2011) 1( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد IAXAX ةالحظات حول معادلة المؤثرالالخطیم n  * مي محمد هاللو بثینه عبد الحسن احمد جامعة بغداد، كلیة العلوم،قسم الریاضیات دادجامعة بغ،ابن الهیثم ةكلیة التربی ،قسم الریاضیات 2009 نیسان 13استلم البحث في 2009تموز 7قبل البحث في ةالخالص IAXAXوالكافیـــة لمعادلــة المــؤثر ةوریالشــروط الضــر n  *، ذاتــي الترافـــق حقیقــي للحصـــول علــى حــل موجـــب Xلبــین الحــ لك العــال قــهوكــذ ،باالعتمــاد علــى هــذه الشــروط وبعــض الخصــائص للمـؤثر قـد اعطیــتXوA قــد اعطیــت .أیضا مؤثر موجب ذاتي الترافق، القطر الطیفي،معادلة المؤثر الالخطیة: الكلمات المفتاحیة IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Notes On The Non Linear Operator Equation IAXAX n  * B.A. Ahme d and M.M. Hilal Departme nt of Mathematics, College of Science, Unive rsity of Baghdad Departme nt of Mathematics, College of Education Ibn Al-haitham, Unive rsity of Baghdad Recei ved in April ,13,2009 Accepted in July,7,2009 Abstract Necessary and sufficient conditions for the op erator equation IAXAX n  * , to have a real positive definite solution X are given. Based on these conditions, some p rop erties of the op erator A as well as relation between the solutions X and A are given. Key words: non-l ine ar operator equati on; spectral radius; posi tive defini te operator. AMS classi fication: 39B42. Introduction Consider the non-linear op erator equation )1(* IAXAX n   where I is identity op erator, and  HBXAA ,, * ; where  HB denotes the Banach algebra of all bounded linear op erators on H; H is an infinite dimensional comp lex Hilbert sp ace. Several authors have st udied the above equation when XA, are matrices and 2,1  nn and they have obtained theoretical p rop erties of these equations. In [1] Equation (1) was st udied in the case X is a self_adjoint p ositive op erator , which arises in many app lications such as in control theory and statist ics and in dynamic p rogramming In this p aper, we study equation (1) where X belongs t o the set; where     TTrHBTTTAAC  ;,: * . Where  Tr is the sp ectral radius of T 1-Preliminaries In this section we p resent notation, lemma and theorem which will be used in the remainder of the p aper. The notation  00  AA means that A is p ositive op erator , and BA  is used as an alternative notation for 0 BA .It is well-known for any op erator   TTHBT *, is p ositive op erator  22.,2 p ,let sp ec A denotes t he sp ectrum of A. Lemma 1.1[3, p. 866]: Let M and N be two arbitrary op erators t hen:    NNMMrMNNMr ****  Proof: By elementary calculus, we have that                          N M OI I NMrMNNMr 0 **** Since the non-zero elements of sp ec MN and sp ec NM are the same [4, P.43]; so for any two op erators, we have: IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011                                             **** 0 0 NM N M I I r N M OI IO NMr Now,   AAr  , where denotes t he operator norm. so            NNMMr N M NMr NM N M r NM N M OI IO NM N M OI I rNM N M OI IO r ** ** ** ** **** 1 0                                                                                Which completes the proof. 2- Ne cessary and suffi cie nt conditi ons of the sol uti on of the equati on We st udy the existence of the solution of equation (1) by the following theorem: Theorem 2.1: the op erator equation (1) has a solution X p ositive op erator if and only if the op erator A takes t he following factorization form     )2( 2 * *2 1 *        evenisnifZWW oddisnifZWWW A n n   where W is an invertible op erator and IZZWW  ** . Proof: sup p ose that equation (1) has a solution X . Then, using the set C we can write X as WWX * . Equation (1) can be written as   IAWWAWW n  *** The prove using mathematical induction:  Sup p ose 1n , then     IAWWAWW IAWWAWW     1*1** 1** * Furt her, we can rewrite the last equations as:      )3(1***1* IAWAWWW   IBN AL- HAITHAM J. FOR PURE & APPL. S CI. VOL.24 (1) 2011 Equation  3 can be rewritten in the equivalent form [5, p.171]: )4( * * * I AW W AW W              Now, set AWZ * ; then ZWA * as desired,  Sup p ose it is t rue when pn  to show that it is true when 1 pn         IAWWWWAWW IAWWAWW P P     1**** 1*** If p is odd, then               )5(**1***1*1** *1*11*1** 1*1*1*1*1*** IAWWWWAWWWWWWW IAWWWWWWWAWW IAWWWWWWWWWWAWW          Equation (5) can be rewritt en in the equivalent form:              AWWWW W AWWWW W **1* * **1*  Now, set AWWWWZ **1*   , then ZWWWWWA ***  , as form   ZWWW p *2 1 *  If p is even, then:             )6(*1*1**1*1* *1*1*1*1** 1*1*1*1*** IAWWWWAWWWWWW IAWWWWWWWWAWW IAWWWWWWWWAWW          Equation (6) can be rewritt en in the equivalent form: I AWWWW W AWWWW W              *1*1 * *1*1  New, set AWWWWZ *1*1   ; then  ZWWWWWWWWA ****  , as form   ZWW P 2* Conversely ,assume that the operator A admits the factorization  ZWWWWWA ***  , if n is odd, and set WWX * , we then need to show that X (which is p ositive op erator ) is a solution to t he operator equation (1), we have:             I Z W Z W ZZWW ZWWWWWWWWWWWWZWW ZWWWWWWWWWWWWWZWW ZWWWWWWWZWWWWWWWAXAX nn                      * ** ****1*1*** ***1*1***** **********    When n is even, then ZWWWWWWWA ***  , and set WWX * , we then need to show that X (which is p ositive definite) is a solution to the operator equation (1) .we have. IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1 ) 2011                 I Z W Z W ZZWW WZWWWWWWWWWWWWWWWZWW ZWWWWWWWWWWWWWWWWWWZWW ZWWWWWWWWZWWWWWWWWAXAX nn                      * ** ****1*1***** ***11*1****** **********    which completes the proof of the theorem. 3- Relation betwee n soluti on X and operator A : In this section, we will st udy the relations between X and A in equation (1) Theorem 3.1: If equation (1) has a solution X , then for all Nn  the following hold: (i) 12 1 2*2 1 2               nn XAAXr . (ii)     *2*2 AAXX nn  . Proof: (i) Using theorem (2.1), when n is even. We obtain:                                          2 1 **2 1 * 2 1 2*2**2*2 1 2*2 1 2*2 1 2 WWZZWWr WWWWZZWWWWrXAAXr nnnnnn We set  2 1 *: WWM  ; then app lying lemma (1.1), we obtain:       1 ** **2 1 2*2 1 2                  Ir ZZMMr MZZMrXAAXr nn Now, when n is odd; we obtain          WZZWr WWWWWZZWWWWWrXAAXr nnnnnn ** 2 1 2*2 1 ***2 1 *2 1 2*2 1 2*2 1 2                         IBN AL- HAITHAM J. FO R PURE & APPL. SC I. VO L.24 (1 ) 2011 then ap p lying lemma (1.1) we obtain:       1 ** **2 1 2*2 1 2                  Ir ZZWWr WZZWrXAAXr nn (ii) If n is even, then from theorem (2.1), we have                 2**2* 2**2*2*2**2*2 nn nnnnnn WWZZIWW WWZZWWWWWWAAXX   Since IZZWW  ** ,    ZZspecZZspec **  and, 0*  ZZI , therefore,      02**2*  nn WWZZIWW . If n is odd, then. From theorem (2.1), we have                               WWWZZIWWW WWWZZWWWWW WWWZZWWWWWWW WWWZZWWWWWWWAAXX nn nn nn nnnnnn 2 1 ***2 1 * 2 1 ****2 1 * 2 1 ***2 1 *2 1 *2 1 * 2 1 ***2 1 *2*2**2*2               Since IZZWW  ** and     0, ****  WWZZIZZspecZZspec , and thus,, 0*  ZZI , therefore,,      02**2*  nn WWZZIWW Re ferences 1. Ahmed, B.A. and Hilal ,M .M ., (2008) ,On Solvability of an Op erator Equation, Proceeding of the 3rd conference on M athematical science in united Arab Emirates university , in the icm, 2. Feintuch, A. (1998), Robust Control Theory in Hilbert sp ace, Sp ringer-Verlag, New York, Inc. 3. Ramadan, M . A. (2007),Necessary and Sufficient Conditions for the Existence of Positive Definite Solution of the M atrix Equation, Nanyang University of Technology . 4. Halmos ,P. R. (1982), A Hilbert Space Problem Book, Sp ringer-Verlag, New York, Heidelberg, New York, Berlin,. 5. Conway , J.B. (1985), A course in functional analysis, Sp ringer- Verlage, Berlin Heidelberg, New York.