Algebra of Pseudo-Differential Operators Over C

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-•••' \ 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 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 . 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 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

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) = ^v.u^J Vu,v«X (iv) /U1,T) = l*

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

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. f, (x), f, (x) = p(x) F, (x) which proves lemma- K Jt K it How come back to the proof of Theorem U.I we have

He have to prove the inequality ;

without lops of generality, it is sufficient to prove inequality (U.l) for the case when 3{?-ri,n) is positive self-conjugate operatorsfor £, n. Then put °(?-n,n) = [t>(£,ri)] and b(£,r|) = t>?5,n). I*t us prove that such a supposition is Justified.

Lemma A.I t .2) Iiet A be C -algebra,, non-commutative infinite. Suppose f : X •+ A is s continuous mapping where X < fjtn. Then Let us show that and Hff^(s-ri)a^,ah))('+ifr)J^*ii

*t - constant , X eftf 1 4 c II ff(uh),u(TjH + IT')' (It .3)

Proof of lemma U.I We have Let X be compact. Then f(x) f A can ^e written as

f(x) • i fa(x) 0+hiT f , f_ are hermitlans.

Since X is compact =^| | f..()0 |[ * C < », It is sufficient to prove

the statement for f7(x). Now f U) = [C + f (x)] - C, C-scalar As

fx(x) = fx (x), then spect(C + f^xDcto, «) So C + (x) and C = ^(x). Hence f(x) = Z ^ f^tx). Since 3(e-n,Tj) € and from

-11- -12- Feeters ' inequality Substituting now (1+.3) and (It.5) in (U.2) we get((U.l). So | |^u| | ,

large vhole number N.

Then norm of integral J KU.n)1^ = B(n) is bounded by a constant which does not depend upon TJ. For the estimate of right side of {h ,lt) we shall use the representation of all elements K(C,n) , u(n) as operators on some Hilbert is pseudo-differential operator. How we want to show the completeness of L space H. Then putting u (n) - u(n) (l + |g| )3'2 ve obtain with respect to L -norm. n n n D We have shown above that H0((R , A )tfL2(fR , A ). In [[6] lemma n n ll.l)it was proved that H0(|R , A ) and ^(A) are isomorphic where *g(A) • MLl is the apace of sequences x = (» , x x, , . .) x

Then

Hence the proof of Theorem (It .1) Is complete . 11*11=/

Theorem It.2 C Algebra of all P.D.O. of order zero with coefficients in C -algebra is i itself C -algebra.

c Proof c Let us denote by cL , algebra of all P.D.O. of order zero. Following Theorem (k.l) we have only to show that | |\A ^ I [ = I I

u,^ e £(**, A*)

-13- Certainly l l|AH = ACKNOWLEDGMENTS

The author acknowledges with great pleasure his gratitude to Professor AS; Mishchenko for his guidance, helful advice and encouragement. The • author also wishes to express his hearty thanks to Dr. M.A. Shiibln and \WtU\ Dr. Ju.P. Solorev for the invaluable assistance he received in the form of suggestions.

The author is also grateful to Professor M.S. Naraslmhaa who read the final manuscript and of his help in mating it a preprint.

The author wishes to express his thanks to Professor Abdus Salem and to the International Centre for Theoretical Physics, Trieste for the hospitality So 11 A* U = ISA extended to him.

u\u

y \\u\\* \

But on the other hand ve have

- uu\

Ttms

which proves theorem.

-16- -15-

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