cayley transform
ψ: Π D, w 7−wi
w+i,
the corresponding unitary operator
U:A2(D)→ A2(Π), f 7−(fψ)ψ0,
and observe that UTgU=Tgψ1.
The next elementary lemma gives a criterion for a function on Rto be almost everywhere
constant. We use there the Lebesgue measure in Rnfor various dimensions (n= 1,2,3),
indicating the dimension as a subindex: µn.
Lemma 3.2. Let f:RCbe a measurable function. Then the following conditions are
equivalent:
(a) There exists a constant cCsuch that f(x) = cfor almost all xR.
(b) µ2(D) = 0, where D:= (x, y)R2|f(x)6=f(y).
(c) µ1(Dx) = 0 for almost all xR, where Dx:= yR|f(x)6=f(y).
Proof. (a)(b). Let C={xR|f(x)6=c}. The condition (a) means that µ1(C) = 0.
Since D(C×R)(R×C), we obtain µ2(D) = 0.
(b)(c). Apply Tonelli’s theorem to the characteristic function of D.
(c)(a). Choose a point x0Rsuch that µ1(Dx0) = 0 and set c:= f(x0). Then f=c
almost everywhere.
Old proof of (b)(c). Denote by Φ the characteristic function of D. By Tonelli’s theorem,
ZR
µ1(Dx)dx =ZRZR
Φ(x, y)dydx =ZR2
Φ=µ2(D) = 0,
and µ1(Dx) = 0 for almost all xR.
7
Proposition 3.3. Let gL(Π). The operator Tgis vertical if and only if there exists a
function aL(R+)such that g(w) = a(Im(w)) for almost every wΠ.
Proof. Sufficiency. For every hR, define gh: Π Cby gh(w) = g(w+h). Then for
almost all wC
gh(w) = g(w+h) = a(Im(w+h)) = a(Im(w)) = g(w).
Applying the formula HhTgHh=Tghwe see that Tgis invariant with respect to horizontal
translations.
Necessity. Since Tgis vertical, for every hRwe have Tg=HhTgHh=Tgh. By Lemma
3.1, g=ghalmost everywhere. It means that for all hRthe equality µ2(Eh) = 0 holds
where
Eh:= (u, v)R2|g(u+h+iv)6=g(u+iv).
Define Λ: R2×R+Cby
Λ(u, x, v) := (0, g(x+iv) = g(u+iv);
1, g(x+iv)6=g(u+iv).
Then for all hR(u, v)Π|Λ(u, u +h, v)6= 0=Eh
and by Tonelli’s theorem
ZR2×R+
Λ(u, x, v)3(u, x, v) = ZR2×R+
Λ(u, u +h, v)3(u, h, v)
=ZRZΠ
Λ(u, u +h, v)2(u, v)dh =ZR
µ2(Eh)dh = 0.
Therefore ZR+ZR2
Λ(u, x, v)2(u, x)dv =ZR2×R+
Λ(u, x, v)3(u, x, v) = 0,
and for almost vR+
µ2({(u, x)R2|g(x+iv)6=g(u+iv)}) = ZR2
Λ(u, x, v)(u, x)=0.
For such v, by Lemma 3.2, there exists a constant c(v) such that g(u+iv) = c(v). Then for
a:R+Cdefined by
a(v) = (c(v),if µ2({(u, x)R2|g(x+iv)6=g(u+iv)})=0,
0,otherwise,
we have g(w) = a(Im(w)) for almost all wΠ.
8
We say that a measurable function g: Π Cis vertical if there exists a measurable
function a:R+Csuch that g(w) = a(Im(w)) for almost all win Π.
The next result was proved in [11, Theorem 3.1] (see also [14, Theorem 5.2.1]).
Theorem 3.4. Let g(w) = a(Im(w)) Lbe a vertical symbol. Then the Toeplitz operator
Tgacting on A2(Π) is unitary equivalent to the multiplication operator Mγa=RTgRacting
on L2(R+). The function γa=γa(s)is given by
γa(s) := 2sZ
0
a(t) e2ts dt, s R+.(3.1)
In particular, this implies that the C-algebra generated by vertical Toeplitz operators
with bounded symbols is commutative and is isometrically isomorphic to the C-algebra
generated by the set
Γ := γa|aL(R+).
4 Very slowly oscillating functions on R+
In this section we introduce and discuss the algebra VSO(R+) of very slowly oscillating
functions, and show that for any vertical symbol aL(R+), the associated “spectral
function” γabelongs to VSO(R+).
We introduce the logarithmic metric on the positive half-line by
ρ(x, y) := ln(x)ln(y):R+×R+[0,+).
It is easy to see that ρis indeed a metric and that ρis invariant under dilations: for all
x, y, t R+,
ρ(tx, ty) = ρ(x, y).
