 Applied Mathematics, 2010, 1, 283-287 doi:10.4236/am.2010.14036 Published Online October 2010 (http://www.SciRP.org/journal/am) Copyright © 2010 SciRes. AM Analyticity of Semigroups Generated by Singular Differential Matrix Operators Ould Ahmed Mahmoud Sid Ahmed1, Adel Saddi2 1Department of Mat hem at ic s, C oll e ge of Scie nce, Al j ouf University, Aljouf, Sa u di Arabia 2Department of Mathematics, College of Educa tion for Girls in Sarat Ebeidah, King Khalid University, Abha, Saudi Arabia E-mail: sidahmed@ju.edu.sa, adel.saddi@fsg. rnu.tn Received March 9, 2010; August 3, 2010; August 6, 2010 Abstract In this paper we prove the analyticity of the semigroups generated by some singular differential matrix op- erators of the form 22()=,dBxdxdxdx  in the Banach space ([0,],( )),dCMC with suitable boundary con- ditions. To illustrate the work an example is discussed. Keywords: Dissipative Operators, Positive Operators, Spectrum, Analytic Semigroups, Evolution Equations 1. Introduction As we will see in the sequel the problem of characte- rizing operator matrices generating strongly continuous semigroups or analytic semigroups is quite difficult. Th e main problem consists in finding appropriate assumptions on the matrix entries allowing general results but still including the concrete examples we have in mind. The evolution of a physical system in time is usually described in a Banach space by an initial value problem of the form 0() ()=0, 0(0) =.dU tUt tdtUU (1.1) Problem of type (1.1) is well posed in a Banach space X if and only if the operator (,())D generates a 0C-semigroup >0()ttTon X. Here the solution ()Ut is given by 0()= tUt TU for the initial data 0()UD. For operator semigroups we refer to [1-5] and to  for the theory of op erator matrices. The harmonic analysis for a class of differential operators with matrix coefficients was treated in [7,8]. In this work we are interested in a generalization of the analyticity and the positivity of the semigroup generated by a matrix singular differential operator (,())D. A similar study was realized in  for a class of differential operators with matrix coefficients and interface. For the scalar case we refer to . This paper is organized as follows. In the second section we introduce some notations and give prelimina- ries results. In the third section we investigate some properties of the operator (,())D in particular we prove that it is closed, densely defined, dissipative and satisfies the positive minimum principle. Some spectral properties of the operator (,())D are obtained in this section. In section fourth we established the analyticity of the semigroup generated by an operator in the form (,())D. In this case the problem (1.1) has a unique solution for all 0UX. In section five we present a concrete example of application of the results obtained. 2. Notations and Preliminary Results Let ()dMC be the space of dd matrices with complex coefficients and dI the identity matrix. The norm of a matrix 1,=()ij dijdAaMC wil l b e defi ned by 1,|| ||=||max ijijdAa (2.1) The reason of this special choice will be justified in Lemma 2.1 and Lemma 2.2. A matrix 1,=()ij dijdAaMC is called nonnega- tive (resp positive) if all entries ija are real numbers and nonnegative (resp positive). In this case we write 0A, (.>0)respA and we call 1,||=| |ij ijdAa O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 284 the absolute value of A. For 1,=()ij dijdAaMC and 1,=ij ijdAa ()dMC, we will write 0.ABifAB and >>0.ABifAB A matrix-valued functions is said to be continuous, differentiable or integrable if all its elements are continuous, differentiable or integ r able fu n ctions. If th e matrix ()Ax is integrable over [,]ab, then ()().bbaaAxdxAx dx For two matrix functions 1,()=()ij ijdAxa x and 1,()=()ij ijdBxb x, we shall write the negligibility 0()=(()),AxoBx xx if, 0()= ()ij ijaxobxxx for all 1, .ij d Similarly we define the notions of domination ()=(())AxOBx and the equivalence ()~ ()AxBx (see for more details ). Let X be the space ([0,],( ))dCMC of continuous, matrix-valued functions on [0, ] . On the space X we define the norm ||.|| ,X by 0||||= sup ||()||,Xxffx for all .fX Note that the normed space (,||.