Advances in Pure Mathematics, 2011, 1, 221-227
doi:10.4236/apm.2011.14039 Published Online July 2011 (http://www.SciRP.org/journal/apm)
Copyright © 2011 SciRes. APM
Multiplication and Translation Operators on the Fock
Spaces for the q-Modified Bessel Function*
Fethi Soltani
Higher College of Technology and Informatics, Street of the Employers, Tunis, Tunisia
E-mail: fethisoltani10@yahoo.com
Received February 28, 2011; revised April 8, 2011; accepted April 20, 2011
Abstract
M
by and the -Bessel operator
2
z
We study the multiplication operator q,q
,q
on a Hilbert spaces
of entire functions on the disk 1
q
,1
Do



0< <1q
,q
, ; and we prove that these operators are
adjoint-operators and continuous from
,q
into itself. Next, we study a generalized translation operators
on
qq I
.
Keywords: Generalized -Fock Spaces, -
Modified Bessel Function, -Bessel Operator,
Multiplication Operator, -Translation Operators
q
q
=0
n
n
n
1. Introduction
In 1961, Bargmann [1] introduced a Hilbert space of
entire functions

=
f
z
az
on such that
2
0
:!
n
n
fan

2
On this space the author studied the differential
operator =dD
dz and the multiplication operator by
, and proved that these operators are densely defined,
closed and adjoint-operators on (see [1]).
z
Next, the Hilbert space is called Segal-Bargmann
space or Fock space and it was the aim of many works
[2].
In 1984, Cholewinski [3] introduced a Hilbert space
of even entire functions on , where the inner
product is weighted by the modified Macdonald function.
On
the Bessel operator
2
d21
2
d
:=,>1 2
d
dzz
z

2
z
q
I
and the multiplication by are densely defined,
closed and adjoint-operators.
In this paper, we consider the -
modified Bessel
function:
 
2
2
2
=0 2
;: ;
n
nn
x
Ixq bq
where
2
2;
n
bq
q,q
are given later in Section 2. We
define the -Fock space

2
=0
=n
n
n
as the Hilbert space of
even entire functions zaz
on the disk
1
,1
Do q



of center o and radius 1
1q
, and such
that

,
2
22
2
0
:;
qnn
n
fabq

,q
Let f and g be in
, such that n

2
=0
=n
n
f
zaz

2
=0
=n
n
n
and zcz
, the inner product is given by
g

,
2
2
0
,= ;
qnn n
n
fgacb q

Next, we consider the multiplication operator
M
by
and the q-Bessel operator ,q
2
z
on the Fock space
,q
,q
, and we prove that these operators are continuous
from
*Author partially supported by DGRST project 04/UR/15-02 and CMCU
p
rogram 1 0 G 1 50 3 .
into itself, and satisfy:
222 F. SOLTANI
,
1
1q
ff
q
,
,q
q

,,
1
1
qq
f
q
Mf

Then, we prove that these operators are adjoint-
operators on ,q
:
,,
,=,
qq
Mf gfg
,,
;,
qq
f g

,q
Lastly, we define and study on the Fock space
q
,
the -translation operators:



1/2 2
,
:= ;; ,
zq
Tf wIzqf wwz


1
,1
D oq



and the generalized multiplication operators:



12 2
:;
;
z
MfwI zMqfwwz

1
,,.
1
Do q



Using the previous results, we deduce that the operato rs
z
T and
z
M
, for 1
,1
zDo q



,q
, are continuous from
into itself, and satisfy:
,
||
1
q
z
z
Tf I,
2
;q
q f
q




,,
2
;
qq
q f
q
||
1
z
z
Mf I





qq,α
aq0< <1q
,=1,2,,n 
2. The -Fock Spaces
Let and be real numbers such that ; the
q-shifted factorial are defined by
 

1
0=0
;:1, ;:1
ni
n
i
aq aqaq
 
Jackson [5] defined the q-analogue of the Gamma
function as
 


1
;
:=1 ,
;
x
qx
qq
xq
qq

0, 1, 2,x

It satisfies the functional equation
 
1
1=1
x
q
q
xx
q
 

, 1=1
q
q

and tends to
x
 when tends to 1. In particular,
for , we have q
=1,2,

n

;
1=1
n
n
qq
q
,nk
qn
The q-combinatorial coefficients are defined for
=0,,kn
,  , by



 
;1
:=
;;1 1
q
n
qq
qknk
qq n
n
kqqqqkn k


 

qq
Df

(1)
The -derivative of a suitable functio n f (see [6])
is given by


:,0
1
q
fx fqx
Df xx
qx
0= 0
q
Df f
provided exists.