Recall that the modulus of continuity of a function f:R+Cwith respect to the metric
ρis defined for all δ > 0 as
ωρ,f (δ) := sup|f(x)f(y)| | x, y R+, ρ(x, y)δ.
Definition 4.1. Let f:R+Cbe a bounded function. We say that fis very slowly
oscillating if it is uniformly continuous with respect to the metric ρor, equivalently, if the
composition fexp is uniformly continuous with respect the usual metric on R. Denote by
VSO(R+) the set of such functions.
Proposition 4.2. VSO(R+)is a closed C-algebra of the C-algebra Cb(R+)of bounded
continuous functions R+Cwith pointwise operations.
9
Proof. Using the following elementary properties of the modulus of continuity one can see
that VSO(R+) is closed with respect to the pointwise operations:
ωρ,f+gωρ,f +ωρ,g, ωρ,f g ≤ kfkωρ,g +kgkωρ,f ,
ωρ,λf =|λ|ωρ,f , ωρ,f=ωρ,f .
The inequality ωρ,f (δ)2kfgk+ωρ,g(δ) and the usual “ ε
3-argument” show that VSO(R+)
is topologically closed in Cb(R+).
Note that instead of the logarithmic metric ρwe can use an alternative one:
Let ρ1:R+×R+[0,+) be defined by
ρ1(x, y) := |xy|
max(x, y).
It is easy to see that ρ1is a metric. To prove the triangle inequality ρ1(x, z) + ρ1(z, y)
ρ1(x, y)0, use the symmetry between xand yand consider three cases: x<y<z,
x<z<y,z<x<y. For example, if x<y<z, then
ρ1(x, z) + ρ1(z, y)ρ1(x, y) = (zy)(x+y)
yz >0.
The other two cases are considered analogously.
Lemma 4.3. For every x, y R+the following inequality holds
ρ1(x, y)ρ(x, y).(4.1)
Proof. The metrics ρand ρ1can be written in terms of max and min as shown below:
ρ(x, y) = ln max(x, y)
min(x, y), ρ1(x, y) = 1 min(x, y)
max(x, y).
Since ln(u)11
ufor all u1, the substitution u=max(x, y)
min(x, y)yields (4.1).
It can be proved that ρ(x, y)2 ln(2)ρ1(x, y) if ρ1(x, y)<1/2. Thus VSO(R+) could
be defined alternatively as the class of all bounded functions that are uniformly continuous
with respect to ρ1.
Theorem 4.4. Let aL(R+). Then γaVSO(R+). More precisely,
kγak≤ kak,
and γais Lipschitz continuous with respect to the distance ρ:
|γa(y)γa(x)| ≤ 2ρ(x, y)kak,(4.2)
that is
ωγa(δ)2δkak.(4.3)
10
Proof. The upper bound kγak≤ kakfollows directly from the definition (3.1) of γa. The
proof of (4.3) written below is based on an idea communicated to us by K. M. Esmeral
Garc´ıa. First, we bound |a(v)|by kak:
|γa(x)γa(y)| ≤ kakZ
02vx e2vx 2vy e2vydv
v.
Without lost of generality assume y > x, so the inequality
2vx e2vx 2vy e2vy 0
is true if and only if vv0:= 1
2
1
yxln y
x. Then
|γa(x)γa(y)| ≤ kakZv0
0
(2vy e2vy 2vx e2vx)dv
v
+kakZ
v0
(2vx e2vx 2vy e2vy )dv
v
= 2kake2v0x1e2v0(xy)
2kakρ1(x, y)2kakρ(x, y),
where the last inequality uses Lemma 4.3.
5 Density of Γin VSO(R+)
The set R+provided with the standard multiplication and topology is a commutative locally
compact topological group, whose Haar measure is given by (s) := ds
s.
For each nN:= {1,2, . . .}, we define a function ψn:R+Cby
ψn(s) = 1
B(n, n)
sn
(1 + s)2n,
where B is the Beta function.
Proposition 5.1. The sequence (ψn)
n=1 is a Dirac sequence, i.e., it satisfies the following
three conditions:
(a) For each nNand every sR+,
ψn(s)0.
(b) For each nN,Z
0
ψn(s)ds
s= 1.
11
(c) For every δ > 0,
lim
n→∞ Zρ(s,1)
ψn(s)ds
s= 0.
Proof. The property (a) is obvious, and (b) follows from the formula for the Beta function:
B(x, y) = Z
0
sx1
(1 + s)x+yds.
We prove (c). Fix a δ > 0. The function s7→ sn1
(1 + s)2nreaches its maximum at the point
sn:= n1
n+1 . It increases on the interval [0, sn] and decreases on the interval [sn,). Since
sn1, there exists a number NNsuch that eδ< sN. Let nNwith nN. Then
eδsNsn, and for all s(0,eδ] we obtain
sn1
(1 + s)2n(eδ)n1
(1 + eδ)2n.
