Advances in Pure Mathematics, 2011, 1, 325-333
doi:10.4236/apm.2011.16059 Published Online November 2011 (http://www.SciRP.org/journal/apm)
Toeplitz and Translation Operators on the q-Fock Spaces*
Fethi Soltani
Higher College of Technology an d Informatics, Tunis, Tunisia
Received June 25, 2011; revised July 12, 2011; accepted July 28, 2011
Abstract
In this work, we introduce a class of Hilbert spaces of entire functions on the disk
q
1
,1 Do q





,
, with reproducing kernel given by the q-exponential function 0< <1q


q
ez; and we prove some proper-
ties concerning Toeplitz operators on this space. The definition and properties of the space extend
naturally those of the well-known classical Fock space. Next, we study the multiplication operator by
and the q-Derivative operator
q

Q z
q
D
on the Fock space ; and we prove that these operators are ad-
joint-operators and continuous from this space into itself. Lastly, we study a generalized translation operators
and a Weyl commutation relations on .
q

q

Keywords: q-Fock Spaces, q-Exponential Function, q-Derivative Operator, q-Translation Operators,
q-Toeplitz Operators, q-Weyl Commutation Relations
1. Introduction
In 1961, Bargmann [1] introduced a Hilbert space
of entire functions


=0
=n
n
n
f
za

z
on  such that
22
=0
:=! <.
n
n
fan



On this space the author study the differential operator
=d dDz and the multiplication operator by , and
proves that these operators are densely defined, closed
and adjoint-operators on (see [1]). Next, the Hilbert
space is called Segal-Bargmann space or Fock
space and it was the aim of many works [2,3].
z


In this paper, we consider the q-exponential function:
 

=0
1
:= ,
;
n
n
qnn
q
ez z
qq


where


11
=0
;:=1,=1,2,,.
ni
ni
qqq n



We discuss some properties of a class of Fock spaces
associated to the q-exponential function and we give
some applications.
In the first part of this work, building on the ideas of
Bargmann [1], we define the q-Fock space q as the
space of entire functions


=0
=n
n
n
f
za

z on the disk
1
,1 Do q






 of center oand radius 1
1d such
that
q, an


2 2
=0
;
:=< .
1
n
nn
qn
qq
fa
q




f
and
g
Let be in, such that q 

=n
n=0
n
f
zaz
 and

=0
=n
n
n


g
zbz

, the inner pro-
duct is given by


=0
;
,=
1
n
nn n
qn
qq
fgab q




.
The q-Fock space has also a reproducing kernel
q

q given by 
 
1
,=;,,
1
qq
wzewzwzD oq






.
*Author partially supported by DGRST project 04/UR/15-02 and
CMCU program 10G 1503.

F. SOLTANI 326
Then if , we have
q
f
 
1
,,.= ,,
1
q.fw fwwDoq






Using this property, we prove that the space is a
Hilbert space and we give an Hilbert basis.
Next, we define and study the Toeplitz operators of
th
nsider the
m
he Focpaceand we prove that
th
q

q

e q-Fock space q .
In the second part of this work, we co
ultiplication operator Q by z and the q-Derivative
operator q
D on tk s q ,
ese operators are continuous from q
 into itself, and
satisfy:
11
,.
q
Dff Qff
 

11
qq q
qqq


Then, we prove that these operators are adjoint-ope-
rators on :
q

,=,;, .
qq
qq
Qf gf Dgf g


Next, we define and study on the Fock space , the
q-translation operators:
q

  
1
:=; ,,,
zqq
fwezDfwwzDo 



1q




and the generalized multiplication operators:
 
