Inversion problem for singular integral operators: C ... - Europe PMC

Proc. Natl. Acad. Sci. USA Vol. 75, No. 10, pp. 4668-4670, October 1978

Mathematics

Inversion problem for singular integral operators: C*-approach (C*-algebras/Fredholm theories/Mihlin-Calderon-Zygmund operators/Wiener-Hopf operators/Folland-Stein operators)

ALEXANDER DYNINt School of Mathematics, Institute for Advanced Study, Princeton, New Jersey 08540

Communicated by I. M. Singer, July 5, 1978

Note that the C*-algebra of a solvable Lie group could not be solvable because a solvable C*-algebra is always a postliminary algebra (1). The spectrum 2fA of a solvable C*-algebra .1 has the following properties: (iv) The E2j are topologically imbedded in .1 A. (V) 5.[A = U 2j (vi) Closure of 2j is u 2k (k > j). Symbols and Fredholm hierarchy Choose *-isomorphisms

ABSTRACT The inversion problem is solved for a wide class of singular integral operators, in particular for Wiener-Hopf operators in several variables, Mihlin-Calderon-Zygmund operators on bounded domains, and Folland-Stein operators on compact nondegenerate Cauchy-Riemann manifolds.

Introduction The inversion problem for a bounded linear operator A on a Hilbert space H is twofold: how (i) to recognize that A is invertible and (ii) to construct its inverse. The problem is regular for Riesz-Shauder operators 1 + T with T e @L(H), the algebra of compact operators on H. A regularization of the inversion problem for arbitrary A means its reduction to some regular cases. The inversion problem for A as an element of L(H), the C*-algebra of all bounded linear operators on H, is equivalent to the inversion problem for A as an element of any C*-algebra 21 which contains A and identity. Solvable C*-algebras are chosen. In this framework it is possible to derive a hierarchy of Fredholm properties. I postpone a relevant Index theory for future publications. Solvable C*-algebras A separable C*-algebra is defined to be solvable if there exists a finite filtration 21

Pj Zj +I/ 3j -Co[1j, IL(Hj)], 0 < j < N. The general C*-theory (5) defines their unique extensions

3j -Map[ 2j, _L(Hj)] pj 21/be the mapping of factorization. Then the 7rj i- 5.1/sj j-symbol aj on 21 is defined to be Let

j7rj: .1 Map[1j, L((Hj)], 0 < j < N. PROPOSITION. Ker oj = Sj and for every A e 51, the evaluation mapping of aj(A) aj

=

ev(oj)(A): Ij X Hi - Hi, ev(aj)(A)(t, u) = uj(A)(t)(u), is continuous.

Note that the general C*-theory (cf. ref. 1) implies that all are C*-ideals even in 51. 2i The immediate examples of solvable C*-algebras are commutative C*-algebras (their length is 0), C*-algebras with symbol of Breuer and Cordes (2) (their length is 1), and

Remark: In terms of ref. 4, rj(A) for A e 51 are continuous families of operators on Ij (cf. ref. 5). Note that in all examples below, j-symbols of operators are even continuous with respect to the operator norm in L(Hj). Suppose now that 51 contains a unit 1. Then element A of 51 is defined to be j-Fredholm if it is invertible modulo Sj. Therefore, A is j-Fredholm if and only if its j-symbol aj(A) is a continuous family of invertible operators. The set Fredj(2.) of j-Fredholm operators is open in .1 and stable urnder !aj-perturbations. Note that Fredj(51) c Fredk(5.) for j < k and Fredo(21) is the group of invertible elements of 21. An element B is defined to be a j-parametrix for A if B is an inverse of A modulo !j. We have Uj(A)uj(B) = oj(B)aj(A) = 1 on 2j, Uj-1(A )j-l(B) = 1 -T, aj l(B)uj_(A) = 1 -T on Ij-1, where T' and T" are compact families of operators on Xj-j (the latter means local compactness of their evaluation mappings). In terms of ref. 4 (cf. ref. 5) the ji-I(A) is the Fredholm family of operators on 1j. In this case we can construct its index in Kc(2j-i) (cf. ref. 6) and define j-index of A as indj(A) = ind[aj-1(A)].