0f
f

q
Df x
and
If is differentiable then tends to
f
x
as 1q
q
.
Taking account of the pap er [4] and the same way, we
define the -
I
modified Bessel functio n by
 
2
2
2
=0 2
;: ;
n
nn
x
Ixq bq

 

where
2
22
2
22
1(1) 1
;: 1
n
qq
n
q
qn n
bq
 


(2)
If we put 2
2
1
=;
n
n
Ubq
, then

2
1
1,1
1
n
n
Uq
Uq

q
I
Thus, the -
modified Bessel function is defined
on

2
1
,1
Do
q




I
and tends to the
modified
Bessel function as 1q
q
 
.
In [4], the authors study in great detail the -Bessel
operator denot ed by

2
,
[2 1]
:q
qq q
f
xDfx Dfqx
x
 
where
21
1
[21] :1
q
q
q

q
The -Bessel operator tends to the Bessel operator
1q
as

2
.;
.
Lemma 1: 1) The function
I
q
,1
,1
Do q



,
is the unique analytic solution of the q-problem:

2
,=,0=1 0=0
qq
yxyxyand Dy
(3)
Copyright © 2011 SciRes. APM
F. SOLTANI
Copyright © 2011 SciRes. APM
223
n
2) For , we have


2
2
,2
2( 1)
;
;
n
q
n
bq
zz
bq
22
(1)
=,
1
nn
n


2n
3) The constants , satisfy the
following relation: 2;
n
bq


22
2
2 ;
nn
qq
bq
22 ;=2222bqn n


Let 12
 ,
. The q-Fock space q

is the Hilbert
space of even entire functions 2
=0
=n
n
n
f
zaz
on
1
,1
Do q



, such that

2
;<
nn
bq

2
,
2
2
2
=0
:
qn
fa
(4)
where is given by (2) .
;
n
b
q
2
The inner product in ,q
2
is given for
n

==0 n
n
f
z2
=0
=n
n
n
az

and
g
zcz
by

2
2;
nnn
,=0
,=
qn
f
ga
cbq
1q
(5)
Remark 1: If , the space ,q
,q
agrees with
the generalized Fock space associated to the Bessel
operator (see [3]).
Theorem 1: The function
given for
1
,,1
D oq


wz 

, by

2
,= ;I wzq
,q
,
qwz
is a reproducing kernel for the -Fock space
q
, that
is:
1) For all 1
,1
wDo q
, the function
,,
q
z
wz
belongs to ,q
.
2) For all 1
,1
oq

wD

 ,q
f
and
, we have
 
,
,,.=
q,q
f
wfw
,q
f
Remark 2: From Theorem 1, 2), for
and
1
,1
wDo q




, we have


,
,
2
,,.= ;
q
q
q
fwwfI w,
1/2
2
q
q f



q
q,α
,q
3. Multiplication and -Bessel Operators on
On
, we consider the multiplication operators
M
and given by
q
N

2
:
M
fz zfz
 
:==1
qq
fz fqz
Nf zzDf zq
,q
We denote also by
the -Bessel operator
defined for entire functions on
q
1
,1
Do q



,,,
,=
qqq
MMM
.
We write




By straightforward calculation we obtain the following
result.
Lemma 2: 2
,,
,122
qq
qq
M
qBW


 ,
where

:
q
Bz fqz
and
 
2
,:1 1
qq
Wfzqq qzDfqz
 
1q
,,
q
(6)
Remark 3: The Lemma 2 is the analogous commu-
tation rule of Cholewinski [3]. When ,
then
M

tends to
d
414
d
I
zz

I
, where
is the identity operator.
Lemma 3: If ,q
f
q
B then
f
, and Wf
belong to
q
Nf ,q
,q
, and
,
,q
q
q
Bf f
1) ,
,
,
1
1q
q
q
Nf f
q
2) ,
3) ,
,
2
,
11
1q
q
q
qq
Wf f
q


2,
=0
=n
nq
n
fz az
.
Proof. Let
 
22
=0
==nn
qn
n
Bfzf qzaqz

, then
(7)


=0
==2
1
n
qn
q
n
fz fqz
Nf zanz
q
(8)
and from (4), we obtain
F. SOLTANI
Copyright © 2011 SciRes. APM
224


,
22
2
=0
2
2
=0
q
n
qn
n
nn
n
Bfaq b
ab
,
42
2
2
=;
;q
n
q
q f

and

,
22
=0
=2
q
qn
q
n
Nfa n


22
2;
n
b q

Using th e fact that 1
2q
n1q, we deduce



,
22
22
11
1
q
f
q
,
22
2
=0 ;=
1
q
qnn
n
Nfa bq
q

222
2nn
q
a nqz
On the oth er hand from (6), we have
 

,=1
=1 1
qn
n
Wfzqq
 (9)
and




,
22
,
2
2
=1
=1 1
q
q
q
n
Wfqq
an




2
42
2
2;
n
nn
qbq

Using th e fact that 1
21
q
nq
, we deduce that




22
,2
;
n
ab q
,
2
2
2
2=1
11
1
q
qn
n
qq
Wf
q



Therefore, we conclude that

,
2
q
qq
f
q
,
,
11
1
q
q
Wf

f
which completes the proof of the Lemma.
Theorem 2: If ,q
then ,q
f
and
M
f
belong to ,q
, and we have
1) ,
1
1q
ff
q
,
,q
q