1
:=; ,,.
1
zq
Mfwe zQfwwzDo




q



Using the previous results, we deduce that the ope-
rators
z

and
z
M
, for 1
,1 zDoq





from into itself, and satisfy:
, are continu-
ous q

,
1
zq
q
fe f
q
 




.
1
q
zq q
q
z
z
Mf ef
q







Lastly, we establish Weyl commutation relations be-
tween the translation operators a

and the multipli-
cation operators b
M
, where 1 ,,
1
abD oq 



. These
relations are realized on the Fock s

pace
2. The q-Fock Spaces and the Toeplitz
Le e real numbers such that ; the
-shifted factorial are defined by
alogue of the Gamma
function as
q
.
q
Operators
2.1. Preliminaries
t a and q b0< <1q
q
 

1
0
;:=1, ;:=1, =1,2,,.
ni
n
aqaqaq n


=0i
Jackson [4] defined the q-an





1
;
:=1,0, 1, 2,
;
qx
qq

It satisfies th
x
qq
xqx



e functional equation




 
1=, 1=1,
qqq
q
xxx 
where

1
:= ;
1
x
qq
q
x
and tends to


x

, we have
when tends to. In particular,
for
q 1
=1,2,n





;
1== !.
1
n
qnq
q
Tbinatorial
qq
nn
he q-comcoefficients are defined for
n

 and , by =0,,kn


 
!
:= .
!!
qqq
kkn k
 

q-derivati of a s
q
n
n

The ve uitable function
q
Df
f
(see
[5]) is given by




:=, 0,
1
qqx
and
fx fqx
Df xx





0
q0=Df f

provided exists.
If

0f
f
is differentiable then tends to

Df x
q


f
x

as 1q


There
.
two important qs of the e
l function [5]:
are-analoguexpon-
entia



1/2
=0 !
qnq
n
:= ,
n
z
Ez q

nn


 =0
:= .
!
n
qnq
z
ez n


Note that the first series converges for <z

and
the second series converges for 1 <1 zq.

F. SOLTANI327
gral rTherefore the function q  has the q-inteepre-
sentation [6]:
 
1
1
1
0
=d,>0,
q
qq q
xr qrrx


 (1)
where the q-integ
xE 
ral (introduced by Jackson [4]) is
defined by
Lemma 1. The function
 

0=0
d=1 .
qn
fx xqaqfaq

ann




.
q
e

, 1 ,1 Do q 





, is
the unique analytic solution of the q-problem:
1.
he form
Replacing in (2ain

=
q
Dyzy  (2)
 
,0=yz
Proof. Searching a solution of (2) in t

=n yz az

. Then
=0 n n
 
=1
=.
qn
q
n
Dyza nz

), we obt
1n


1
=1 =1
=.
nn
nn
q
nn
anz ax 



Thus,


1
=,=1,2,
nn
q
anan 

We deduce that

1
=.
nn
q
aa
n


We get

=.
!
n
n
q
an

Therefore,
 


=0
==
!
n
q
nq
z
yze z
n


 ,
which completes the proof of the lemma. □
2.2. The q-Fock Spaces
e denote by
q
W
1


,1 HDo q






the space of entire functions on


1
,1 Do q




 efined on


.
the measure d
q
m 1 ,1 Do q





by
 
1
:=,=e.
2π
i
qqq
dmzEqrdzrrd




21
,,
1q
LDo m
q









functions
the space of measurable
f
on 1
,Do



satisfying
1q

 
2
2
1
, 1
2,,
:= <
q
Do
LDom
ffzdmz 

 

 1
1
.
q
q
q



 





Definition 1. We define the prehilbertian space
to be the space of functions in
q, 
2
11
,,,
1q 1
H
DoL Dom
q
 


 
 


 

, equippe
q

 


d
with the inner product
  1
,1
,=
qq
Do q
,
f
gfzgzd






 mz
and the norm
 
1/2
2
1
,1
=.
q
f

q
Do q
fzdmz












Remark 1. If

1q


’s space
, the space agrees with the
Segal-Bargmann (see [1]).
Proposi t ion 1. 1) For all such that
q

q

=0
=n
n
n
f
f
zaz

ve we ha,

22
=0
=!.
qn
fa


nq
n (3)
2) For all

,q fg



=0
=n
n
n
f
zaz


such that
and

==0
n
n
n
g
zbz
, we
 have

=0
,=
qn
fg 
!.
nn q
abn
 (4)
3) For ,q
fg

, we have

  
,=0, =
qq.
f
gfDggzgz


Proof. Given and
ed convergence theoremave

=0
=n
nq
n
fz az

n
nq
z.


