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 Advances in Pure Mathematics, 2011, 1, 325-333 doi:10.4236/apm.2011.16059 Published Online November 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Toeplitz and Translation Operators on the q-Fock Spaces* Fethi Soltani Higher College of Technology an d Informatics, Tunis, Tunisia E-mail: fethisoltani10@yahoo.com 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1,1Do q, , with reproducing kernel given by the q-exponential function 0< <1qqez; 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 zqD 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=nnnfzaz on  such that 22=0:=! <.nnfan 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:  =01:= ,;nnqnnqez zqq where 11=0;:=1,=1,2,,.niniqqq 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=nnnfzaz on the disk 1,1Do q of center oand radius 11d such that q, an22=0;:=< .1nnnqnqqfaq f and gLet be in, such that q=nn=0nfzaz and =0=nnngzbz, the inner pro- duct is given by =0;,=1nnn nqnqqfgab q .The q-Fock space has also a reproducing kernel q q given by  1,=;,,1qqwzewzwzD oq .*Author partially supported by DGRST project 04/UR/15-02 and CMCU program 10G 1503. F. SOLTANI 326 Then if , we have qf 1,,.= ,,1q.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 thnsider the mhe Focpaceand we prove that thqqe q-Fock space q. In the second part of this work, we coultiplication operator Q by z and the q-Derivative operator qD on tk s q, ese operators are continuous from q into itself, and satisfy: 11,.qDff Qff  11qq qqqqThen, we prove that these operators are adjoint-ope- rators on : q,=,;, .qqqqQf gf Dgf g Next, we define and study on the Fock space , the q-translation operators: q1:=; ,,,zqqfwezDfwwzDo 1qand the generalized multiplication operators:  1:=; ,,.1zqMfwe zQfwwzDoq Using the previous results, we deduce that the ope- rators z and zM, for 1,1zDoq from into itself, and satisfy: , are continu- ous q ,1zqqfe fq.1qzq qqzzMf efq Lastly, we establish Weyl commutation relations be- tween the translation operators a and the multipli- cation operators bM, where 1,,1abD 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< <1qq 10;:=1, ;:=1, =1,2,,.ninaqaqaq n =0iJackson [4] defined the q-an1;:=1,0, 1, 2,;qxqqIt satisfies thxqqxqx e functional equation  1=, 1=1,qqqqxxx  where 1:= ;1xqqqx and tends to x, we have when tends to. In particular, for q 1=1,2,n;1== !.1nqnqqTbinatorial qqnn he q-comcoefficients are defined for n and , by =0,,kn !:= .!!qqqkknk  q-derivati of a sqnnThe ve uitable function qDf f (see [5]) is given by :=, 0,1qqxand fx fqxDf xx 0q0=Df f provided exists. If 0f f is differentiable then tends to Df xqfx as 1qThere. two important qs of the el function [5]: are-analoguexpon- entia1/2=0 !qnqn:= ,nzEz q nn=0:= .!nqnqzez n Note that the first series converges for 0,qqqqxr qrrx (1) where the q-integxEral (introduced by Jackson [4]) is defined by Lemma 1. The function  0=0d=1 .qnfx xqaqfaq ann.qe, 1,1Do q, is the unique analytic solution of the q-problem: 1. he form Replacing in (2ain =qDyzy(2)  ,0=yz Proof. Searching a solution of (2) in t=nyz az. Then =0 nn=1=.qnqnDyza nz ), we obt1n1=1 =1=.nnnnqnnanz ax Thus, 1=,=1,2,nnqanan We deduce that 1=.nnqaan We get =.!nnqan Therefore,  =0==!nqnqzyze zn ,which completes the proof of the lemma. □ 2.2. The q-Fock Spaces e denote by q W1,1HDo q the space of entire functions on 1,1Do q efined on . the measure dqm 1,1Do q by  1:=,=e.2πiqqqdmzEqrdzrrd  21,,1qLDo mqfunctions the space of measurable f on 1,Do satisfying 1q 221,12,,:= za. Then for ,nk, we get   12,,1=dnkLDo q ,1|| .!!mqknqzaqqzzz mznk Thus, Copyright © 2011 SciRes. APM F. SOLTANI 330    12,,1||2()/20110, d!! ()d!! d.!!nkLDo mqqnkqzaqqank qqqqnkqqqqqKzmznkKrEqrnkKa Eqrrnk  rBut from (1), we have  110d= 1=1.qqqqEqrr Hence  12,,1,.nkLDoq !!nkmqqqKank Thus, we obtain 222212,,,=0 1,