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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

y→1f(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 dµ =dxdy be

the standard Lebesgue plane measure on Π. Recall that the Bergman space A2(Π) is the

(closed) subspace of L2(Π, 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(w−z)2;

thus the Bergman (orthogonal) projection of L2(Π, dµ) onto A2(Π) is given by

(P f )(w) = hf, KΠ,wi.

Given a function g∈L∞(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|xm−xn|= 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

y→1f(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 h∈R, let Hh∈ L(A2(Π)) be the horizontal translation operator deﬁned by

Hhf(z) := f(z−h).

We call an operator S∈ L(A2(Π)) vertical (or horizontal translation invariant) if it com-

mutes with all horizontal translation operators:

∀h∈R, 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

e−ist f(t)dt,

being a unitary operator on L2(R).

For each h∈R, the translation operator τh:L2(R)→L2(R) is deﬁned by

τhf(s) := f(s−h).

An operator Son L2(R) is called translation invariant if τhS=Sτh, for all h∈R. 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

σ∈L∞(R) such that

S=F−1MσF. (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 MΘ−hare related via the Fourier transform,

MΘ−hF=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 h∈R:

MMΘh=MΘhM.

(b) Mis the multiplication operator by a bounded measurable function:

∃σ∈L∞(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

F−1M F τh=F−1MMΘ−hF=F−1MΘ−hMF =τhF−1MF,

which implies that F−1MF commutes with translations. Then (2.1) implies

F−1MF =F−1MσF.

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 h∈R:

MMΘ+

h=MΘ+

hM.

(b) Mis the multiplication operator by a bounded function:

∃σ∈L∞(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 x≤0.

For every h∈Rthe 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 h∈R:

J MP MΘh=J MMΘ+

hP=JMΘ+

hMP =MΘhJMP,

and by Lemma 2.2 there exists a function σ1∈L∞(R) such that JM P =Mσ1. Set σ=σ1|R+.

Then for all f∈L2(R+) and all x∈R+,

(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+),

(Rφ)(x) := √x

√πZΠ

φ(w) e−iwx dµ(w).

The operator Ris unitary, and its inverse R∗:L2(R+)→ A2(Π) is given by

(R∗f)(z) = 1

√πZR+pξf (ξ) eizξ dξ.

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:

∀h∈RSHh=HhS.

(b) RSR∗is invariant under multiplication by Θ+

hfor all h∈R:

∀h∈RRSR∗MΘ+

h=MΘ+

hRSR∗.

(c) There exists a function σ∈L∞(R+)such that

S=R∗MσR.

(d) The Berezin transform of Sis a vertical function, i.e., depends on Im(w)only.

Proof. (a)⇒(b). Follows from the formulas R∗MΘ+

h=HhR∗and RHh=MΘ+

hR.

(b)⇒(c). Follows from Lemma 2.2.

(c)⇒(d). Using the residue theorem we get

(RKΠ,w)(x) = −i√x

√πe−iRe(w)xe−Im(w)x.

Therefore

B(S)(w) = hMσRKΠ,w , RKΠ,w i

hKΠ,w, KΠ,w i= (2 Im(w))2Z+∞

0

xσ(x) e−2 Im(w)xdx,

and B(S)(w) depends only on Im(w).

(d)⇒(a). Compute the Berezin transform of H−hSHhusing the formula HhKΠ,w =

KΠ,w+h:

B(H−hSHh)(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], H−hSHh=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 g∈L∞(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−→ w−i

w+i,

the corresponding unitary operator

U:A2(D)→ A2(Π), f 7−→ (f◦ψ)ψ0,

and observe that U∗TgU=Tg◦ψ−1.

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:R→Cbe a measurable function. Then the following conditions are

equivalent:

(a) There exists a constant c∈Csuch that f(x) = cfor almost all x∈R.

(b) µ2(D) = 0, where D:= (x, y)∈R2|f(x)6=f(y).

(c) µ1(Dx) = 0 for almost all x∈R, where Dx:= y∈R|f(x)6=f(y).

Proof. (a)⇒(b). Let C={x∈R|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 x0∈Rsuch 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

Φdµ =µ2(D) = 0,

and µ1(Dx) = 0 for almost all x∈R.

7

Proposition 3.3. Let g∈L∞(Π). The operator Tgis vertical if and only if there exists a

function a∈L∞(R+)such that g(w) = a(Im(w)) for almost every w∈Π.

Proof. Suﬃciency. For every h∈R, deﬁne gh: Π →Cby gh(w) = g(w+h). Then for

almost all w∈C

gh(w) = g(w+h) = a(Im(w+h)) = a(Im(w)) = g(w).

Applying the formula H−hTgHh=Tghwe see that Tgis invariant with respect to horizontal

translations.

Necessity. Since Tgis vertical, for every h∈Rwe have Tg=H−hTgHh=Tgh. By Lemma

3.1, g=ghalmost everywhere. It means that for all h∈Rthe 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 h∈R(u, v)∈Π|Λ(u, u +h, v)6= 0=Eh

and by Tonelli’s theorem

ZR2×R+

Λ(u, x, v)dµ3(u, x, v) = ZR2×R+

Λ(u, u +h, v)dµ3(u, h, v)

=ZRZΠ

Λ(u, u +h, v)dµ2(u, v)dh =ZR

µ2(Eh)dh = 0.