Integration of both sides from 0 to eδyields
Zeδ
0
sn1
(1 + s)2nds eδ
(1 + eδ)2n
=1
4 cosh2(δ/2) n
.
Applying Stirling’s formula ([3, formula 8.327]), we have
1
B(n, n)=Γ(2n)
(Γ(n))2n
2
4n
πn .
Since cosh(δ/2) >1,
Zeδ
0
ψn(t)dt
t1
B(n, n)1
4 cosh2(δ/2) n
n
2πcosh2n(δ/2) 0.
To prove a similar result for the integral from eδto , make the change of variable s= 1/t:
lim
n→∞ Z
eδ
ψn(t)dt
t= lim
n→∞ Zeδ
0
ψn(s)ds
s.
Let
Rn,δ := Zρ(s,1)
ψn(s)ds
s,(5.1)
then
lim
n→∞ Rn,δ = lim
n→∞ Zeδ
0
ψn(s)ds
s+ lim
n→∞ Z
eδ
ψn(s)ds
s= 0.
12
Introduce now the standard Mellin convolution of two functions aand bfrom L1(R+, dν):
(ab)(x) := Z
0
a(y)bx
ydy
y, x R+,(5.2)
being a commutative and associative binary operation on L1(R+, dν).
Note that (5.2) is well defined also if one of the functions aor bbelongs to L(R+)
and the other belongs to L1(R+, dν). In that case abL(R+) and ab=ba. The
associativity law also holds for any three functions a, b, c such that one of them belongs to
L(R+) and the other two belong to L1(R+, dν).
The next result is a special case of a well–known general fact on Dirac sequences and
uniformly continuous functions on locally compact groups. For the reader’s convenience we
write a proof for our case.
Theorem 5.2. Let σVSO(R+). Then
lim
n→∞ kσψnσk= 0.(5.3)
Proof. For every nN,δ > 0 and xR+,
|(σψn)(x)σ(x)|=Z
0
σx
yψn(y)dy
yZ
0
σ(x)ψn(y)dy
y
Z
0σx
yσ(x)ψn(y)dy
y=I1+I2,
where
I1=Zρ(y,1)δσx
yσ(x)ψn(y)dy
y,
I2=Zρ(y,1)σx
yσ(x)ψn(y)dy
y.
If ρ(y, 1) δ, then ρ(x/y, x) = ρ(x, xy) = ρ(y, 1) δ. Thus
I1ωρ,σ(δ)ZR
ψn(y)dy
y=ωρ,σ(δ).
For the term I2we obtain an upper bound in terms of Rn,δ, see (5.1):
I22kσkZρ(y,1)
ψn(y)dy
y= 2kσkRn,δ.
Therefore
kσψnσkωρ,σ(δ)+2kσkRn,δ .
13
Given ε > 0, first apply the hypothesis that σVSO(R+) and choose δ > 0 such that
ωρ,σ(δ)<ε
2. Then use the fact that Rn,δ 0 and find a number NNsuch that Rn,δ <
ε
4kσkfor all nN. Then for all nN
kσψnσk<ε
2+ε
2=ε.
Recall now that, for each m, n N, the generalized Laguerre polynomial (called also
associated Laguerre polynomial) is defined by
L(m)
n(t) = 1
n!tmetdn
dtnettn+m=
n
X
j=0
(1)j(n+m)!
(nj)! (m+j)! j!tj, t R+.
Then, for each nN, we introduce the function φn:R+Cby
φn(t) = 1
(n1)! tnetL(n)
n1(t).(5.4)
Each function φnis obviously bounded and continuous on R+, and admits the following
alternative representation
φn(t) = 1
(n1)!2
dn1
dtn1ett2n1.
The next lemma relates the functions ψnand φnvia the Laplace transform L, which is
defined by
L(f)(s) := Z
0
f(t) est dt.
Lemma 5.3. For each nN,
ψn(s)
s=L(φn)(s), s R+.(5.5)
Proof. The function t7→ ett2n1and its first 2n2 derivatives vanish at 0 and +.
Integrating by parts n1 times we get
Z
0
dn1
dtn1ett2n1est dt =sn1Z
0
ett2n1est dt =sn1Γ(2n)
(1 + s)2n.
Therefore
L(φn)(s) = Γ(2n)
Γ(n)Γ(n)
sn1
(1 + s)2n=ψn(s)
s.
14
Given a function a:R+C, we define ea:R+Cas ea(t) = a(1/t).