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tors.

C = ° C 3 C ..*CN+1. =

such that for every j with 0 < j < N. (i) %j is a closed self-adjoint ideal in + (ii) !3j+ 1/% is C*-isomorphic with a C*-algebra Co[Ij, @L (H;)] of all continuous mappings of a locally compact space EJ to the C*-algebra of compact linear bounded operators on a complex Hilbert space Hi. (iii) Let d; = dim Hi. Then di > d +1 The minimal possible N is called the length of W and denoted

Gohberg-Krupnik (3) C*-algebras generated by singular integral operators on singular curves (their length can be arbitrary). Other examples can be provided by tensoring of solvable C*-algebras and by PROPOSITION. Let 51, and 52 be solvable C*-subalgebras in a C*-algebra such that the C*-ideals [Wh, 2] generated by commutators of elements from 51h and 2 are solvable C*algebras. Then the C*-algebra generated by 51, and 212 is solvable as well.

Abbreviation: MCZ-operators, Mihlin-Calderon-Zygmund

opera-

t Present address: Department of Mathematics, State University of New York, Stony Brook, New York 11794.

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Mathematics: Dynin

Proc. Natl. Acad. Sci. USA 75 (1978)

Regularization of the inversion problem I suggest the following algorithm of step-by-step regularization for the inversion problem. Suppose that A is an invertible element of 2[. We have dN < o because of 1 E 21. Therefore, 'N(A) is an invertible continuous family of (dN X dN)-matrices on 2N. Let lrN-l be a continuous section of the factorization TrN. Take BN = WrN IPN1[UN(A)'1]. Then BN is an N-parametrix for A. Suppose that we already have a (j + 1)-parametrix Bj+ 1 for A. Then Kercrj(Bj+i) = Ker[oj(A)aj(Bj +)] = Ker(I -T),

Keroj(Bj+l)* = Ker[aj(A)*aj(Bj+1)*] = Ker(l -T), where Tj and T belong to Co[Ej, l 2L (Hj)]. Therefore, by solution of the Riesz-Shauder equations we can construct two continuous finite rank projector families P1 and Qj from Co[42, _Ls(H1)] such that oj(Bj+ 1) defines a 1 - 1 mapping from Ker P1 onto Ker Qp. Moreover, indj(Bj+ 1) + indj(A) = ind( -T'); hence, indj(Bj+ 1) = 0. Therefore, we can construct a vector bundle isomorphism Lj Im Pj Im Qj and define a continuous finite rank operator family Fj = [L - j(Bj+ 1)]Pp Now the aj(Bj+ 1) + F1 is a continuous invertible operator o

family on 2j and T; = aj(A)[aj(Bj+1) + Fj] -1 e Co[ 1,e2L(Hj)] We see that 1 - Tj is an invertible Riesz-Shauder operator family from Co[ l1,2L(H)]. Choose a continuous section -rj1 of the canonic mapping 21 ?I/S. Then the element of ?1

gr

Bj = Bj+ i1[j-1[ -pi(T-Mis a j-parametrix of A. Take j = N-1, N -2,. 1. Then Bo = A-1. Actually we have the following PROPOSITION. The regularization above is possible if and only if A is an invertible operator.

Wiener-Hopf operators in several variables (cf. ref. 7) Consider Wiener-Hopf operators on H = L2(Rn ), where R+ =(R+)n,

Au(x) = au(x) + 3' b(x - y)u(y)dy, u e H, where a = constant and b(x) e L l(Rn). Take the C*-algebra (n) generated with Wiener-Hopf operators in _L(H). PROPOSITION. The C*-algebra J(n) is solvable with length nand dj = o for j < n, dn = 1. For j < n, 2y can be realized as the disjoint union of (7) coordinate subspaces RJ of Rn, J c 1, nJ, card(J) = j. Then aj(A) is the (n-Di)-valued function. Indeed, if {' E RJ, x" e R 'J, then for v e L2(RCJ)

aj(A)(Q')v(x") = av(x") +

RCJ b(, y"

-X")V(Y" )dy"

4669

where

bA~t'x

)=

(2)n,- J b(x', x")e-ix'T'dx'.