,
2) ,,
1
1
qq
f
q
Mf

2,
=0
n
nq
az
.
Proof. Let

n
fz



.
1) From Lemma 1, 2),


Then from (10), we get




2
2( 1)
2
2
2
2
;
;
;
;
n
n
bq
z
bq
bq
az
bq
2
,=1 21
22
1
=0 2
=
=
n
qn
nn
n
n
nn
fz a
(10)

,
2
222
22
,122
2
=0 2
;
=;
;
q
n
qn n
nn
bq
f
abq
bq

Using Lemma 1, 3), we obtain
 

,
222
,1 22
=0
=22222;
q
qn n
qq
n
f
annb q



and consequently,
 

,
222
,2
=1
=222;
q
qn n
qq
n
f
annb q


 

(11)
Using the fact that

2
1
2221
qq
nn
q

, we
obtain

,
,
1/2
22
,2
=1
11
;=
11
q
q
qnn
n
fabqf
qq






2
1
=1
=n
n
n
2) On the other hand, since
M
fza z
(12)
then
 
,
22
222
12 22
=1 =0
=;= ;
qnn nn
nn
M
fabqabq




By Lemma 1, 3), we deduce


,
222
2
=0
=22222;
qnn
qq
n
M
fannbq




(13)
Using the fact that 2
1
22222 1
qq
nn
q

,
we obtain
,,
1
1
qq
Mf f
q

,q
f
We deduce also the following norm equalities.
Theorem 3: If
then

,,
2
,
,=11 ,
qq
qqq
fWfqqN f Bf
1)


,

,, ,
22
,=2,
qqq
qq qq
q
fNf NfBf
2)


 
,


,,,
22
2=122
qqq
qq
q
MfNfqB f
3)
 


,
122 ,
q
qq
q
qNfBf,
 


,,,
22
2
,
=122
qqq
qq
q
MffqB f
4)


,
,
,
q
q
fW f.
F. SOLTANI
Copyright © 2011 SciRes. APM
225
2,
=0
n
nq
az

=n
fzProof. Let

.
1) Follows from (7), (8) and (9).
2) From (11), we get


,
22
=0
=22
q
qn
n

2
,2
2 ;
n
qq
f
ann

b q

Using the fact
22
n
qq q
q22 =2nn
 , we
deduce

,,
22
,=2
qq
qqq
fNf ,
,
q
qq
NfBf



3) By (13) and using the fact that




2
4
22222
212
122
qq
n
q
nn
nq
qq


 



2
22
n
qq
q
qn
we obtain




,
2=1
122
q
q
MfN fq
qN
,,
,
22
22
,
qq
q
qq
q
qq
B f
fBf

2,
=0
n
nq
az


4) Follows directly from 1), 2) and 3).
Remark 4: Let

=n
fz
. Since
,
,0
q
f
,q
fW
, then


,
212
q
Mf q
,
2
2
q
q
qB f

=0Mf=0f
,q
Therefore implies that . Then
,q
:M

q
is an injective continuous operator on
,
.
Proposition 1: The operators M and ,q
are
adjoint-operators on ,q
,
,q
fg; and for all
, we
have
,
,=
q
Mf g
,
,
,
q
q
f g
2
=0
=n
n
n
Proof. Consider

zaz
and

2
=0
=n
n
n
g
zcz
q
in ,
. From (10) and (12),



2
22 2
2
;
;
n
n
bq
,1
=0 2
=n
qn
n
g
zc
z
bq

2
1
=1
=n
n
n
and
M
fza z
Thus from (5), we get
,
,
2
12
=1
2
12 2
=0
,
,= ;
=;
=,
q
q
nnn
n
nn n
n
q
Mfgac bq
ac bq
fg

q,α
,q
which gives the result.
4. Generalized Multiplication and
Translation Operators on
In this section, we study a generalized multiplication and
translation operators on
,q
f
.
Definition 2: For
, and 1
,,
1
wzDoq



q,q
,
we define:
-The -translation operators on
 

, by
,2
2
=0 2
:;
n
qn
z
nn
fw
f
wz
bq
,q
(14)
-The generalized multiplication operators on
 

, by
2
2
=0 2
:;
n
n
z
nn
Mfw
fw z
bq
M
(15)
For 1
,,
1
wzDoq




2
.;
I
, the function q satis-
fies the following product formulas:
 
