For A, B and Y operators in B(H) it's well known the importance of sylvester equation AX - XB= Y in control theory and its applications. In this paper -using integral calculus- we were able to give a priori estimate of the solution of famous sylvester equation when A and B are selfadjoint operators, some other results are also given

Bekkar, L. Hariz

Gaaya, H

Mansour, Abdelouahab

English

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                                                             ISSN 2347-1921                                                        3633 | P a g e                                                       M A Y  2 9 ,  2 0 1 5   A PRIORI ESTIMATE FOR THE SOLUTION OF SYLVESTER EQUATION Abdelouahab Mansour, Camille Jordan Institute, Claude Bernard University-Lyon1, France. amansour@math.univ-lyon1.fr L. Hariz Bekkar Mathematics departement, Eloued University, Algeria. harizlnd@yahoo.fr H. Gaaya Camille Jordan Institute, Claude Bernard University-Lyon1, France. gaaya@math.univ-lyon1.fr  ABSTRACT For A , B  and Y  operators in )(HB  it’s well known the importance of sylvester equation YXBAX =  in control theory and its applications. In this paper -using integral calculus- we were able to give a priori estimate of the solution of famous sylvester equation when A  and B  are selfadjoint operators, some other results are also given.  Mathematics Subject Classification 2010: Primary:47A10, 47A62, 35B45,Secondary:97I50 Keywords: Sylvester equation; A priori estimate; selfadjoint operator; spectrum                       Council for Innovative Research Peer Review Research Publishing System Journal: JOURNAL OF ADVANCES IN MATHEMATICS Vol.10, No.7 www.cirjam.com , editorjam@gmail.com                                                              ISSN 2347-1921                                                        3634 | P a g e                                                       M A Y  2 9 ,  2 0 1 5   1  INTRODUCTION  Let H  be a separable infinite dimensional complex Hilbert space, and )(HB  denote the algebra of all bounded linear operators on .H  In general we define the generalized derivation on )(HB  by XBAXXBA =, . It’s well known the importance of sylvester equation   YXBAXXBA ==,   (1)  in control theory and its applications in quantum mechanics. The study of these operator equations generated many work, and several problems with these equations are still unanswered. The mathematical key is the elementary operator BA, , that is find an operator )(HBX   satisfying the operator equation. Solving operator equations of this kind returns to some tens of years, but giving the best priori estimate of the solution with simple integral calculation, easily manipulated by hand in our knowledge is not exist. In our present work we used the selfadjoiness of operators A , B  (their spectra are real) and some technics of integral calculation. Also, our work has its roots in physics, the Heisenberg uncertainly principle may be mathematically formulated as saying that there exists a pair XA,  of linear transformations and a non-zero scalar   for which IXAAX = . In  7  W.E.Roth has shown for finite matrices A  and B  over a field that CXBAX =  and CYBAX =  are solvable if and only if the matrices CBAandBA000are similar. In  6  Rosenblum showed that the result remains true when A  and B  are bounded selfadjoint operators in )(HB . In general if A  is an invertible operator in )(HB  and YA 1  is contraction we can easly see that |1|11  PPPPBAX , in this paper we give the best priori estimate of the solution of operator equation (1)  under some suitable conditions, and some other results are also given. 2  Preliminaries Theorem 2.1 For A  and B  operators in )(HB  such that the spectrum of A , B  given by }<||,{=)(  zzB , }>||,{=)(  zzA , where   is real number, the solution of sylvester equation (1) is given by nnnYBAX 10==  . Lemma 2.2 For A  and B  operators in )(HB  such that 0}<||,{=)( zzB , 0}>||,{=)( zzA , the integral dtYee tBtA0  is convergent where )(HBY .  Theorem 2.3 (BHATIA[2]) For A  and B  operators in )(HB  such that 0}<||,{=)( zzB , 0}>||,{=)( zzA , the solution of the sylvester equation (1) is given by dtYeeX tBtA0= .  3  Main results Proposition 3.1 Let be A  a selfadjoint operator in )(HB  such that ],[)( MmA   , where )(inf= )(   Am  , )(sup= )(   AM  , for any z  in the circle ),2( rC Mm  with the radius 2> mMr , and rmM2= we have the following estimation   ]1211[2)( 1  ArccoslnrdzIA  Lemma 3.2 Let be A  a normal operator in )(HB , then 1)(  zIA  is normal for any complex number z  and we have for any )(Az   the resolvent of A , ))(,(1=)( 1AzdzIAPP  , where                                                              ISSN 2347-1921                                                        3635 | P a g e                                                       M A Y  2 9 ,  2 0 1 5   |)(|inf=))(,( )(    zAzd A .  