=0
=n gz b


1) By dominat’s, we h

2
1
,
,=0 1
qmn q
Do
mn q
=.
n
m
aazzdm z








f

F. SOLTANI 328
We put =e
i
zr

, then we deduce

1
221
0
=0
nq
=d.
q
n
qq
n
f
arEqr
r




But from (1), we have
  
1
1
0d=
qq
rE qr
1= !.
n
qqq
r nn

Thus,

22
=0
=!
qnq
n
fan


 .
2) We obtain the result from (1) by polarization.
3) Since


1
=,1,
kk
qq
Dzk zk

then



!
=,
!
q
nk kn
q
q
k
Dzzk n
kn


,
(5)
and




=
!
=.
!
q
n
qk
kn q
k
Dgz bz
kn



k
n
Thus,



0
=,
!
n
q
n
q
Dg
bn
and
 
 =0
0
=.
!
n
qn
nq
Dg
g
zz
n

 (6)
Using (4) and (6), we get
 
=0
qnq
n

=0
,=0= 0.
nn
nq
n
fg aDgaDg



Thus



,= 0
qq
fgfD g

 ,
whi the desired result. □
The following theorem prove that is a repro-
ducing kernel space. iven for
ch gives
q

Theorem 1. The function  g
q
1
,,wzD o



, b
1q

y
 
,=
q
wz ,
q
e wz
is a reproducing kernel for the q-Fock space , that is:
1) for all
q

1
,1q



wDo

 , the function


,zw
qz belongs to
2) For all
q .
1
,1



wDo q

 and, we have q f


=

,,. .
q
q
f
wf


Proof. 1) Since
w


=0
,=
qn
wz n

1
; ,,,
!1
nn
q
wz zwD oq





(7)
then from (3), we deduce that



2
22
=< ,
n
q
ww

which proves 1).
2) If , from (4) and7), we
deduce
=0
,. =!
q
q
nq
we
n




=0
=n
nq
n
fz az


(
 
=0
, .==1
q
n
n
awf q





This completes the proof of the theorem. □
Remark 2 . From Theorem 1 (2), for and
1
,,, .
qn
fw wwDo




q
f
1
,Do

, we have
1q



w
 

1/2
2
= .
q
qq
fwe wf




 (8)
Proposition 2. The space equipped with the
inner product
,. qq w f


q 
.,. q
 is anace; and th Hilbert spe set



nn



given by

1
=,, ,
n
nz
zzDo 



1
!
qq
n

forms an Hilbert basis for the space
Proof. Let
q
.


nn


 be a Caucequence in
We put
), we have
hy s q .
=,
l
ffin .
im nq
n

From (8
 

1/2
2.
q
npnqnp n
fwfw ewff


 

 


nn
f

This inequality shows that the sequence  is
pointwise convergent to . Since the function f


1/2
2
q
ew 1
,1
Do q





hen w






 is continuous on 


, t

F. SOLTANI329
converge compact
of

nn
fs to f uniformly on allset
1
,1q
. Consently, f is an entire Do




que
ction on fun1 ,1q
, then belongs to theDo




space
er hand, fromti
f
q
On the oth the relaon (4), we get
.
,
,=
q
nm nm

 ,
where ,nm

is show
is the Kronecker symbol.
Ths that the family


nn

 is an orthonormal
set in q .
Let

=n az

 be an eleme

=0 n n
fz nt of q  such
that
,=0, .
q
n
fn 

From the relation (4), we deduce that
2.3. Toepliz Operators on
this paragraph we study the Toeplitz operators on
cla sical Toeplitz op
rs [2].
=0,.
n
an
This completes the proof. □
q
In q .
era- These operators generalize thes
to
First we define the orthogonal projection operator
from
P
21
L