The mapping a7→ eais obviously an involution:
e
ea=a, (5.6)
and, for all aL(R+) and bL1(R+, dν), we have
g
ab=eae
b. (5.7)
The change of variable t=1
uyields
Z
0
a(t)b(st)dt
t= (eab)(s).(5.8)
The next lemma relates “spectral functions” γawith Mellin convolutions.
Lemma 5.4. Let α(u)=2ue2u, then for each aL(R+),
γa=eaα. (5.9)
Proof. Rewrite γain the form
γa(s) = Z
0
a(t)2st e2stdt
t
and apply (5.8).
Introduce the function m2(s) := 2s, then (5.5) and (5.9) imply that the elements ψnof
the Dirac sequence are in fact certain “spectral functions”:
ψn=^
(φnm2)α=γφnm2.
Now we are ready to prove the main result of the paper.
Recall first that, by Theorem 3.4, the C-algebra generated by vertical Toeplitz operators
with bounded symbols is isometrically isomorphic to the C-algebra generated by the set
Γ = γa|aL(R+).
Theorem 5.5. We have that Γ = VSO(R+).
Proof. Let σVSO(R+). For each nN, we define an:R+Cby
an:= eσ(φnm2).
From (5.4) it follows that φnL1(R+, dν), and thus anL(R+). Then equations (5.7),
(5.6) and the associativity of Mellin convolutions yield
γan=eanα=e
eσ^
(φnm2)α=σ^
(φnm2)α=σψn,
which means that σnψnΓ. To finish the proof apply Theorem 5.2.
15
Let us mention some important corollaries of the theorem. First of all it implies that
the C-algebra VT (L) generated by Toeplitz operators with bounded vertical symbols is
isometrically isomorphic to VSO(R+). Moreover it shows that the set of initial generators of
VT (L) (i.e., the Toeplitz operators with bounded vertical symbols) is dense in VT (L).
That is, the two quite different types of the closures, the C-algebraic closure and the topo-
logical closure, of the set of initial generators end up with the same result: the C-algebra
VT (L) generated by Toeplitz operators with bounded vertical symbols.
Then, the theorem permits us to compare and realize the difference between the algebra
generated by general vertical operators and its subalgebra generated by special vertical
operators, Toeplitz operators with bounded vertical symbols. The first one is isomorphic to
L(R+), while the second, its subalgebra, is isomorphic to VSO(R+).
In this connection it is interesting to consider “intermediate”, in a sense, operators,
the bounded vertical Toeplitz operators whose defining symbols are unbounded. As it turns
out such operators do not necessarily belong to the algebra VT (L) generated by vertical
Toeplitz operators with bounded symbols.
The next section is devoted to an example of such an operator.
6 Example
Note that γacan be defined by the formula (3.1) not only if aL(R+), but also if
aL1(R+,eηt dt) for all η > 0.
In this section we construct a non-bounded function a:R+Csuch that aL1(R+,eηt dt)
for all η > 0 and γaL(R+), but γa/VSO(R+). This implies that the corresponding
vertical Toeplitz operator is bounded, but it does not belong to the C-algebra generated by
vertical Toeplitz operators with bounded generating symbols.
The idea of this example is taken from [4].
Proposition 6.1. Define f:{zC|Re(z)0} → Cby
f(z) := 1
z+ 1 exp i
3πln2(z+ 1),(6.1)
where ln is the principal value of the natural logarithm (with imaginary part in (π, π]).
Then there exists a unique function A:R+Csuch that AL1(R+,eηu du)for all η > 0
and fis the Laplace transform of A:
f(z) = Z+
0
A(u) ezu du.
Proof. For every zCwith Re(z)0 we write ln(z+ 1) as ln |z+ 1|+iarg(z+ 1) with
16
π
2<arg(z+ 1) <π
2. Then
|f(z)|=1
|z+ 1|exp i
3πln |z+ 1|+iarg(z+ 1)2
=1
|z+ 1|exp 2 arg(z+ 1)
3πln |z+ 1|
=1
|z+ 1|1+ 2 arg(z+1)
3π
.
Since |z+ 1| ≥ 1 and 1
3<2 arg(z+1)
3π<1
3,
|f(z)| ≤ 1
|z+ 1|2/3.
Therefore for every x > 0,
ZR|f(x+iy)|2dy ZR
dy
((x+ 1)2+y2)2/3<ZR
dy
(1 + y2)2/3<+,
and fbelongs to the Hardy class H2on the half-plane {zC|Re(z)>0}. By Paley–Wiener
theorem (see, for example, Rudin [6, Theorem 19.2]), there exists a function AL2(R+)
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
older’s inequality we easily see that AL1(R+,eηu du) for all η > 0:
Z+
0|A(u)|eηu du ≤ kAk2Z+
0
e2ηu du1/2
=kAk2
2η.