Therefore ZR+ZR2

Λ(u, x, v)dµ2(u, x)dv =ZR2×R+

Λ(u, x, v)dµ3(u, x, v) = 0,

and for almost v∈R+

µ2({(u, x)∈R2|g(x+iv)6=g(u+iv)}) = ZR2

Λ(u, x, v)dµ(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 w∈Π.

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)) ∈L∞be a vertical symbol. Then the Toeplitz operator

Tgacting on A2(Π) is unitary equivalent to the multiplication operator Mγa=RTgR∗acting

on L2(R+). The function γa=γa(s)is given by

γa(s) := 2sZ∞

0

a(t) e−2ts 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|a∈L∞(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 a∈L∞(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 f◦exp 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+g≤ωρ,f +ωρ,g, ωρ,f g ≤ kfk∞ωρ,g +kgk∞ωρ,f ,

ωρ,λf =|λ|ωρ,f , ωρ,f∗=ωρ,f .

The inequality ωρ,f (δ)≤2kf−gk∞+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) := |x−y|

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) = (z−y)(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)≥1−1

ufor all u≥1, 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 a∈L∞(R+). Then γa∈VSO(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∞≤ kak∞follows 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)| ≤ kak∞Z∞

02vx e−2vx −2vy e−2vydv

v.

Without lost of generality assume y > x, so the inequality

2vx e−2vx −2vy e−2vy ≥0

is true if and only if v≥v0:= 1

2

1

y−xln y

x. Then

|γa(x)−γa(y)| ≤ kak∞Zv0

0

(2vy e−2vy −2vx e−2vx)dv

v

+kak∞Z∞

v0

(2vx e−2vx −2vy e−2vy )dv

v

= 2kak∞e−2v0x1−e2v0(x−y)

≤2kak∞ρ1(x, y)≤2kak∞ρ(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 dν(s) := ds

s.

For each n∈N:= {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 n∈Nand every s∈R+,

ψn(s)≥0.

(b) For each n∈N,Z∞

0

ψn(s)ds

s= 1.

11

(c) For every δ > 0,

lim

n→∞ Zρ(s,1)>δ

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

sx−1

(1 + s)x+yds.

We prove (c). Fix a δ > 0. The function s7→ sn−1

(1 + s)2nreaches its maximum at the point

sn:= n−1

n+1 . It increases on the interval [0, sn] and decreases on the interval [sn,∞). Since

sn→1, there exists a number N∈Nsuch that e−δ< sN. Let n∈Nwith n≥N. Then

e−δ≤sN≤sn, and for all s∈(0,e−δ] we obtain

sn−1

(1 + s)2n≤(e−δ)n−1

(1 + e−δ)2n.

Integration of both sides from 0 to e−δyields

Ze−δ

0

sn−1

(1 + s)2nds ≤e−δ

(1 + e−δ)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))2∼n

2

4n

√πn .

Since cosh(δ/2) >1,

Ze−δ

0

ψn(t)dt

t≤1

B(n, n)1

4 cosh2(δ/2) n

∼√n

2√πcosh2n(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→∞ Ze−δ

0

ψn(s)ds

s.

Let

Rn,δ := Zρ(s,1)>δ

ψn(s)ds

s,(5.1)

then

lim

n→∞ Rn,δ = lim

n→∞ Ze−δ

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):

(a∗b)(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 a∗b∈L∞(R+) and a∗b=b∗a. 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 σ∈VSO(R+). Then

lim

n→∞ kσ∗ψn−σk∞= 0.(5.3)

Proof. For every n∈N,δ > 0 and x∈R+,

|(σ∗ψn)(x)−σ(x)|=Z∞

0

σx

yψn(y)dy

y−Z∞

0

σ(x)ψn(y)dy

y

≤Z∞

0σx

y−σ(x)ψn(y)dy

y=I1+I2,

where

I1=Zρ(y,1)≤δσx

y−σ(x)ψn(y)dy

y,

I2=Zρ(y,1)>δ σx

y−σ(x)ψn(y)dy

y.

If ρ(y, 1) ≤δ, then ρ(x/y, x) = ρ(x, xy) = ρ(y, 1) ≤δ. Thus

I1≤ωρ,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):

I2≤2kσk∞Zρ(y,1)>δ

ψn(y)dy

y= 2kσk∞Rn,δ.

Therefore

kσ∗ψn−σk∞≤ωρ,σ(δ)+2kσk∞Rn,δ .

13

Given ε > 0, ﬁrst apply the hypothesis that σ∈VSO(R+) and choose δ > 0 such that

ωρ,σ(δ)<ε

2. Then use the fact that Rn,δ →0 and ﬁnd a number N∈Nsuch that Rn,δ <

ε

4kσk∞for all n≥N. Then for all n≥N

kσ∗ψn−σk∞<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!t−metdn

dtne−ttn+m=

n

X

j=0

(−1)j(n+m)!