||)XX is a Banach space. We denote by (]0,],( )),=1,2,kdCMCk  the space of all ktimes continuously differentiable matrix- valued functio ns U defined on ]0, [ such that ()lim( )pxUx exists and finite forall0pk. Consider a singular second order differential operator (,())D with matrix coefficients defined on X by 22()=,dBxdUU Uxdxdx  where Uand Bare matrix-valued functions, with the domain 20()= {([0,],())(]0,],()),()= 0}.limddxDUCMCMC Ux  We assume that B is a real valued continuous and bounded matrix-valued function on [0,[ and if (0) =dBI we add the assumption |() (0)|DRxBx B (2.2) in a neighborhood of 0, for nonnegative constants diagonal matrices R and D. Here Dx is the diagonal matrix with diagonal entries =,=1,2,diixid, where , =1,2,,idi d are the diagonal entries of D. We will now recall some results needed in the sequel. Lemma 2.1 Let ()dAMC. The following properties hold 1) 0A if and only if 0AB  for all 0.B 2) ||||||,ABAB hence | |||||ABAB if 0A. Lemma 2.2 Let ,()dABMC. The following proper- ties hold 1) ||||AB implies .AB 2) ||= .AA 3) |||||| ||,AEdA where 1,=,=1,1,.ij ijijdEe eijd  Proposition 2.1 Let A be a nonnegative matrix with spectral radius ()rA. 1) The resolvent (,)RA is positive whenever >().rA 2) If ||>()rA, then |(,)|(| |,).RAR A Proof. 1) We use the Neumann series representation for the resolvent 11 1110(,)=() =()=||>().dnnnRAAI AAforrA  If 0A then 0,nA for all n, hence for >()rA, we have 1=0(,)= 0lim knknkARA  since the finite sums are positive and convergence holds in every entry. 2) For ||>(),rA we have 1=01=0|(,)|=lim=(||,).lim ||knknkknknkARAARA Theorem 2.1 (, Perrons Theorem) If A is a nonnegative matrix, then ()rA is an eigenvalue of A with positive eigenvector. Definitio n 2 .1 1) An operator (, ())SDS on a Banach Lattice (,)X is called positive if 0()={(),0}.SuforalluD SvD Sv 2) A semigroup >0()ttT on X is positive if and only if tT is positive for all >0t. Remark 2.1 An operator (, ())SDS defined on a ()Kspace(Kcompact) satisfies the positive minimum principle if for every (),uDSu positive and xK,()=0ux , then ()()0.Sux  Theorem 2.2 () O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 285Let (, ())SDS be the generator of a semigroup >0()ttT on ()CK , then the semigroup is positive if and only if (, ())SDS satisfies the positive minimum principle. 3. The Diagonal Case In this section we assume that the matrix function B is diagonal. 3.1. Characterization of the Operator (,())D The proofs of Proposition 3.1 and Proposition 3.2 can be deduced from the scalar case given in Proposition 2.2 and Prop osi tion 2.3 from . Proposition 3.1 Let >(0)dIB and 2(]0,],( ))dUXC MC . Then ()UD if and only if 2([0, ],())dUMC for some >0 and (0) =(0)= 0UU. Proposition 3.2 Let (0) =dBI and 2(]0,],( ))dUXC MC . Then ()UD if and only if 1([0,]),( ))dUC MC for some >0, (0) = 0U and 001(() ())=0.lim xxUxUtdtx  Moreover, if ()UD then ()= (log)Ux oxE and 1()=Uxo logxEx as 0,x where E is the constant matrix introduced in Lemma 2.2. Proposition 3.3 The operator (,())D is a densely defined, closed, dissipative and satisfies the positive minimum princi ple. Proof. Put 12() 000()()= 000()dbxbxBxbx and for =1,2, ,id let ()=iibxuu ux  with 20()={([0, ],)(]0, ],),()=0}.limixiDuC CCux  Then1,=()ij ijdUuDif and only if ()ij iuD for all ,=1,2,.ij d Hence from (. Lemma 2.4), the operator (,())D is a densely defined and closed. Let us show that (,())D is dissipative: ||||||||,for>0and()XXUUU UD Let 1,=()ij ijdUuD and >0. According to , for all 1,ijd we have |||| ||||ijiji ijuuu then ||||||||,XXUUU and hence the dissipativity holds. In order to prove that (, ())D satisfies the positive min i mu m p r in ci p le , a ss u me t h at 1,=()ij ijdUuD and positive such that 0()=0Ux for some 0[0, [.x If 0>0xthen0ijuand 0()=0,ijux for all i, j = 1, 2, ,d. Hence 0'() =0ijux and 0'( )0ijux for all i, j = 1, 2,,d. That means 0()0Ux. If 0=0x then 0()=0Ux. 3.2. Spectral Analysis of the Operator (,())D The purpose of next th eorem is to deduce under re asonable hypothesis on the coefficients of B a precise description of the spectrum of the operator (,())D. Theorem 3.1 If (0)dIB, then the spectrum of (,( ))D is contained in ],0]. Proof. Let 1,,= ()ij ijdCU uD and 1,=( ).iji j dVv X We have()=()=for all,=1,2,,iij ijUVu vij d 1==1=(), =1,2,,().ijiijidiiuvfor allijdand This is sufficient to note that the spectrum of (,( ))D verifies ==1()( )idii and then the result hold by (, Lemma 2.6). Theorem 3.2 The operator (,())D with (0)dIB, generates an analytic semigroup of angle .2 Moreover, the semigroup is positive and contractive. Proof. For 0< <, put *={;|()|}zC argz. It is clear from theorem 3.1 that () and then for  and =1,2, ,id, the operator ()i is invertible and verifies 1|| ()||||iiiC for some positive cons- tant iC (see ). So () is invertible and veri- O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 286 fies 11max|| ()||||iidXC. Hence the operator (,())D with (0)dIB, generates an analytic semigroup of angle .2 For the positivity using the rela- tion between the operators , =1,2,,iid and  and from the fact that each operator (,( )),iiD =1,2, ,id, generates an analytic positive and contrac- tive semigroup, we deduce the result. 