REFERENCE -•••' \ IC/82/109 IHTERHAL REPORT 7 *.-. (Limited distribution) International Atomic Energy Agency and United nations Educational Scientific and Cultural Organization INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS ALGEBRA OF PSEUDO-DTFFEKEHTIAL OPERATORS OVER C -ALGEBRA « Noor Mohammad •• International Centre for Theoretical Physics, Trieste, Italy. MIRAMARE - TRIESTE August 1982 * To tie submitted for publication. •• Permanent address: Ph.D Scholar Moacov State University, USSR, IWl I. INTRODUCTION Pseudo-differential.operators have-been extensively studied recently, Kumano-go, [It ,5] studied algebra of pseudo-differential operators complex valued m n through the notion of Hormander class 12] S fi(n)> 0 < 4 « p < 1, n = fR _ + ; Mishchenko and Fomenko [2] defined pseudo-differential operators over C -algebras • • • "n in a classical way. For TG r we defined the Fourier transform of <o by In the present paper ve shall atudy algebras of pseudo-differential operators over C -algebras for the special case when in Honaander class S ,(£)) P n * »" n ™ IR ; p • li S = 0, m any real number, and the C -algebra is infinite (2.2) dimensional non-commutative. The results obtained in this paper, have been *. i * i j. x. 1 already obtained in tbe case when pseudo-differential operators are complex valued by Hormander [1,2], Kohn and Nirenberg [3], Kumano-go [U,5], Mishchenjco The inverse Fourier transform is defined as and Fomenko [6] and Shubin [8]. In the present paper the space B i.e. the set of A-valued C -functions n n in f£( (or (K iR°) whose derivatives are all bounded, plays an important (2.3) role. Here A denotes 0 -algebra. where = JL_ J J , Cf £ S ((R > ) . • In Sec .1 we define the operator class S _ and through it, the class L „ of pseudo-differential operators. We now take End(a,n)-space of all n x n matrices with elements in A which In SecJI ve state the basic asymptotic expansion theorems concerning is obviously C -algebra and allow the functions a(x,S) defined on R n jR" adjoint and product of operators of class S „. to take values In End{a,n)_ i,u n Sec ,111 is devoted to the proofs of Lg-continuity theorem and to the main Let A = A A denote the direct sum of n-copies of A_ theorem which states that algebra of all pseudo-differential operators over * • C -algebras, is itself C - algebra. Definition S.I We say that a C°-function a(x,C) belonging to n ifin, End(A.n)) n II. DEFINITIONS AND NOTATIONS belongs to the class S1 (R xS2 ) when for any multi-indices a, 8 3 a constant C s.t. Let x = (:*,».. ,x )£lRn and let A be infinite dimensional, non- commutative C -algebra and let C = s(|Rn, A) denote the space of all A-valued C°-functidns which together with all their derivatives decrease faster than any i\ 1 2 (2.U) power of \x\ = (i|1 xf) ^ as |x| -* - i.e. f (i)6 5 if for any pair of -2vl/2 multi-indices a,S, a=(a1>,. , ,ccn) and B=(B1>. • ,6n) such that Here we used the Frledrichs notation ^Z/ =(l +.E £ ) and a is any real number. {See [U,5i). -2- -1- Definition 2_g s n X D For any a U,y,C) < if0(R ^ ". End(A,n)) we define an operator Called pseudo-differential operator by V (2.5) So we have shown the existence of integral (2.6) and similarly it can be. shown n n n where u€S{fR , A ) and x,y,£ «(R . for a(x,y,5). m We set L* Q = L -class of all pseudo-differential operators of type (2 .5). Theorem 2.1 a Remark Let Jkj= t a (x)D be a differential operator of order m with The operators ,jb : 5(RD,An) — 5(lRn,AQ) defined by (2.6), defines |a|«m coefficients a (x)*B-space of all A-valued C -functions, defined In f$n whose a continuous linear mapping and can be continuously extended of space a n n derivatives are all bounded. B = -Tu(x) iC°(fR ,A ) : Sup || a" u(x) || < Caj into Itself. Then jk- € L I.e. fie is a pseudo-differential operator of order m. Let us show that Integral (2-5) converges. Here convergence is taken Proof In the sense that firstly integration with regard to y and then with regard to £ (after partial differentiation) . where U £ S(ft*, A'} Firstly let a(x,y,O = a(x,£). (2.6) Since By easy calculation we have where N is sufficiently large whole number. Substituting for (2,6) and integrating