Proposition 6.2. The function σ:R+Cdefined by
σ(s) := s
s+ 1 exp i
3πln2(s+ 1),(6.2)
belongs to L(R+)\VSO(R+). Moreover there exists a function a:R+Csuch that
aL1(R+,eηt dt)for all η > 0and σ=γa.
Proof. The function σis bounded since |σ(s)| ≤ s
s+1 1 for all sR+. Let Abe the
function from Proposition 6.1. Define a:R+Cby
a(s) = A(2s).
17
Then for all η > 0
Z+
0|a(t)|eηt du =1
2Z+
0|A(t)|eηt/2dt < +,
and
γa(s)=2sZ+
0
a(t) e2st dt = 2sZ+
0
A(2t) e2st dt
=sZ+
0
A(t) est dt =s
s+ 1 exp i
3πln2(s+ 1)=σ(s).
Let us prove that σ /VSO(R+). For all s, t R+
|σ(s)σ(t)|=11
s+ 1exp i
3πln2(s+ 1)
11
t+ 1exp i
3πln2(t+ 1)
exp i
3πln2(s+ 1)exp i
3πln2(t+ 1)
1
s+ 1 1
t+ 1
=exp i
3πln2(s+ 1) ln2(t+ 1)11
s+ 1 1
t+ 1.
Replace sby the following function of t:
s(t) := t+t+ 1
ln1/2(t+ 1).
Then
ln(s(t) + 1) = ln(t+ 1) + ln 1 + 1
ln1/2(t+ 1)
= ln(t+ 1) + 1
ln1/2(t+ 1) 1
2 ln(t+ 1) +O1
ln3/2(t+ 1).
Denote ln2(s(t)+1)ln2(t+ 1) by Ltand consider the asymptotic behavior of Ltas t+:
Lt:= ln2(s(t) + 1) ln2(t+ 1) = 1 + 2 ln1/2(t+ 1) + O1
ln(t+ 1).
Since Ltis continuous and tends to +as t+, for every T > 40 there exists an integer
tTsuch that Lt+ 1 is equal to an integer multiple of 6π2, say to 62:
Lt+ 1 = 62.
18
For such t,
exp i
3πLt1=exp i
3π(621)1
=exp i
3π10.106 >1
10
and
|σ(s(t)) σ(t)| ≥ exp i
3πLt12
T+ 1 >1
10 1
20 =1
20.
It means that |σ(s(t)) σ(t)|does not converge to 0 as tgoes to infinity. On the other hand,
ρ(s(t), t) = ln s(t)
tt+ 1
tln1/2(t+ 1) 0.
Thus σ /VSO(R+).
References
[1] F. A. Berezin, Covariant and contravariant symbols of operators. Mathematics of the
USSR Izvestiya 6(1972), 1117–1151.
[2] F. A. Berezin, General concept of quantization. Communications in Mathematical
Physics 40 (1975), 153–174.
[3] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products. Academic
Press, New York, 1980.
[4] S. M. Grudsky, E. A. Maximenko, and N. L. Vasilevski, block Radial Toeplitz operators
on the unit ball and slowly oscillating sequences. Comm. Math. Anal. 14 (2013), 77–94.
[5] L. H¨ormander, Estimates for translation invariant operators in Lpspaces. Acta Math-
ematica 104 (1960), 93–140.
[6] W. Rudin, Real and Complex Analysis. McGraw-Hill, New York, 3rd Edition, 1987.
[7] R. Schmidt, ¨
Uber divergente Folgen and lineare Mittelbildungen. Math. Z. 22 (1925),
89–152.
[8] K. Stroethoff, The Berezin transform and operators on spaces of analytic functions.
Linear Operators, Banach Center Publications 38 (1997), 361–380.
[9] D. Su´arez, Approximation and the n-Berezin transform of operators on the Bergman
space. J. reine angew. Math. 581 (2005), 175–192.
19
[10] D. Su´arez, The eigenvalues of limits of radial Toeplitz operators. Bull. London Math.
Soc. 40 (2008), 631–641.
[11] N. L. Vasilevski, On Bergman-Toeplitz operators with commutative symbol algebras.
Integr. Equ. Oper. Theory 34 (1999), 107–126.
[12] N. L. Vasilevski, On the structure of Bergman and poly-Bergman spaces. Integr. Equ.
Oper. Theory 33 (1999), 471–488.
[13] N. L. Vasilevski, Bergman space structure, commutative algebras of Toeplitz operators,
and hyperbolic geometry. Integr. Equ. Oper. Theory 46 (2003), 235–251.
[14] N. L. Vasilevski, Commutative Algebras of Toeplitz Operators on the Bergman Space,
volume 185 of Operator Theory: Advances and Applications. Birkh¨auser, Basel–Boston–
Berlin, 2008.