,,
Do m
1q
q




into , by




q

 
1
2
,. LD m
q


,,
1
:= ,,
1
,,
1
qoq
PfwfKw
wDo q




 

 





where q
K
finiti is the reproducing kernel given by (7). Deon 2. Let

be a measurable function on
1
, Do



. The Toeplitz operator T
1q 


is the operator
given by


:= ,TfP f


for every

21
:=:, ,
1
qq
fDT

 f fLDom
q





 









.
Remark 3 . Let 1
,1
LDo q 


 





.


1) The operator is bounded and T



. T

2) By derivation a under the integral signnd using (2),
we have =
z
q
TD.
Theorem 2. If 1
,1 LDo q


 
hen







has compact
support, tT

is a compact operator.
Proof. For 1
,1
LDo q 
have






, we


 
1
2,,
1
1
,1
,
=d
nk
LDo m
q
q
nk q
Do q
T
Tw wmw






 











.
Since


 
1
,1
=,
n
nq q
Do q
Tw
zzKwzmz


 





d.
Applying Fubini's theorem and Theorem 1, we obtain
1
2,,
1
1
2
=, .
nk
 
 

,,
1
,
nk
LDo m
q
q
LDo m
q
q
T



 




 
 

 
Thus,
2
1
2,,
,=0 1
2
1
2,,
,=0 1
,
=,
nk
LDo m
q
nk q
nk
LDo m
q
nk q
T


 



 













 


.
Since 1 ,1 LDo q 


 





with compact support,
there are positive constants and a
K
so that


,..zK


z  =0, ae 
and for all >za . Then for
,nk

, we get
 
 
1
2,
,
1
=d
nk
LDo q
 

,
1
|| .
!!
m
q
k
nq
za
qq
zzz mz
nk








Thus,





F. SOLTANI 330
 

 
 

1
2,,
1
||
2()/2
0
1
1
0
,
d
!!
()d
!!
d.
!!
nk
LDo m
q
q
nk
q
za
qq
ank qq
qq
nk
qqq
qq
Kzmz
nk
KrEqr
nk
Ka Eqrr
nk
 
 


 














r
But from (1), we have
 
1
1
0d= 1=1.
qqqq
Eqrr


Hence
 
1
2,,
1
,.
nk
LD oq
 

!!
nk
m
qqq
Ka
nk






Thus, we obtain








2
2
22
1
2,,
,=0 1
,<
nk q
LDo m
q
nk q
TKe

 .a







 




Then,

T

is an Hilbert-Schmidt operator [7], and
consequepact. □
3. The Multiplication and Translation Operators
on
3.1. The Derivative and Multiplication
n , we consider the multiplication operator
Lq
ntly it is com
q
Operators on q
Oq 
given by
Q
 
:= .Qf zzf z
By straightforward calculation we obtain.
emma 2. ,DQ D
= =
qqq
QQD



 , where q

is thift operator given by
 
e q-sh
:= .
q
f
zfqz
This lemma is the q-analogous cmmutation rule of
[1]. When , then tends to the identity
op
o
1q ,DQ

q

erator
I
.
We now study the continuousy of the ope
rators q , q D and Q on q 
propert-
.
If th Theorem 3. q fen q
f
, and
bel q
Df Qf
ong to q , and we have
1) q
q
2) 1
1
qq
q
Df f
q

,
3) 1
1
Qf q


.
qq
f
of. Let
1) We have
,
Pro n



=0
=n
nq
fz az.
 