En can be realized as Rn u

arn(A)() = a

+

o

and

(2) 3' b(x)e-ixtdx, t e Rn

Mihlin-Calderon-Zygmund operators (MCZ-operators)

on manifolds with boundaries (cf. ref. 8) Let X be a Riemannian compact C'-manifold with boundary bX. Take H = L2(X). Consider the C*-algebra E9Yx generated in L(H) by singular integral MCZ-operators. The Mihlin symbol of a MCZ-operator A we denote u(A). This is a C'function on the manifold with boundary S(X) of all unit tangent vectors of X. PROPOSITION. The C*-algebra Ex is solvable of length 2 with do = di = o= D2 = 1. We have 20 = point, Ho = H, o - Id. 11 = S(bX), H1 = L2(R+). if A is a MCZ-operator, then for r e S(bX) the a1(A)(r) is the Wiener-Hopf operator with symbol a(A)(r + sv), where v is the inward unit normal and s e R. 2 = S(X) u N, (bX), where N(bX) = Isv: IsI < 11 is the manifold of normal vectors to boundary with length < 1. H2 = C and for a MCZ-operator A we have

U2(A)|S(X) = o(A), o2(A)(sv) (s2 + 1)o(v) +~~~2 (s - 1)a(-v).

Singular integral operators on contact manifolds In ref. 9 I have defined a class of pseudo-differential operators on contact manifolds. It embraces the Folland-Stein singular integral operators on nondegenerate Cauchy-Riemann manifolds (10). Let M be a compact contact oriented smooth manifold of dimension 2n + 1, n > 0. Let C(M) be the 2n-bundle of contact tangent vectors on M, and let E(M) be the dual bundle of T(M)/C(M). We denote by Ee(M) the bundle of unit vectors in this bundle (with respect to some Riemannian metric on M). Let H = L2(M). Consider the C*-algebra (CM generated in L(H) by the zero order pseudo-differential operators of ref. 9. The (o)-symbol defined for them in ref. 11 can be extended on all CM; for every A e £M the u(A) is a continuous Ee(M)-family of singular integral operators of the type discussed in ref. 12 with phasespaces Cm(M), m e M. PROPOSITION. The CM is a solvable C*-algebra of length 2 withdo=di = co,d2= 1.

Moreover, 10 = point, Ho = H, io = Id. 1= Ee(M), H1 = L2(Rn), a, = a. -2 = S(M), H2 = C, a2 is the restriction of the principal symbol of A (9) on S(M), the manifold of unit cotangent vectors of M. This research was supported by National Science Foundation Grant MCS77-18723.

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Mathematics: Dynin

1. Dixmier, J. (1964) Les C*-algebres et leurs Representations (Gauthier-Villars, Paris). 2. Breuer, M. & Cordes, H. 0. (1974) J. Math. Mech. 13, 313324. 3. Gohberg, I. & Krupnik, N. Ja. (1972) in Hilbert Space Operator Algebras, Colloquia on Mathematics, ed. Sz-Nagy, B. (NorthHolland, Amsterdam), Vol. 5, pp. 239-264. 4. Dolganova, S. (1972) Uspekhi 27,211-212 (in Russian). 5. Hormander, L. (1970) Matematika 14, 78-97 (in Russian). 6. Segal, G. (1970) Q. J. Math., Ser. 2, 21, 385-402.

Proc. Nati. Acad. Sci. USA 75 (1978) 7. Douglas, R. G. & Howe, R. (1971) Trans. Am. Math. Soc. 158, 203-217. 8. Boutet de Monvel, L. (1970) Acta Math. 126, 11-51. 9. Dynin, A. (1975) Sov. Math. (Engi. Transi.) 16, 1608-1612. 10. Folland, G. B. & Stein, E. M. (1974) Commun. Pure Appl. Math. 27,429-522. 11. Dynin, A. (1976) Sov. Math. (Engi. Transl.) 17,508-512. 12. Grossman, A., Toupias, G. & Stein, E. M. (1969) Annu. Inst. Fourier 18, 343-368.