[15] N. Zorboska, The Berezin transform and radial operators. Proceedings of the American
Mathematical Society 131 (2003), 793–800.
Crispin Herrera Ya˜nez
Departamento de Matem´aticas, CINVESTAV
Apartado Postal 14-740, 07000, M´exico, D.F., M´exico
cherrera@math.cinvestav.mx
Egor A. Maximenko
Escuela Superior de F´ısica y Matem´aticas, Instituto Polit´ecnico Nacional,
07730, M´exico, D.F., M´exico
maximenko@esfm.ipn.mx
Nikolai Vasilevski
Departamento de Matem´aticas, CINVESTAV
Apartado Postal 14-740, 07000, M´exico, D.F., M´exico
nvasilev@math.cinvestav.mx
20
... , such that each Toeplitz operator T a with G-invariant symbol a is unitarily equivalent to the multiplication operator by a certain function γ a , acting on L 2 (X G ; σ G ), R G T a R * G = γ a I. This function γ a is usually (see, e.g., [9]) called the spectral function of the Toeplitz operator T a . We recall the description of the spectral functions for the above three model cases. ...
... Returning to Toeplitz operators with bounded vertical symbols, the complete description of the those ϕ for which the operator ϕ(K) belongs to the C * -algebra generated by Toeplitz operator with such symbols is given in [9]. This happens (see e.g. ...
... Since VSO(R + ) is a proper subset of L ∞ (R + ), there exist bounded vertical operators which, although being Toeplitz, still do not belong to the C * -algebra generated by Toeplitz operator with bounded vertical symbols. An example of such operator for the unweighted Bergman space case, λ = 0, is given in [9,Section 6]. Let us recall some details. ...
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.
... (i) the C * -algebra generated by the set of Toeplitz operators with bounded radial symbols is isometrically isomorphic to the C * -algebra VSO(N) of slowly oscillating sequences, see [2,9,31]; (ii) the C * -algebra generated by the set of Toeplitz operators with bounded vertical symbols is isometrically isomorphic to the C * -algebra VSO(R + ) of very slowly oscillating functions on R + , see [11,12]; (iii) the C * -algebra generated by the set of Toeplitz operators with bounded angular symbols is isometrically isomorphic to the C * -algebra VSO(R), see [5]. ...
... Furthermore, in that case Theorem 1 gives an isometric characterization of the "time-frequency" case commutative algebra of TLOs as an analogy of the description of commutative C * -algebras of Toeplitz operators with radial, vertical and angular symbols acting on Bergman spaces. The proof of Theorem 1 is more general, but not so explicit in comparison with the constructions used in the proofs of density in [11] and [12]. Note that the method used in the proof of Theorem 1 yields an elegant and short proof of the main result of [11]. ...
... There is another connection between [12] and Theorem 1. Surprisingly, it is described in [18] that the case of Toeplitz operators acting on true poly-analytic Bergman spaces can be viewed as TLOs related to a special case of wavelets, namely, to the wavelets constructed from Laguerre functions on the half-line. It is also known from [18] that the C * -algebra generated by Toeplitz operators with bounded vertical symbols acting on true poly-analytic Bergman space of order k ∈ Z + = N ∪ {0} over the upper half-plane is commutative, and is isometrically isomorphic to the C * -algebra generated by the set ...
Article
Full-text available
We consider two classes of localization operators based on the Calderón and Gabor reproducing formulas and represent them in a uniform way as Toeplitz operators. We restrict our attention to the generating symbols depending on the first coordinate in the phase space. In this case, the Toeplitz localization operators (TLOs) exhibit an explicit diagonalization, i.e., there exists an isometric isomorphism that transforms all TLOs to the multiplication operators by some specific functions—we call them spectral functions. We show that these spectral functions can be written in the form of a convolution of the generating symbol of TLO with a kernel function incorporating an admissible wavelet/window. Using the Wiener’s deconvolution technique on the real line, we prove that the set of spectral functions is dense in the C\(^*\)-algebra of bounded uniformly continuous functions on the real line under the assumption that the Fourier transform of the kernel function does not vanish on the real line. This provides an explicit and independent description of the C\(^*\)-algebra generated by the set of spectral functions. The result is then applied to the case of a parametric family of wavelets related to Laguerre functions. Thereby we also provide an explicit description of the C\(^*\)-algebra generated by vertical Toeplitz operators on true poly-analytic Bergman spaces over the upper half-plane.
... Regarding the operator theory, our motivation to introduce the approximate invertibility concept is related to projects dealing with density of the range of certain convolution operator arising in the study of Toeplitz and Toeplitz-type operators acting on various function spaces (usually, the weighted Bergman spaces over the upper half-plane, or the unit disk in the complex plane, [18,20,19,28,29], as well as wavelet function spaces on the affine group [31]). In these cases approximate inverses for some particular convolutions have been constructed. ...