=0
==
nn
qn
n

f
zfqz aqz


anom (3), we obtain d fr
 
2=
qn
f
22 2
2
0 =0
!!=.
n
n
qq
q
aqnanf



2) We have
1. (9)
Then from (9), we get
=
qnn

  
1
1
=0
==
nn
qn n
qq
n
Dfzanza nz




=1 n



2
22
1
=0
=1
q
qn
qq
n
Dfann



 !.
Since






1!= 1!,
qq
nnn
q
w
(10)
e obtain

22
1
=0
=1
q
qn
qq
n
Dfann



1!,
equently, and cons

22
=0
=!. (11)
q
qn
qq
n
Dfan n



Using the fact that

1
1
q
nq , we obtain

1/2
2
=0
1
Df  1 !=.
11
q
qn
q
n
a nf
qq






3) On the other hand, since
(12)
then
q

,
n
Qf
1
=1
=
n
n
zaz



 
22 2
1
=1
qn
By (10), we deduce
=0
=!=1!.
nn
qq
n
Qfana n





22
=0
=1!.n (13)
qnqq
n
Qfan



Using the fact that

1
11
q
nq 
, we obtain
q
f
f
,


F. SOLTANI331
1.
1q
q
Qf f
q


□
We deduce also the following norm equality.
Theorem 4. 1) If then
q
f
2
2
2=.
qq
q
Qf f

q
q
D f

2) The operator is injective on
Proof. Let
1) By (13) and using the fact that
:qq Q
n
q
.

=0
=nq
n
fz az
.




1= n
qq
nn , q
we obtain
 
2
=0
.
qq
nq
q qq
n
QqD ff



2) From (1), we have
2
2
2=!=
qn
fann




2
.f
2
qq
q
Qf 

. Then
ator on □
e operators ad-
joint-ope rators on and for all e have
Therefore =0Qf implies that f
:qq Q is injective continuous op
Propositi
=0
er
Q and q
.
re on 3. Th q D a
q
, w
q
; ,fg
,=,
qq
q
QfgfD g
 .
Proof. Consider

=0
=n
n
n
f
za

z and

=0
=n
n
n
g
zb

z in q
. From (9) and (12),
.
n

=0
=Qf zz


, =1
n
azDgzbn

11
=1
nq n
q
nn

Thus from (4), we get




1
=1
=0
,=!
=
qnn q
n
n
Qf gabn
a n





which gives the result. □
3.2. The Translation Operators on
In this section we study a generalized translation ope-
rators on We begin by the following definition.
Definition 3. For and
11
!=,,
q
nn q
q
bf Dg


q
q
.
q
f1 ,,
1
wzD oq





,
perators on , by we define the q-translation oq 


 

=0
:== n
zq q q
n
fw e zDfwDfn


For 1
,,
1
wz Doq




, the function q e satisfies

the following product formula:


=.
zqq q
e w

ew ze 

Proposi t ion 4. Let

=0
=n
nq
n
fz az

 and
1
,,
1
zwD oq





. Then

=0
=.
nnk k
zn
n
n
=0 kq
f
wa 


wz
k



Proof. Let

=0
=n
n
n
fz az

q
. From (14), we
have




 =0
1
= .
1
q
zn
Df
fw nq



; ,,
!
nn
q
wzwzDo




But from (5), we have



 =
!
=.
!
q
nk
qk
kn q
k
Dfwaw
kn


n
in Thus we obta



 
=0 =0
.
!
n
q
z
w

(14)
=0 =0
n
nk
!
=!!
=.
nqnk k
zn
nk qq
nnk k
q
n
f
wa wz
knk
n
awz
k









□
Definition 4. For and 1 ,,
1
wzD oq
q
f

 