4. The Non Diagonal Case In this section we consider the operator ()='' 'BxUU Ux Assume that 1()=() ()()BxPxDxPx, where D is diagonal matrix function and P is nonsingular matrix function. If P is constant, put 1=VPU, then =,= =' ''' ''UPVU PVandUPV so we obtain 1()='' 'DxPPVV Vx Hence, from Theorem 3.2 we can easily verify the following Proposition 4.1 The operator (,())D with (0)dID , generates an an alytic semi g rou p of a ngle .2 Remark that the condition (0)dID is equivalent to the fact that the spectrum ((0))B of the constant matrix (0)B verifies ((0))],1].B We turn now to the general case in which we proceed with a perturbation argument. For this we recall the following definition. Definiti on 4.1 (.Definition 2.1). Let :()ADA XX be a linear operator on the Banach space X. Any operator :()BDB X X is called A-bounded if )()DAD B and if there exist constants ,ab in  such that ||||||||||||( ).BU aAU bUforallUDA  (5.1) The A-bound of B is 0=inf{ 0:0(5.1)}aathere existsbsuch thatholds Proposition 4.2 (.Theorem 2.10). Let the operator (, ())ADA generates an analytic semigroup on a Banach space X. Then there exists a constant >0 such that (,())ABDA generates an analytic semigroup for every A-bounded operator B having A-bound 0<.a Introduce now the operators 00(,())D and (,( ))D given by 202(0)( )(0)==dBd BxBdandxdxx dxdx with, 0200()=( )={([0,],())(]0, ],()),()=0}.limddxDD UCMC CMCUx Then we have 0=  and if we choose 0()=()DD, we obtain the principal result of the paper. Theorem 4.1 Assume that (0)B is diagonalizable and ((0))],1]B or (0) =dBI and (2.2) holds. Then the operator (,())D, generates an analy- tic semigroup of angle .2 Proof. Let ()UD and observe that 22220()=(0)( )(0)==,dBxdUU UxdxdxdBdBxBdUU UxdxxdxdxUU Let >0, there exists >0 such that for all 0< x and for all ()UD there exists a constant >0C such that 0||()||||||||||(0)|||||||||||| .XXXXXRU xUCUBUUCUx  Since (0)B is diagonalizable and ((0))],1[B or B satisfies the condition (2.2) for (0) =,dBI the map (0) ,BdUUxdx from ()D into ([0,],( ))dCMC is continuous (see . Remark 2.5), so we deduce that the operator  is -bounded with -bound is equal to zero. Hence, the desired result follows by applying Theorem 3.2 and Proposition 4.2. O. A. M. S. AHMED ET AL. Copyright © 2010 SciRes. AM 2875. Application and Example Assume now that the operator (,( ))D satisfies the assumptions of Theorem 4.1, it generates so an analytic semigroup, and consider the evolution equation problem 0()()=(), 0(0) =dU tUtfttdtUU (5.2) Corollary 5.1 If =0f then the problem (5.2) has a unique solution for all 0UX. This solution is of infinitely continuously differentiable on ]0, [. For general case we have by Pazy  and R.autry . Corollary 5.2 If 1,=( )iji jdff , and for all 1,,(]0,[, )ijijfLTC and for every ]0, [tT there is a ,ijt and a continuous function ,:[0,[ [0,[ijt  such that ,,0()()()(||)< .ijijtij tij ijtdftfsts and  Then the problem (5.2) has a classical solution. EXAMP LE Let the Banach space =([0,],( ))dXCMC and 0. Put ()=xx and define the linear transforma- tion P on X in itself by =.PUUo It is clear that P is invertible and 11()= .PP Consider now the operator (, ())D defined on X by 22 12=()Ux Ux BxU  with 20()= {(]0,],()),,( )()=0},limdxDUXC MCUX oPUx  and BX is a diagonal matrix real valued function satisfying (0)<(12).dBI A simple calculus gives 21=()PP (5.3) where 111()=.BxUU Ux  Put 20()={([0, ], ())(]0, ],()), ()=0}.limddxDUC MCCMC Ux From Theorem 3.2 the operator (,())D generates an analytic semigrou p of angle 2 moreover, the semi- group is positive and contractiv e. Hence the relation (5.3) implies that the operator (, ())D generates an analytic semigroup of angle 2 and the semigroup is contractive if 1. 6. Acknowledgements The Authors wish to thank Professor A. Rhandi for many helpful discussions and comments on the manuscript. 7. References  R. 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