... In these cases approximate inverses for some particular convolutions have been constructed. In particular, the main step in recent papers [18,28,29,32] was to construct an appropriate Dirac net, and using this net to show a density result of a function algebra under consideration in certain C*-algebra (e.g., the algebra SO(N), or the algebra VSO(R + )). An idea of Wiener deconvolution technique on the real line has already been elaborated in [32]. ...
Preprint
Full-text available
We introduce a concept of approximately invertible elements in non-unital normed algebras which is, on one side, a natural generalization of invertibility when having approximate identities at hand, and, on the other side, it is a direct extension of topological invertibility to non-unital algebras. Basic observations relate approximate invertibility with concepts of topological divisors of zero and density of (modular) ideals. We exemplify approximate invertibility in the group algebra, Wiener algebras, and operator ideals. For Wiener algebras with approximate identities (in particular, for the Fourier image of the convolution algebra), the approximate invertibility of an algebra element is equivalent to the property that it does not vanish. We also study approximate invertibility and its deeper connection with the Gelfand and representation theory in non-unital abelian Banach algebras as well as abelian and non-abelian C*-algebras.
... Toeplitz operators with vertical symbols, which depend on y = Im z, and acting on Bergman type spaces have been studied. In [4][5][6][7][8][9], the authors proved that the algebra generated by Toeplitz operators with vertical symbols and acting on the weighted Bergman space A 2 λ ðΠÞ is isometrically isomorphic to the algebra VSOðℝ + Þ of all bounded functions that are very slowly oscillating on ℝ + . Taking vertical symbols having limit values at y = 0 and y = ∞, in [10,11], the authors found that ℝ + = ½0, +∞ is the spectrum of the algebra generated by all Toeplitz operators on the true-poly Bergman space A 2 ðnÞ ðΠÞ. ...
... Under this continuity condition, we will see that γ b is continuous on Π ≔ ℝ × ℝ + , where ℝ + = ½0, +∞ is the two-point compactification of ℝ + = ð0,+∞Þ. Apply the change of variable y 2 ↦ 2x 2 y 2 in the integral representation of γ b , then 4 Journal of Function Spaces ...
Article
Full-text available
We describe certain C∗-algebras generated by Toeplitz operators with nilpotent symbols and acting on a poly-Bergman type space of the Siegel domain D2⊂ℂ2. Bounded measurable functions of the form cIm ζ1,Im ζ2−ζ12 are called nilpotent symbols. In this work, we consider symbols of the form aIm ζ1bIm ζ2−ζ12, where both limits lims→0+bs and lims→+∞bs exist, and as belongs to the set of piecewise continuous functions on ℝ¯=−∞,+∞ and having one-side limit values at each point of a finite set S⊂ℝ. We prove that the C∗-algebra generated by all Toeplitz operators Tab is isomorphic to CΠ¯, where Π¯=ℝ¯×ℝ¯+ and ℝ¯+=0,+∞.
... Recall (see [9]) that the vertical operators S (acting on A 2 (Π)) are those that commute with the horizontal translation operators ( ...
... The C * -algebra generated by vertical Toeplitz operators with bounded symbols is isomorphic [9] to the algebra VSO(R + ), which consists of those functions in L ∞ (R + ) that are uniformly continuous with respect to the logarithmic metric ρ(x, y) = |ln x − ln y| . ...
Article
Full-text available
The definition of Toeplitz operators in the Bergman space $A^2(D)$ of square integrable analytic functions in the unit disk in the complex plane is extended in such way that it covers many cases where the traditional definition does not work. This includes, in particular, highly singular symbols such as measures, distributions, and certain hyper-functions.
... Using (52) we compute the spectral functions of vertical Toeplitz operators: [12]. The C*-algebra G in this example consists of all bounded functions on R + , uniformly continuous with respect to the log-distance, see [18,19]. ...
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.
... (see, for example, [6]). On the other hand, let n 2 R þ and k 1 ; k 2 2 Z þ with k 2 ! ...
Article
Let \(D_3\) be the three-dimensional Siegel domain and \({\mathcal {A}}_\lambda ^2(D_3)\) the weight-ed Bergman space with weight parameter \(\lambda >-1\). In the present paper, we analyse the commutative (not \(C^*\)) Banach algebra \({\mathcal {T}}(\lambda )\) generated by Toeplitz operators with parabolic quasi-radial quasi-homogeneous symbols acting on \({\mathcal {A}}_\lambda ^2(D_3)\). We remark that \({\mathcal {T}}(\lambda )\) is not semi-simple, describe its maximal ideal space and the Gelfand map, and show that this algebra is inverse-closed.
... Following [17] we introduce the logarithmic metric on the positive halfline: 12) and mention that the metric ...