222
.;= ;;
z
I
qwIzqIwq


222
.;= ;;
z
M
IqwIwzqIwq

1
Remark 5: If q

2,
=0
=n
nq
n
fz az
, we obtain the generalized
translation operator given in ([3], page 181).
Proposition 2: Let
and
1
,,
1
zwD oq




 
 
. Then
1)
=0=0 2
2
22 2
22
=
11
.
11
n
zn
nk
q
k
qq n
qq
n
fw ak
nzw
knkw





 






2) 22
=0 =02
2
=;
nkn
nk
znk
k
a
fwz w
bq





M
.
226
2,
=0
n
nq
az
F. SOLTANI
Proof. 1) Let

=n
fz
. From (14),
we have ,,
2
;
1
qq
z
z
MfIq f
q
 

,2
2
=0 2
=;
;
n
qn
z
nn
fw
fwz wz
bq
1
,,
1
Do q



Since from Lemma 1, 2),


2
2
22
,2
2( )
;
;
k
nk
q
kn
bq
ww
bq
()
=,
kn
kn



we can write

2
2( )
2
;
;
nk
n
bq
w
bq
2
,=2( )
=k
qk
kn kn
fw a
Thus we o btain




2
2
2
=0=0 22
;
=;;
nn
zn
nk knk
bq 2( ) 2
2
nk k
f
wa
bqb

 wz
q
On the other hand from (1) and (2), we get






 
 
2
2
2
22
22
22
22
;
;;
=
n
knk
qq
qqq
bq
bqb q
n
k



 

11
11
n
kn
k




which gives the 1).
2) From (15), we have
 

2
2
=0 2
=;
;
n
n
z
nn
Mfw
Mfwz wz
bq
1
,,
1
Do q



2
=
=
nk
kn
kn
But from (12), we have

M
fwa w

Thus we o btain

22
2
kn
=0=0 2
=;
n
nk
z
nk k
a
M
fw z


w
bq



According to Theorem 2 we study the continuous
property of the operators
z
T and
z
M
on ,q
.
Theorem 4: If ,q
f
and 1
,1
zDo q



, then
and
z
Tf z
M
f belong to ,q
, and we have
1) ,,
2
;
qq
q f
q1
z
z
Tf I





,
2)





.
Proof. From (14) and Theorem 2, 1), we deduce



,,
,
2
,2
=0 2
2
2
=0 2
;
1;
qq
q
n
n
zq
nn
n
n
nn
z
Tff bq
zf
qb q

Therefore,
,,
2
;
1
qq
z
z
TfIq f
q





,
,q
fg
which gives the first inequality, and as in the same way
we prove the second inequality of this theorem.
From Proposition 1 we deduce the following results.
Proposi t ion 3: For all
, we have
,,
,=,
qq
zz
Mfg fTg
,,
,=,
qq
zz
Tf gf Mg
We denote by
z
R
,q
the following operator defined on
by




1/221/2 2
,
1/2 21/22
,
:=;;
;;
zzz q
zz
q
RTMMTIz qIzMq
I
zMq Izq
 

 

,q
f
Then, we prove the followi n g theorem.
Theorem 5. For all
, we have
,, ,
22
=,
qqq
zz z
Mf TffRf


Proof. From Proposition 3, we get

,,
,
,,
2
2
=,
=,
=,
qq
q
qq
zz
z
zz
z
zz
Mf fTMf
fMT Rf
Tff Rf


5. References
[1] V. Bargmann, “On a Hilbert Space of Analytic Functions
and an Associated Integral Transform, Part I,” Commu-
nications on Pure and Applied Mathematics, Vol. 14, No.
3, 1961, pp. 187-214. doi:10.1002/cpa.3160140303
[2] C. A. Berger and L. A. Coburn, “Toeplitz Operators on
the Segal-Bargmann Space,” Transactions of the Ameri-
can Mathematical Society, Vol. 301, 1987, pp. 813-829.
doi:10.1090/S0002-9947-1987-0882716-4
Copyright © 2011 SciRes. APM
F. SOLTANI
Copyright © 2011 SciRes. APM
227
[3] F. M. Cholewinski, “Generalized Fock Spaces and Asso-
ciated Operators,” SIAM Journal on Mathematical Analy-
sis, Vol. 15, No. 1, 1984, pp. 177-202.
doi:10.1137/0515015
[4] A. Fitouhi, M. M. Hamza and F. Bouzeffour, “The q-jα
Bessel Function,” Journal of Approximation Theory, Vol.
115, No. 1, 2002, pp. 144-166.
doi:10.1006/jath.2001.3645
[5] G. H. Jackson, “On a q-Definite Inte gral s,” The Quarterly
Journal of Pure and Applied Mathematics, Vol. 41, 1910,
pp. 193-203.
[6] T. H. Koornwinder, “Special Functions and q-Commut-
ing Variables,” Fields Institute communications, Vol. 14,
1997, pp. 131-166.