2011) 1( 24مجلة ابن الهیثم للعلوم الصرفة والتطبیقیة المجلد IAXAX ةالحظات حول معادلة المؤثرالالخطیم n  * مي محمد هاللو بثینه عبد الحسن احمد د، كلیة العلوم،قسم الریاضیات جامعة بغدا دادجامعة بغ،ابن الهیثم ةكلیة التربی ،قسم الریاضیات 2009 نیسان 13استلم البحث في 2009تموز 7قبل البحث في ةالخالص ــة لمعادلــة المــؤثر ةوریالشــروط الضــر IAXAXوالكافیـ n  *، ــق حقیقــي للحصـــول علــى حــل موجـــب ذاتــي الترافـ Xـاد علــى هــذه الشــروط وبعــض الخصــائص للمـؤثر قـد اعطیــت ـهوكــذ ،باالعتمـ قــد اعطیــت AوXلبــین الحــ لك العــال قـ .أیضا مؤثر موجب ذاتي الترافق، القطر الطیفي،معادلة المؤثر الالخطیة: الكلمات المفتاحیة IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (1) 2011 Notes On The Non Linear Operator Equation IAXAX n  * B.A. Ahmed and M.M. Hilal Department of Mathematics, College of Science, University of Baghdad Department of Mathematics, College of Education Ibn Al-haitham, University of Baghdad Received in April,13,2009 Accepted in July,7,2009 Abstract Necessary and sufficient conditions for the operator equation IAXAX n  * , to have a real positive definite solution X are given. Based on these conditions, some properties of the operator A as well as relation between the solutions X and A are given. Key words: non-linear operator equation; spectral radius; positive definite operator. AMS classification: 39B42. Introduction Consider the non-linear operator equation )1(* IAXAX n   where I is identity operator, and  HBXAA ,, * ; where  HB denotes the Banach algebra of all bounded linear operators on H; H is an infinite dimensional complex Hilbert space. Several authors have studied the above equation when XA, are matrices and 2,1  nn and they have obtained theoretical properties of these equations. In [1] Equation (1) was studied in the case X is a self_adjoint positive operator , which arises in many applications such as in control theory and statistics and in dynamic programming In this paper, we study equation (1) where X belongs to the set; where     TTrHBTTTAAC  ;,: * . Where  Tr is the spectral radius of T 1-Preliminaries In this section we present notation, lemma and theorem which will be used in the remainder of the paper. The notation  00  AA means that A is positive operator , and BA  is used as an alternative notation for 0 BA .It is well-known for any operator   TTHBT *, is positive operator  22.,2 p ,let spec A denotes the spectrum of A. Lemma 1.1[3, p. 866]: Let M and N be two arbitrary operators then:    NNMMrMNNMr ****  Proof: By elementary calculus, we have that                         N M OI I NMrMNNMr 0 **** Since the non-zero elements of specMN and specNM are the same [4, P.43]; so for any two operators, we have: IBN AL- HAITHAM J. FOR PURE & APPL. SCI. VOL.24 (1) 2011                                          **** 0 0 NM N M I I r N M OI IO NMr Now,   AAr  , where denotes the 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- Necessary and sufficient conditions of the solution of the equation We study the existence of the solution of equation (1) by the following theorem: Theorem 2.1: the operator equation (1) has a solution X positive operator if and only if the operator A takes the following factorization form     )2( 2 * *2 1 *        evenisnifZWW oddisnifZWWW A n n   where W is an invertible operator and IZZWW  ** . Proof: suppose 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:  Suppose 1n , then     IAWWAWW IAWWAWW     1*1** 1** * Further, we can rewrite the last equations as:      )3( 1* **1* IAWAWWW   IBN AL- HAITHAM J. FOR PURE & APPL. SCI. 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,  Suppose it is true 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 rewritten 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 rewritten 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 positive operator ) is a solution to the 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 positive 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 between solution X and operator A : In this section, we will study 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 applying 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 applying 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 References 1. Ahmed, B.A. and Hilal ,M.M., (2008) ,On Solvability of an Operator Equation, Proceeding of the 3rd conference on Mathematical science in united Arab Emirates university, in the icm, 2. Feintuch, A. (1998), Robust Control Theory in Hilbert space, Springer-Verlag, New York, Inc. 3. Ramadan, M. A. (2007),Necessary and Sufficient Conditions for the Existence of Positive Definite Solution of the Matrix Equation, Nanyang University of Technology. 4. Halmos ,P. R. (1982), A Hilbert Space Problem Book, Springer-Verlag, New York, Heidelberg, New York, Berlin,. 5. Conway, J.B. (1985), A course in functional analysis, Springer- Verlage, Berlin Heidelberg, New York.