Article
Full-text available
We give a detailed description of a \(C^*\)-algebra generated by Toeplitz operators acting on the weighted Bergman space over three-dimensional Siegel domain. The symbols of these generating Toeplitz operators belong to a certain class of functions being invariant under the action of a subgroup of the nilpotent group of biholomorphisms of the Siegel domain. In particular, we describe the canonical representation of each element of the \(C^*\)-algebra in question as well as the list of all it is irreducible representations.
Chapter
The separately radial Fock-Carleson type measures for derivatives of order k are introduced and characterized on the Fock space. Also, we study the separately radial Toeplitz operators generated by derivatives of k-FC type measure and give a criterion for Toeplitz operators to be separately radial. Finally, we show that the C*-algebra generated by these Toeplitz operators is isometrically isomorphic to a C*-subalgebra of the bounded sequences.
Chapter
We study Toeplitz operators acting on the harmonic Fock space and consider two classes of symbols: radial and horizontal. Toeplitz operators with radial symbols behave quite similar in both settings, namely, the Fock and the harmonic Fock spaces. In fact, these operators generate a commutative C∗-algebra which is isomorphic to the algebra of uniformly continuous sequences with respect to the square root metric. On the contrary, Toeplitz operators with horizontal symbols on the harmonic Fock space do not commute in general. Nevertheless, up to compact perturbation, they have a similar behavior to the corresponding Toeplitz operators acting on the Fock space. In fact, we prove that the Calkin algebra of the C∗-algebra generated by Toeplitz operators with horizontal symbols is isomorphic to the algebra consisting of bounded uniformly continuous functions with respect to the standard metric on ℝ, which, at the same time, is isomorphic to the C∗-algebra generated by Toeplitz operators with horizontal symbols acting on the Fock space.
Article
Full-text available
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.
Article
Full-text available
Let \mathbbD\mathbb{D} be the unit disk in A2 (\mathbbD)\mathcal{A}^2 (\mathbb{D}) be the Bergman space, consisting of all analytic functions from L2 (\mathbbD)L_2 (\mathbb{D}) , and B\mathbbD B_\mathbb{D} be the Bergman projection of L2 (\mathbbD)L_2 (\mathbb{D}) onto A2 (\mathbbD)\mathcal{A}^2 (\mathbb{D}) . We constructC *-algebras A Ì L¥ (\mathbbD)\mathcal{A} \subset L_\infty (\mathbb{D}) , for functions of which the commutator of Toeplitz operators [T a ,T b ]=T a T b –T b T a is compact, and, at the same time, the semi-commutator [T a ,T b )=T a T b –T ab is not compact.It is proved, that for each finite set AL\mathcal{A}_\Lambda of the above type, such that the symbol algebras Sym T(AL )\mathcal{T}(\mathcal{A}_\Lambda ) of Toeplitz operator algebras T(AL )\mathcal{T}(\mathcal{A}_\Lambda ) arecommutative, while the symbol algebras Sym R(AL ,B\mathbbD )\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} ) of the algebras R(AL ,B\mathbbD )\mathcal{R}(\mathcal{A}_\Lambda ,B_\mathbb{D} ) , generated by multiplication operators a Î ALa \in \mathcal{A}_\Lambda and B\mathbbD B_\mathbb{D} , haveirreducible representations exactly of dimensions n 0,n 1,..., n m .
Article
We illustrate how the Berezin transform (or symbol) can be used to study classes of operators on certain spaces of analytic functions, such as the Hardy space, the Bergman space and the Fock space. The article is organized according to the following outline. 1. Spaces of analytic functions; 2. Definition and properties Berezin transform; 3. Berezin transform and non-compact operators; 4. Commutativity of Toeplitz operators; 5. Berezin transform and Hankel or Toeplitz operators; 6. Sarason’s example.
Article
We analyze the connection between compactness of operators on the Bergman space and the boundary behaviour of the corresponding Berezin transform. We prove that for a special class of operators that we call radial operators, an oscilation criterion is a sufficient condition under which the compactness of an operator is equivalent to the vanishing of the Berezin transform on the unit circle. We further study a special class of radial operators, i.e., Toeplitz operators with a radial L1 (double-struck D sign) symbol.
Article
To any bounded operator S on the Bergman space La ² we associate a sequence of linear transforms Bn(S ) ∈ L ∞(D), where n ≧ 0, and prove that the Toeplitz operators
Article
We consider the covariant and contravariant symbols of operators that are a generalization of the Wick and anti-Wick symbols introduced earlier. With the help of these symbols we investigate the spectral characteristics of operators, and construct a method of successive approximations to calculate the exponential of an operator. In an application to anti-Wick symbols this method provides a basis of one of the variants of the Feynman functional integral along trajectories in phase space.