Proof. It’s clear that if A  is normal, then zIA  is normal and consequently 1)(  zIA  is normal. from this we can write ))((=)( 11   zIAzIA PP  the spectrum radius. Then |)(|sup=)()(1    zzIAAPP  that’s ))(,(1=)( 1AzdzIAPP  . Proof (of proposition (3.1)). Let be A  a selfadjoint operator in )(HB , by lemma (3.2) we have ))(,(1=)( 1AzdzIAPP  , for any )(Az  . If C  is the circle with center 2Mm   and radius 2> mMr , we consider the parametrization  iMm rez 2=  and  =2=rcos mM. We will estimate ))(,( Azd   the distance between the point z  of the circle C  and )(A  the spectrum of A , we have three different cases: If  |<| : |=|))(,( MzAzd   . If  |<<| : |=|))(,( mzAzd   . If  |<<| : ||))(,(  sinrAzd  . Then   dzIAdzIAdzIAdzIA 1|<<|1|<<|1|<|1 )()()(=)(    we will estimate the three integrals in the right side of equality  |2|=|2|=||=)(=|<||<||<|1|<|  iMmMiMmM redredzddzIAI  since 2|2| mMiMm rre  , we get   22)(= 1|<|mMrdzIAI     |2|=|2|=|<<||<<|  imMmiMm redredJ    it is easy to prove that the integral J  is equal to I  as follows   .|2||2|=<<<<  imMimM redredJ    With change of variables   =  we get                                                               ISSN 2347-1921                                                        3636 | P a g e                                                       M A Y  2 9 ,  2 0 1 5    .|2|=|2|0<<<<    iMmimM redred With change of variables   =  we get   ,|2|=|2|<<0<<    iMmimM redred by summation the last two equalities term by term we get IJ = . Then   .22)(= 1||mMrdzIAJ   PP  On the interval   ||  we have ||))(,(  sinrAzd     ,..=)( 1||  sinrdsinrddzIA   PP  then   ..2=)( 1||  sinrddzIA   PP  A simple calculation gives us   )2(1=.2tanlnrsinrd   And since coscostan11=22, then rrlnrsinrdmMmM21211=.     with rmM2=  i.e.    111=.lnrsinrd. Then      112=)( 1||lnrdzIA PP  Therefore   ]1112[2=112)222()( 1   lnArccosrlnrrdzIAmMPP  Remark 3.3 Let be   ]1112[2=)(lnArccosrr  where rmM2= and .2> mMr  By proposition (3.1)   )()( 1 rdzIA   PP                                                               ISSN 2347-1921                                                        3637 | P a g e                                                       M A Y  2 9 ,  2 0 1 5   We can examine the rate of this estimate as the values â€‹â€‹of r  which will be best when r  tends to infinity.  Theorem 3.4 If A  and B  are selfadjoint operators in )(HB  such that }<||,{=)( bzzB , }>||,{=)( azzA  where a  and b  are real numbers and 0>a  and 0<b , then we have the estimations tatA eCe   1PP  and tbtB eCe 2PP  where 1C  and 2C  are positif constants.  Proof. It well known that if f  is an analytic function on a neighborhood of )(T  the spectrum of T , for any )(T  the resolvent of T :   dTRfiTf )()(21=)(   where )(TR  is the mapping resolvent of T  for  ,   is closed cantour of Cauchy. for tAeTf =)(  we have    dTRfie tA )()(21=  Theorem 3.5 Let be A  and B  self adjoint operators in ),(HB  )(A  and )(B  their spectrum respectively such that ],[)( MmA    and ],[)( MmB   , where   ,sup=,inf=,sup=,inf=)()()()( BMBmAMAm with mM  < , then we can estimate the solution of Sylvester equation (1) by   ]1112][1112[.212   lnarccoslnarccosaXPP  where rmM2= and rmM2=   Proof. Since the conditions of Bhatia theorem are satisfying then the solution of the equation (1) is given by   .=0dtYeeX tBtA  Then .0dteeYX tBtA PPPPPPPP   Using theorem (3.4) and proposition (3.1) we get   ...].1112][1112[.44202dteLKlnarccoslnarccosrrYX taPPPP  with )2(= aK mM  and )2(= mMaL  . Since the spectra of A  and B  are isolated, we can always put amr =  and mar  = , then   ]1112][1112[2 2   lnarccoslnarccosaYXPPPP  with rmM2= and rmM2=  and 0>a .  Corollary 3.6 If A  and B  self adjoint operators in )(HB  with spectrum reduced to one point i.e. )(A  =   , )(B  =    and  <  then the estimation in thorem (3.5) comes PPPP YaX21                                                                ISSN 2347-1921                                                        3638 | P a g e                                                       M A Y  2 9 ,  2 0 1 5   Aknoweldgments The authors thank Mr. Gilles Cassier for his valuable comments and for his help. References  1 W.Arendt, F.Rabiger, A.Sourour: Spectral Properties of the operator equation         YXBAX = , Quart. J. Math.Oxford, 45:2(1994) 133-149.  2 R.Bhatia,Matrix Analysis, Springer-Verlag, Newyork,  1997 , Graduate Texts in Mathematics.  3 N. Dunford and J.T. Shcwarz ,Linear Operators, part I, II et III, Interscience, Newyork, 1964.  4 P.R.Halmos, A Hilbert Space Problem Book, D.Van Nostrand Company, Inc.Princeton, New Jerse,  .1967   5 A.Mansour, Solvability of ECXDAXB =  in the operators algebra B(H), Lobachevskii journal of mathematics, vol.31, NÂ°3  2010  257-261.  6 M.Resenblum, On the operator equation AX - XB = Q with self-adjoint A and B, Proc. Amer. Math.Soc., 20  1969 , 115-120.  7 W.E.Roth, The equations AX - YB = Q and AX - XB = Q in matrices, Proc. Amer.     Math.Soc., 3  1952 , 302-316.    

A priori Estimate for the solution of Sylvester equation

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A priori Estimate for the solution of Sylvester equation

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