(n−j)! (m+j)! j!tj, t ∈R+.

Then, for each n∈N, we introduce the function φn:R+→Cby

φn(t) = 1

(n−1)! tne−tL(n)

n−1(t).(5.4)

Each function φnis obviously bounded and continuous on R+, and admits the following

alternative representation

φn(t) = 1

(n−1)!2

dn−1

dtn−1e−tt2n−1.

The next lemma relates the functions ψnand φnvia the Laplace transform L, which is

deﬁned by

L(f)(s) := Z∞

0

f(t) e−st dt.

Lemma 5.3. For each n∈N,

ψn(s)

s=L(φn)(s), s ∈R+.(5.5)

Proof. The function t7→ e−tt2n−1and its ﬁrst 2n−2 derivatives vanish at 0 and +∞.

Integrating by parts n−1 times we get

Z∞

0

dn−1

dtn−1e−tt2n−1e−st dt =sn−1Z∞

0

e−tt2n−1e−st dt =sn−1Γ(2n)

(1 + s)2n.

Therefore

L(φn)(s) = Γ(2n)

Γ(n)Γ(n)

sn−1

(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 a∈L∞(R+) and b∈L1(R+, dν), we have

g

a∗b=ea∗e

b. (5.7)

The change of variable t=1

uyields

Z∞

0

a(t)b(st)dt

t= (ea∗b)(s).(5.8)

The next lemma relates “spectral functions” γawith Mellin convolutions.

Lemma 5.4. Let α(u)=2ue−2u, then for each a∈L∞(R+),

γa=ea∗α. (5.9)

Proof. Rewrite γain the form

γa(s) = Z∞

0

a(t)2st e−2stdt

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=^

(φn◦m2)∗α=γφn◦m2.

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|a∈L∞(R+).

Theorem 5.5. We have that Γ = VSO(R+).

Proof. Let σ∈VSO(R+). For each n∈N, we deﬁne an:R+→Cby

an:= eσ∗(φn◦m2).

From (5.4) it follows that φn∈L1(R+, dν), and thus an∈L∞(R+). Then equations (5.7),

(5.6) and the associativity of Mellin convolutions yield

γan=ean∗α=e

eσ∗^

(φn◦m2)∗α=σ∗^

(φn◦m2)∗α=σ∗ψn,

which means that σn∗ψn∈Γ. 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 a∈L∞(R+), but also if

a∈L1(R+,e−ηt dt) for all η > 0.

In this section we construct a non-bounded function a:R+→Csuch that a∈L1(R+,e−ηt dt)

for all η > 0 and γa∈L∞(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:{z∈C|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 A∈L1(R+,e−ηu du)for all η > 0

and fis the Laplace transform of A:

f(z) = Z+∞

0

A(u) e−zu du.

Proof. For every z∈Cwith 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 {z∈C|Re(z)>0}. By Paley–Wiener

theorem (see, for example, Rudin [6, Theorem 19.2]), there exists a function A∈L2(R+)

such that for all x > 0

f(x) = Z+∞

0

A(u) e−ux du.

The uniqueness of Afollows from the injective property of the Laplace transform. Applying

H¨older’s inequality we easily see that A∈L1(R+,e−ηu du) for all η > 0:

Z+∞

0|A(u)|e−ηu du ≤ kAk2Z+∞

0

e−2ηu 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

a∈L1(R+,e−ηt dt)for all η > 0and σ=γa.

Proof. The function σis bounded since |σ(s)| ≤ s

s+1 ≤1 for all s∈R+. 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)|e−ηt du =1

2Z+∞

0|A(t)|e−ηt/2dt < +∞,

and

γa(s)=2sZ+∞

0

a(t) e−2st dt = 2sZ+∞

0

A(2t) e−2st dt

=sZ+∞

0

A(t) e−st 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)|=1−1

s+ 1exp i

3πln2(s+ 1)

−1−1

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)−1−1

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

t≥Tsuch that Lt+ 1 is equal to an integer multiple of 6π2, say to 6mπ2:

Lt+ 1 = 6mπ2.

18

For such t,

exp i

3πLt−1=exp i

3π(6mπ2−1)−1

=exp −i

3π−1≈0.106 >1

10

and

|σ(s(t)) −σ(t)| ≥ exp i

3πLt−1−2

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)

t≤t+ 1

tln1/2(t+ 1) →0.

Thus σ /∈VSO(R+).

References

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USSR Izvestiya 6(1972), 1117–1151.

[2] F. A. Berezin, General concept of quantization. Communications in Mathematical

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[3] I. S. Gradshteyn and I. M. Ryzhik, Tables of Integrals, Series, and Products. Academic

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[5] L. H¨ormander, Estimates for translation invariant operators in Lpspaces. Acta Math-

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[6] W. Rudin, Real and Complex Analysis. McGraw-Hill, New York, 3rd Edition, 1987.

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[10] D. Su´arez, The eigenvalues of limits of radial Toeplitz operators. Bull. London Math.

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[12] N. L. Vasilevski, On the structure of Bergman and poly-Bergman spaces. Integr. Equ.

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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