,
we defi
 The generalized multiplication operators on , by
ne:
q



=0
:= =.
!
n
n
zq nq
z
Mfwe zQfwQfwn


 The generalized shift operators on , by
q

   
 =0
zq n
:= =.
!
n
n
q q
q
z
Sf wezf wf wn



g to Theorem 3 we study the continuous Accordin
oprperty of the operators
z

,
z
M
and
z
S on q .
and 1 ,1 zDoq Theorem 5. If q f

 



en , th
z
f

,z
M
f and belong to and we have
1)
z Sf q ,
||z
1
qq
zq
f
ef
q
 




,


F. SOLTANI 332
2) 1
qq
zq
z
M
fef
q






,
3)

qq
zq
Sfe zf

.
Proof. From (14) and Tore3 (2), we deduc hem e

 
/2
qq
zq n
zz
f f



fore,
=0 =0
!1!
q
nn
qq
nqn.
nn
n
fD 



There
|| ,
1
qq
zq
z
fef
q







which gives the first inequality, and as in the same
way we prove the second and the third inequalities of
this theorem. □
s.
Proposi t ion 5. For all
From Proposition 3 we deduce the following result
,q fg

, we have
,=,
qq
zz
Mfgf g 

,
,=,
qq
zz
fg fMg 

.
We denote by
z
R the following opera defined on
by
tor
q


 

:=
=.
zzz
zz
qqqq qq
RMM
ezDezQ ezQezD

 

Then, we prove the following theorem.
Theorem 6. For all f have
q
, we
22
=,
qq
zz z
Mff fRf  
 .
q
Proof. From Proposition 5, we get


2=,=,
qqq
zz zz
z
2
=,.
qq
z
zz
M
ffMffMRf

ff Rf




□
3.s on
Let
3. The Weyl Commutation Relationq
1
,,
1
abD oq





. In this paragraph e establish
Wn the translation
operators
w
eyl commutation relations betwee
a

and the multiplication operators b
M
.
These relations are realized on the Fock space
Lemma 3. For
q
.
1
,,
1
abD oq





, we have
1)


1
,=,= 1,2,
nn
qq
q
DQ nQn

 
 .
2) ,=
qb bq
DM bM


 .
Proof. 1) From Lemma 2, for we deduce that =1,2,n,
11
,= ,=
nn
nk
qq q
DQQ DQQQQ

 



11
.
nk knk  
=0 =0 kk
Since
=,
qq
QqQ



we get


1
,=
nn
qq
q
DQ nQ

 
 .
Which proves the first equality.
e have 2) W

=1
,= ,
!
nn
qb q
nq
.DM DQ
n
 

Using (1), we obtain
b



1
=1
=0
,= 1!
==.
[]!
n
n
qb q
nq
n
nqb
nq
b
DMQ n
b
bQb q
n


 
 



□
Theorem 7. For
M
1
,,
1
abD oq





, we have
=.
ab baab
M
MS


Proof. From Lemma 3 (2), we have
Then, for

=.
qbbqq
DMM Db
0,1, 2,n

, we deduce
Multiplying by

=.
n
n
qbbqq
DMM Db

!
n
q
a
n and summing, we get
Since D

=.
abbq qq
MMeaDab  
=Dq
qq qq

, from [5] we get




 

==
qqq qqqq aab
eaD abeaDeabS  ,
which completes the proof of the theorem. □
. If Remark 41q

, we obtain the classical commu-
tation relations [8]:


,=,=;,
aD bQab bQ aD
DQIeee eeab .
4. References
[1] V. Bargmann, “On a Hilbert Space of Analytic Functions

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an Associated Intart I,” Commu-
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[2] C. A. Berger and L. A. Coburn, “Toeplitz O
rgmann S
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] G. H. Jackson, “On a q-Definite Integrals,” Quarterly
nctions,”
[4 Journal of Pure and Applied Mathematics, Vol. 41, 1910,
pp. 193-203.
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Cambridge University Press, Cambridge, 1999.
[7] M. Naimark, “Normed Rings,” Noordhoff, Groningen,
1959.
[8] T. Hida, “Brownian Motion,” Springer-Verlag, Berlin,
1980.