by parts, we obtain where -3- Integrating Toy parts we ofctain vhere and VL and H^ sufficiently large whole numbers tL, M > 0, Putting x - y = z, ve obtain = Ji Therefore «•"««-•<' r JfiK Since <:„, ,_,. Since ) ' 5 From here follows the continuity of operator tA,. Let us 3-how that -Jx : 33 ~+ i5 is also continuous. Let j^; Tae given a and This completes proof. and where u t 3 adtC) * S? 0. It is easy to show the existence of this integral. Let us prove that tA, maps J) to itself continuously. Consider: From here onwards we shall use abbreviation P.D.O. for Pseudo-. differential operator. III. THE SYMBOLIC CALCULUS OF PSEUDO-DIFFERENTIAL OPERATOR Definition 3.1 Let J4; be P.D.O.. By symbol of PJ5.0. cA ve mean the function n ctj( Its" x(R defined by the formula: (3.1) -6- -5- Here a (x,y,0 means the transpose of the matrix function a(x,y,£). From vhere t is a constant which fcelongs to A . Obviously f = const =^ f € B so e f ^ B. With the help of Fourier inverse transform we can vrite (3.It), it is obvious that %k € L™ , Now we state some theorems, their proofs are omitted as they are quite easy and can be proved on the same lines in Kumano-go [5]. For the definition (3.2) of asymptotic expansion see Kohn and Nirenterg [3]. Definition 3.2 Theorem 3.3 Let tft, he F.D.O., a{x,0 be its symbol and a (x,E|) be the symbol Let cA he P.D.O., Since hy theorem (2,1) ^A. defines a continuous 11 perator tA. Then linear map of S {(R^.A ) into itself and of B into itself. The same is true of its transpose tA which can he defined ty (3 5) (3.3) where " •*> " means asyniptotic expansion. Where Therefore if <^£ L? „ ia defined hy Definition 3.U The dual symbol a(x,E) of operator _A ia defined in the following way: II c,E) = o (x,t) where a (x,E) denotes the sysibol of operator c/t. We note that ^ A") ^a then from It follows from here that operator Cti can he written in term of dual symbol we have <Au v> . o(x,Z) "by the formula = J] , j) (3.6) Theorem 3.5 The dual symbol a(x,£) of operator having symbol a(x,5) is where given by L (3.U) Proof: The proof immediately follows from -7- and Theorem U.g Let L CSn,An) =\t-i ! f U) f(x) dx converges t • Then n n 1) L2(iR ,A ) is a-Hilbert A-module. Theorem 3.6 2) Pseudo-differential operator Js.; tS't R Q,A°) ->-J{Rn,An) of order Let (A. and <J4. € L™ . and have symbols a(x,E), b(x,O, Then the zero la bounded operator in the norm of L (!Rn,An) i .e. A can be j 1 , d. 1,0 2 n n symbol of iA . o*tp iB (x,C): continuously extended of Lp(fi^ ,A ) into itself. 3) Pseudo-differentail operator <Jx of order zero admits conjugate operator defined by This theorem is a general form of the Leibniz formula. Proof IV. Lg-COHTINUITY THEOREM n n It is easy to show that L2(|R ,A ) is a pre-Hilbert A-module. We will In this section ve firstly define a Hilbert A-module as given by prove at the end of the proof of this theorem that L (!R ,A } is complete Pasehke [7]. with respect to g 2) Let tA t>e pseudo-differential operator of order zero defined by Definition lt.1 Let A be a C - algebra, A pre-Hilbert A-module is a right A-module X equipped with a conjugate 'bilinear map /. , \ : X x X + A aatiafying the Also we shall suppose that u(x,£) ia a finite function with respect to x, following conditions: Then (i) <(u,u) >, 0 Vu< X (ii) ^u,u> = 0 only if u * 0 (ill) <u,v> = ^v.u^J Vu,v«X (iv) /U1,T) = l*<J\i,v^ Vu.v t I, 1 « 1. We put | |(Al I = Supp | |tAu I I where | \u\\ < 1. The map /. , .\ will be called an A-vaiued inner product on X. For a pre-Hilbert A-module X, define norm [ I * I L on X by | |uj | =| I C^ The norm | |. | | satisfies We define norm of u(x) € 5(!Rn, An) by : (vi) * II t* II ^ - II U, A pre-Hilbert A-module X which is complete with tespect to will be called a Hilbert A-module. = t We denote ty H ((Rn, An)-Hilbert space obtained as the completion of space n, A°) in the norm | | • | I and call it Sobolev space. It is quite -10- easy .to see that | | u | | g = | | u j L Consider now L f(x) —^ Let X = |K and suppose F(x) 1 and ilium, Au<s)> IJ = F(x) (x) where X^-oonstant and F (x) = FAx) - Row f(x) = 1 + ||f(x)H ,l\ P 2{i) also 1 + ||f(x)|| = p2Cx) as it Is real k k 2 valued function then fU) = I X.